A key requirement for the success of supervised deep learning is a large labeled dataset - a condition that is difficult to meet in medical image analysis. Self-supervised learning (SSL) can help in this regard by providing a strategy to pre-train a neural network with unlabeled data, followed by fine-tuning for a downstream task with limited annotations. Contrastive learning, a particular variant of SSL, is a powerful technique for learning image-level representations. In this work, we propose strategies for extending the contrastive learning framework for segmentation of volumetric medical images in the semi-supervised setting with limited annotations, by leveraging domain-specific and problem-specific cues. Specifically, we propose (1) novel contrasting strategies that leverage structural similarity across volumetric medical images (domain-specific cue) and (2) a local version of the contrastive loss to learn distinctive representations of local regions that are useful for per-pixel segmentation (problem-specific cue). We carry out an extensive evaluation on three Magnetic Resonance Imaging (MRI) datasets. In the limited annotation setting, the proposed method yields substantial improvements compared to other self-supervision and semi-supervised learning techniques. When combined with a simple data augmentation technique, the proposed method reaches within 8\% of benchmark performance using only two labeled MRI volumes for …

We introduce Bootstrap Your Own Latent (BYOL), a new approach to self-supervised image representation learning. BYOL relies on two neural networks, referred to as online and target networks, that interact and learn from each other. From an augmented view of an image, we train the online network to predict the target network representation of the same image under a different augmented view. At the same time, we update the target network with a slow-moving average of the online network. While state-of-the art methods intrinsically rely on negative pairs, BYOL achieves a new state of the art without them. BYOL reaches 74.3% top-1 classification accuracy on ImageNet using the standard linear evaluation protocol with a standard ResNet-50 architecture and 79.6% with a larger ResNet. We also show that BYOL performs on par or better than the current state of the art on both transfer and semi-supervised benchmarks.

Normalizing flows and variational autoencoders are powerful generative models that can represent complicated density functions. However, they both impose constraints on the models: Normalizing flows use bijective transformations to model densities whereas VAEs learn stochastic transformations that are non-invertible and thus typically do not provide tractable estimates of the marginal likelihood. In this paper, we introduce SurVAE Flows: A modular framework of composable transformations that encompasses VAEs and normalizing flows. SurVAE Flows bridge the gap between normalizing flows and VAEs with surjective transformations, wherein the transformations are deterministic in one direction -- thereby allowing exact likelihood computation, and stochastic in the reverse direction -- hence providing a lower bound on the corresponding likelihood. We show that several recently proposed methods, including dequantization and augmented normalizing flows, can be expressed as SurVAE Flows. Finally, we introduce common operations such as the max value, the absolute value, sorting and stochastic permutation as composable layers in SurVAE Flows.

In self-supervised learning, a system is tasked with achieving a surrogate objective by defining alternative targets on a set of unlabeled data. The aim is to build useful representations that can be used in downstream tasks, without costly manual annotation. In this work, we propose a novel self-supervised formulation of relational reasoning that allows a learner to bootstrap a signal from information implicit in unlabeled data. Training a relation head to discriminate how entities relate to themselves (intra-reasoning) and other entities (inter-reasoning), results in rich and descriptive representations in the underlying neural network backbone, which can be used in downstream tasks such as classification and image retrieval. We evaluate the proposed method following a rigorous experimental procedure, using standard datasets, protocols, and backbones. Self-supervised relational reasoning outperforms the best competitor in all conditions by an average 14% in accuracy, and the most recent state-of-the-art model by 3%. We link the effectiveness of the method to the maximization of a Bernoulli log-likelihood, which can be considered as a proxy for maximizing the mutual information, resulting in a more efficient objective with respect to the commonly used contrastive losses.

Learning object-centric representations of complex scenes is a promising step towards enabling efficient abstract reasoning from low-level perceptual features. Yet, most deep learning approaches learn distributed representations that do not capture the compositional properties of natural scenes. In this paper, we present the Slot Attention module, an architectural component that interfaces with perceptual representations such as the output of a convolutional neural network and produces a set of task-dependent abstract representations which we call slots. These slots are exchangeable and can bind to any object in the input by specializing through a competitive procedure over multiple rounds of attention. We empirically demonstrate that Slot Attention can extract object-centric representations that enable generalization to unseen compositions when trained on unsupervised object discovery and supervised property prediction tasks.

Density-ratio estimation via classification is a cornerstone of unsupervised learning. It has provided the foundation for state-of-the-art methods in representation learning and generative modelling, with the number of use-cases continuing to proliferate. However, it suffers from a critical limitation: it fails to accurately estimate ratios p/q for which the two densities differ significantly. Empirically, we find this occurs whenever the KL divergence between p and q exceeds tens of nats. To resolve this limitation, we introduce a new framework, telescoping density-ratio estimation (TRE), that enables the estimation of ratios between highly dissimilar densities in high-dimensional spaces. Our experiments demonstrate that TRE can yield substantial improvements over existing single-ratio methods for mutual information estimation, representation learning and energy-based modelling.

Weighted model integration (WMI) is a framework to perform advanced probabilistic inference on hybrid domains, i.e., on distributions over mixed continuous-discrete random variables and in presence of complex logical and arithmetic constraints. In this work, we advance the WMI framework on both the theoretical and algorithmic side. First, we exactly trace the boundaries of tractability for WMI inference by proving that to be amenable to exact and efficient inference a WMI problem has to posses a tree-shaped structure with logarithmic diameter. While this result deepens our theoretical understanding of WMI it hinders the practical applicability of exact WMI solvers to real-world problems. To overcome this, we propose the first approximate WMI solver that does not resort to sampling, but performs exact inference on one approximate models. Our solution performs message passing in a relaxed problem structure iteratively to recover certain lost dependencies and, as our experiments suggest, is competitive with other SOTA WMI solvers.

In neural networks with binary activations and or binary weights the training by gradient descent is complicated as the model has piecewise constant response. We consider stochastic binary networks, obtained by adding noises in front of activations. The expected model response becomes a smooth function of parameters, its gradient is well defined but it is challenging to estimate it accurately. We propose a new method for this estimation problem combining sampling and analytic approximation steps. The method has a significantly reduced variance at the price of a small bias which gives a very practical tradeoff in comparison with existing unbiased and biased estimators. We further show that one extra linearization step leads to a deep straight-through estimator previously known only as an ad-hoc heuristic. We experimentally show higher accuracy in gradient estimation and demonstrate a more stable and better performing training in deep convolutional models with both proposed methods.

The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) – an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

Reconciling symbolic and distributed representations is a crucial challenge that can potentially resolve the limitations of current deep learning. Remarkable advances in this direction have been achieved recently via generative object-centric representation models. While learning a recognition model that infers object-centric symbolic representations like bounding boxes from raw images in an unsupervised way, no such model can provide another important ability of a generative model, i.e., generating (sampling) according to the structure of learned world density. In this paper, we propose Generative Neurosymbolic Machines, a generative model that combines the benefits of distributed and symbolic representations to support both structured representations of symbolic components and density-based generation. These two crucial properties are achieved by a two-layer latent hierarchy with the global distributed latent for flexible density modeling and the structured symbolic latent map. To increase the model flexibility in this hierarchical structure, we also propose the StructDRAW prior. In experiments, we show that the proposed model significantly outperforms the previous structured representation models as well as the state-of-the-art non-structured generative models in terms of both structure accuracy and image generation quality.

This paper re-examines a continuous optimization framework dubbed NOTEARS for learning Bayesian networks. We first generalize existing algebraic characterizations of acyclicity to a class of matrix polynomials. Next, focusing on a one-parameter-per-edge setting, it is shown that the Karush-Kuhn-Tucker (KKT) optimality conditions for the NOTEARS formulation cannot be satisfied except in a trivial case, which explains a behavior of the associated algorithm. We then derive the KKT conditions for an equivalent reformulation, show that they are indeed necessary, and relate them to explicit constraints that certain edges be absent from the graph. If the score function is convex, these KKT conditions are also sufficient for local minimality despite the non-convexity of the constraint. Informed by the KKT conditions, a local search post-processing algorithm is proposed and shown to substantially and universally improve the structural Hamming distance of all tested algorithms, typically by a factor of 2 or more. Some combinations with local search are both more accurate and more efficient than the original NOTEARS.

We propose a novel graph cross network (GXN) to achieve comprehensive feature learning from multiple scales of a graph. Based on trainable hierarchical representations of a graph, GXN enables the interchange of intermediate features across scales to promote information flow. Two key ingredients of GXN include a novel vertex infomax pooling (VIPool), which creates multiscale graphs in a trainable manner, and a novel feature-crossing layer, enabling feature interchange across scales. The proposed VIPool selects the most informative subset of vertices based on the neural estimation of mutual information between vertex features and neighborhood features. The intuition behind is that a vertex is informative when it can maximally reflect its neighboring information. The proposed feature-crossing layer fuses intermediate features between two scales for mutual enhancement by improving information flow and enriching multiscale features at hidden layers. The cross shape of feature-crossing layer distinguishes GXN from many other multiscale architectures. Experimental results show that the proposed GXN improves the classification accuracy by 2.12% and 1.15% on average for graph classification and vertex classification, respectively. Based on the same network, the proposed VIPool consistently outperforms other graph-pooling methods.

Combinatorial optimization (CO) problems are notoriously challenging for neural networks, especially in the absence of labeled instances. This work proposes an unsupervised learning framework for CO problems on graphs that can provide integral solutions of certified quality. Inspired by Erdos' probabilistic method, we use a neural network to parametrize a probability distribution over sets. Crucially, we show that when the network is optimized w.r.t. a suitably chosen loss, the learned distribution contains, with controlled probability, a low-cost integral solution that obeys the constraints of the combinatorial problem. The probabilistic proof of existence is then derandomized to decode the desired solutions. We demonstrate the efficacy of this approach to obtain valid
solutions to the maximum clique problem and to perform local graph clustering. Our method achieves competitive results on both real datasets and synthetic hard instances.

We study the problem of semi-supervised learning on graphs, for which graph neural networks (GNNs) have been extensively explored. However, most existing GNNs inherently suffer from the limitations of over-smoothing, non-robustness, and weak-generalization when labeled nodes are scarce. In this paper, we propose a simple yet effective framework—GRAPH RANDOM NEURAL NETWORKS (GRAND)—to address these issues. In GRAND, we first design a random propagation strategy to perform graph data augmentation. Then we leverage consistency regularization to optimize the prediction consistency of unlabeled nodes across different data augmentations. Extensive experiments on graph benchmark datasets suggest that GRAND significantly outperforms state-of- the-art GNN baselines on semi-supervised node classification. Finally, we show that GRAND mitigates the issues of over-smoothing and non-robustness, exhibiting better generalization behavior than existing GNNs. The source code of GRAND is publicly available at https://github.com/Grand20/grand.

Graph-based neural network models are producing strong results in a number of domains, in part because graphs provide flexibility to encode domain knowledge in the form of relational structure (edges) between nodes in the graph. In practice, edges are used both to represent intrinsic structure (e.g., abstract syntax trees of programs) and more abstract relations that aid reasoning for a downstream task (e.g., results of relevant program analyses). In this work, we study the problem of learning to derive abstract relations from the intrinsic graph structure. Motivated by their power in program analyses, we consider relations defined by paths on the base graph accepted by a finite-state automaton. We show how to learn these relations end-to-end by relaxing the problem into learning finite-state automata policies on a graph-based POMDP and then training these policies using implicit differentiation. The result is a differentiable Graph Finite-State Automaton (GFSA) layer that adds a new edge type (expressed as a weighted adjacency matrix) to a base graph. We demonstrate that this layer can find shortcuts in grid-world graphs and reproduce simple static analyses on Python programs. Additionally, we combine the GFSA layer with a larger graph-based model trained end-to-end on the variable misuse program …

Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the actual task the GNN is solving. In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs. Here we introduce Pointer Graph Networks (PGNs) which augment sets or graphs with additional inferred edges for improved model generalisation ability. PGNs allow each node to dynamically point to another node, followed by message passing over these pointers. The sparsity of this adaptable graph structure makes learning tractable while still being sufficiently expressive to simulate complex algorithms. Critically, the pointing mechanism is directly supervised to model long-term sequences of operations on classical data structures, incorporating useful structural inductive biases from theoretical computer science. Qualitatively, we demonstrate that PGNs can learn parallelisable variants of pointer-based data structures, namely disjoint set unions and link/cut trees. PGNs generalise out-of-distribution to 5x larger test inputs on dynamic graph connectivity tasks, outperforming unrestricted GNNs and Deep Sets.

Graph convolution networks (GCNs) have become effective models for graph classification. Similar to many deep networks, GCNs are vulnerable to adversarial attacks on graph topology and node attributes. Recently, a number of effective attack and defense algorithms have been developed, but certificates of robustness against \emph{topological perturbations} are currently available only for PageRank and label/feature propagation, while none has been designed for GCNs. We propose the first algorithm for certifying the robustness of GCNs to topological attacks in the application of \emph{graph classification}. Our method is based on Lagrange dualization and convex envelope, which result in tight approximation bounds that are efficiently computable by dynamic programming. When used in conjunction with robust training, it allows an increased number of graphs to be certified as robust.

We study properties of Graph Convolutional Networks (GCNs) by analyzing their behavior on standard models of random graphs, where nodes are represented by random latent variables and edges are drawn according to a similarity kernel. This allows us to overcome the difficulties of dealing with discrete notions such as isomorphisms on very large graphs, by considering instead more natural geometric aspects. We first study the convergence of GCNs to their continuous counterpart as the number of nodes grows. Our results are fully non-asymptotic and are valid for relatively sparse graphs with an average degree that grows logarithmically with the number of nodes. We then analyze the stability of GCNs to small deformations of the random graph model. In contrast to previous studies of stability in discrete settings, our continuous setup allows us to provide more intuitive deformation-based metrics for understanding stability, which have proven useful for explaining the success of convolutional representations on Euclidean domains.

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP has impractically slow convergence to high frequency signal components. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

We present a transductive learning algorithm that takes as input training examples from a distribution P and arbitrary (unlabeled) test examples, possibly chosen by an adversary. This is unlike prior work that assumes that test examples are small perturbations of P. Our algorithm outputs a selective classifier, which abstains from predicting on some examples. By considering selective transductive learning, we give the first nontrivial guarantees for learning classes of bounded VC dimension with arbitrary train and test distributions—no prior guarantees were known even for simple classes of functions such as intervals on the line. In particular, for any function in a class C of bounded VC dimension, we guarantee a low test error rate and a low rejection rate with respect to P. Our algorithm is efficient given an Empirical Risk Minimizer (ERM) for C. Our guarantees hold even for test examples chosen by an unbounded white-box adversary. We also give guarantees for generalization, agnostic, and unsupervised settings.

The Information Bottleneck (IB) objective uses information theory to formulate a task-performance versus robustness trade-off. It has been successfully applied in the standard discriminative classification setting. We pose the question whether the IB can also be used to train generative likelihood models such as normalizing flows. Since normalizing flows use invertible network architectures (INNs), they are information-preserving by construction. This seems contradictory to the idea of a bottleneck. In this work, firstly, we develop the theory and methodology of IB-INNs, a class of conditional normalizing flows where INNs are trained using the IB objective: Introducing a small amount of controlled information loss allows for an asymptotically exact formulation of the IB, while keeping the INN's generative capabilities intact. Secondly, we investigate the properties of these models experimentally, specifically used as generative classifiers. This model class offers advantages such as improved uncertainty quantification and out-of-distribution detection, but traditional generative classifier solutions suffer considerably in classification accuracy. We find the trade-off parameter in the IB controls a mix of generative capabilities and accuracy close to standard classifiers. Empirically, our uncertainty estimates in this mixed regime compare favourably to conventional generative and discriminative classifiers. Code is provided in the supplement.

Temporal point process (TPP) models combined with recurrent neural networks provide a powerful framework for modeling continuous-time event data. While such models are flexible, they are inherently sequential and therefore cannot benefit from the parallelism of modern hardware. By exploiting the recent developments in the field of normalizing flows, we design TriTPP - a new class of non-recurrent TPP models, where both sampling and likelihood computation can be done in parallel. TriTPP matches the flexibility of RNN-based methods but permits several orders of magnitude faster sampling. This enables us to use the new model for variational inference in continuous-time discrete-state systems. We demonstrate the advantages of the proposed framework on synthetic and real-world datasets.

We propose a framework for solving high-dimensional Bayesian inference problems using \emph{structure-exploiting} low-dimensional transport maps or flows. These maps are confined to a low-dimensional subspace (hence, lazy), and the subspace is identified by minimizing an upper bound on the Kullback--Leibler divergence (hence, structured). Our framework provides a principled way of identifying and exploiting low-dimensional structure in an inference problem. It focuses the expressiveness of a transport map along the directions of most significant discrepancy from the posterior, and can be used to build deep compositions of lazy maps, where low-dimensional projections of the parameters are iteratively transformed to match the posterior. We prove weak convergence of the generated sequence of distributions to the posterior, and we demonstrate the benefits of the framework on challenging inference problems in machine learning and differential equations, using inverse autoregressive flows and polynomial maps as examples of the underlying density estimators.

Determinantal point processes (DPPs) are a useful probabilistic model for selecting a small diverse subset out of a large collection of items, with applications in summarization, recommendation, stochastic optimization, experimental design and more. Given a kernel function and a subset size k, our goal is to sample k out of n items with probability proportional to the determinant of the kernel matrix induced by the subset (a.k.a. k-DPP). Existing k-DPP sampling algorithms require an expensive preprocessing step which involves multiple passes over all n items, making it infeasible for large datasets. A naïve heuristic addressing this problem is to uniformly subsample a fraction of the data and perform k-DPP sampling only on those items, however this method offers no guarantee that the produced sample will even approximately resemble the target distribution over the original dataset. In this paper, we develop alpha-DPP, an algorithm which adaptively builds a sufficiently large uniform sample of data that is then used to efficiently generate a smaller set of k items, while ensuring that this set is drawn exactly from the target distribution defined on all n items. We show empirically that our algorithm produces a k-DPP sample after observing only a small fraction of …

Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems where they are used, and are simple to evaluate, differentiate and integrate. However, for modeling non-negative functions, which are crucial for unsupervised learning, density estimation, or non-parametric Bayesian methods, linear models are not applicable directly. Moreover, current state-of-the-art models like generalized linear models either lead to non-convex optimization problems, or cannot be easily integrated. In this paper we provide the first model for non-negative functions which benefits from the same good properties of linear models. In particular, we prove that it admits a representer theorem and provide an efficient dual formulation for convex problems. We study its representation power, showing that the resulting space of functions is strictly richer than that of generalized linear models. Finally we extend the model and the theoretical results to functions with outputs in convex cones. The paper is complemented by an experimental evaluation of the model showing its effectiveness in terms of formulation, algorithmic derivation and practical results on the problems of density estimation, regression with heteroscedastic errors, and multiple …

We study three notions of uncertainty quantification---calibration, confidence intervals and prediction sets---for binary classification in the distribution-free setting, that is without making any distributional assumptions on the data. With a focus towards calibration, we establish a 'tripod' of theorems that connect these three notions for score-based classifiers. A direct implication is that distribution-free calibration is only possible, even asymptotically, using a scoring function whose level sets partition the feature space into at most countably many sets. Parametric calibration schemes such as variants of Platt scaling do not satisfy this requirement, while nonparametric schemes based on binning do. To close the loop, we derive distribution-free confidence intervals for binned probabilities for both fixed-width and uniform-mass binning. As a consequence of our 'tripod' theorems, these confidence intervals for binned probabilities lead to distribution-free calibration. We also derive extensions to settings with streaming data and covariate shift.

We propose the use of hyperedge replacement graph grammars for factor graphs, or actor graph grammars (FGGs) for short. FGGs generate sets of factor graphs and can describe a more general class of models than plate notation, dynamic graphical models, case-factor diagrams, and sum-product networks can. Moreover, inference can be done on FGGs without enumerating all the generated factor graphs. For finite variable domains (but possibly infinite sets of graphs), a generalization of variable elimination to FGGs allows exact and tractable inference in many situations. For finite sets of graphs (but possibly infinite variable domains), a FGG can be converted to a single factor graph amenable to standard inference techniques.

Metropolis-Hastings (MH) is a commonly-used MCMC algorithm, but it can be intractable on large datasets due to requiring computations over the whole dataset. In this paper, we study \emph{minibatch MH} methods, which instead use subsamples to enable scaling. We observe that most existing minibatch MH methods are inexact (i.e. they may change the target distribution), and show that this inexactness can cause arbitrarily large errors in inference. We propose a new exact minibatch MH method, \emph{TunaMH}, which exposes a tunable trade-off between its minibatch size and its theoretically guaranteed convergence rate. We prove a lower bound on the batch size that any minibatch MH method \emph{must} use to retain exactness while guaranteeing fast convergence---the first such bound for minibatch MH---and show TunaMH is asymptotically optimal in terms of the batch size. Empirically, we show TunaMH outperforms other exact minibatch MH methods on robust linear regression, truncated Gaussian mixtures, and logistic regression.

A fundamental question in multiclass classification concerns understanding the consistency properties of surrogate risk minimization algorithms, which minimize a (often convex) surrogate to the multiclass 0-1 loss. In particular, the framework of calibrated surrogates has played an important role in analyzing the Bayes consistency properties of such algorithms, i.e. in studying convergence to a Bayes optimal classifier (Zhang, 2004; Tewari and Bartlett, 2007). However, follow-up work has suggested this framework can be of limited value when studying H-consistency; in particular, concerns have been raised that even when the data comes from an underlying linear model, minimizing certain convex calibrated surrogates over linear scoring functions fails to recover the true model (Long and Servedio, 2013). In this paper, we investigate this apparent conundrum. We find that while some calibrated surrogates can indeed fail to provide H-consistency when minimized over a natural-looking but naively chosen scoring function class F, the situation can potentially be remedied by minimizing them over a more carefully chosen class of scoring functions F. In particular, for the popular one-vs-all hinge and logistic surrogates, both of which are calibrated (and therefore provide Bayes consistency) under realizable models, but were previously shown to pose problems for realizable H-consistency, we …

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between \emph{one-dimensional random projections} of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.

Humans flexibly solve new problems that differ from those previously practiced. This ability to flexibly generalize is supported by learned concepts that represent useful structure common across different problems. Here we develop a naturalistic drawing task to study how humans rapidly acquire structured prior knowledge. The task requires drawing visual figures that share underlying structure, based on a set of composable geometric rules and simple objects. We show that people spontaneously learn abstract drawing procedures that support generalization, and propose a model of how learners can discover these reusable drawing procedures. Trained in the same setting as humans, and constrained to produce efficient motor actions, this model discovers new drawing program subroutines that generalize to test figures and resemble learned features of human behavior. These results suggest that two principles guiding motor program induction in the model - abstraction (programs can reflect high-level structure that ignores figure-specific details) and compositionality (new programs are discovered by recombining previously learned programs) - are key for explaining how humans learn structured internal representations that guide flexible reasoning and learning.

A common goal in the analysis of neural data is to compress large population recordings into sets of interpretable, low-dimensional latent trajectories. This problem can be approached using Gaussian process (GP)-based methods which provide uncertainty quantification and principled model selection. However, standard GP priors do not distinguish between underlying dynamical processes and other forms of temporal autocorrelation. Here, we propose a new family of “dynamical” priors over trajectories, in the form of GP covariance functions that express a property shared by most dynamical systems: temporal non-reversibility. Non-reversibility is a universal signature of autonomous dynamical systems whose state trajectories follow consistent flow fields, such that any observed trajectory could not occur in reverse. Our new multi-output GP kernels can be used as drop-in replacements for standard kernels in multivariate regression, but also in latent variable models such as Gaussian process factor analysis (GPFA). We therefore introduce GPFADS (Gaussian Process Factor Analysis with Dynamical Structure), which models single-trial neural population activity using low-dimensional, non-reversible latent processes. Unlike previously proposed non-reversible multi-output kernels, ours admits a Kronecker factorization enabling fast and memory-efficient learning and inference. We apply GPFADS to synthetic data and show that it correctly recovers ground truth phase portraits. GPFADS also …

A core problem in cognitive science and machine learning is to understand how humans derive semantic representations from perceptual objects, such as color from an apple, pleasantness from a musical chord, or seriousness from a face. Markov Chain Monte Carlo with People (MCMCP) is a prominent method for studying such representations, in which participants are presented with binary choice trials constructed such that the decisions follow a Markov Chain Monte Carlo acceptance rule. However, while MCMCP has strong asymptotic properties, its binary choice paradigm generates relatively little information per trial, and its local proposal function makes it slow to explore the parameter space and find the modes of the distribution. Here we therefore generalize MCMCP to a continuous-sampling paradigm, where in each iteration the participant uses a slider to continuously manipulate a single stimulus dimension to optimize a given criterion such as ‘pleasantness’. We formulate both methods from a utility-theory perspective, and show that the new method can be interpreted as ‘Gibbs Sampling with People’ (GSP). Further, we introduce an aggregation parameter to the transition step, and show that this parameter can be manipulated to flexibly shift between Gibbs sampling and deterministic optimization. In an initial study, we show GSP …

Primate vision depends on recurrent processing for reliable perception. A growing body of literature also suggests that recurrent connections improve the learning efficiency and generalization of vision models on classic computer vision challenges. Why then, are current large-scale challenges dominated by feedforward networks? We posit that the effectiveness of recurrent vision models is bottlenecked by the standard algorithm used for training them, "back-propagation through time" (BPTT), which has O(N) memory-complexity for training an N step model. Thus, recurrent vision model design is bounded by memory constraints, forcing a choice between rivaling the enormous capacity of leading feedforward models or trying to compensate for this deficit through granular and complex dynamics. Here, we develop a new learning algorithm, "contractor recurrent back-propagation" (C-RBP), which alleviates these issues by achieving constant O(1) memory-complexity with steps of recurrent processing. We demonstrate that recurrent vision models trained with C-RBP can detect long-range spatial dependencies in a synthetic contour tracing task that BPTT-trained models cannot. We further show that recurrent vision models trained with C-RBP to solve the large-scale Panoptic Segmentation MS-COCO challenge outperform the leading feedforward approach, with fewer free parameters. C-RBP is a general-purpose learning algorithm for any application that can benefit from expansive …

The brain modifies its synaptic strengths during learning in order to better adapt to its environment. However, the underlying plasticity rules that govern learning are unknown. Many proposals have been suggested, including Hebbian mechanisms, explicit error backpropagation, and a variety of alternatives. It is an open question as to what specific experimental measurements would need to be made to determine whether any given learning rule is operative in a real biological system. In this work, we take a "virtual experimental" approach to this problem. Simulating idealized neuroscience experiments with artificial neural networks, we generate a large-scale dataset of learning trajectories of aggregate statistics measured in a variety of neural network architectures, loss functions, learning rule hyperparameters, and parameter initializations. We then take a discriminative approach, training linear and simple non-linear classifiers to identify learning rules from features based on these observables. We show that different classes of learning rules can be separated solely on the basis of aggregate statistics of the weights, activations, or instantaneous layer-wise activity changes, and that these results generalize to limited access to the trajectory and held-out architectures and learning curricula. We identify the statistics of each observable that are most relevant for rule identification, finding …

Generalized linear models are one of the most efficient paradigms for predicting the correlated stochastic activity of neuronal networks in response to external stimuli, with applications in many brain areas. However, when dealing with complex stimuli, the inferred coupling parameters often do not generalise across different stimulus statistics, leading to degraded performance and blowup instabilities. Here, we develop a two-step inference strategy that allows us to train robust generalised linear models of interacting neurons, by explicitly separating the effects of correlations in the stimulus from network interactions in each training step. Applying this approach to the responses of retinal ganglion cells to complex visual stimuli, we show that, compared to classical methods, the models trained in this way exhibit improved performance, are more stable, yield robust interaction networks, and generalise well across complex visual statistics. The method can be extended to deep convolutional neural networks, leading to models with high predictive accuracy for both the neuron firing rates and their correlations.

Group studies involving large cohorts of subjects are important to draw general conclusions about brain functional organization. However, the aggregation of data coming from multiple subjects is challenging, since it requires accounting for large variability in anatomy, functional topography and stimulus response across individuals. Data modeling is especially hard for ecologically relevant conditions such as movie watching, where the experimental setup does not imply well-defined cognitive operations. We propose a novel MultiView Independent Component Analysis (ICA) model for group studies, where data from each subject are modeled as a linear combination of shared independent sources plus noise. Contrary to most group-ICA procedures, the likelihood of the model is available in closed form. We develop an alternate quasi-Newton method for maximizing the likelihood, which is robust and converges quickly. We demonstrate the usefulness of our approach first on fMRI data, where our model demonstrates improved sensitivity in identifying common sources among subjects. Moreover, the sources recovered by our model exhibit lower between-sessions variability than other methods. On magnetoencephalography (MEG) data, our method yields more accurate source localization on phantom data. Applied on 200 subjects from the Cam-CAN dataset, it reveals a clear sequence of evoked activity in sensor and source space.

Diffusion-weighted magnetic resonance imaging (DWI) is the only non-invasive method for quantifying microstructure and reconstructing white-matter pathways in the living human brain. Fluctuations from multiple sources create significant noise in DWI data which must be suppressed before subsequent microstructure analysis. We introduce a self-supervised learning method for denoising DWI data, Patch2Self, which uses the entire volume to learn a full-rank locally linear denoiser for that volume. By taking advantage of the oversampled q-space of DWI data, Patch2Self can separate structure from noise without requiring an explicit model for either. We demonstrate the effectiveness of Patch2Self via quantitative and qualitative improvements in microstructure modeling, tracking (via fiber bundle coherency) and model estimation relative to other unsupervised methods on real and simulated data.

Functional magnetic resonance imaging (fMRI) is a crucial technology for gaining insights into cognitive processes in humans. Data amassed from fMRI measurements result in volumetric data sets that vary over time. However, analysing such data presents a challenge due to the large degree of noise and person-to-person variation in how information is represented in the brain. To address this challenge, we present a novel topological approach that encodes each time point in an fMRI data set as a persistence diagram of topological features, i.e. high-dimensional voids present in the data. This representation naturally does not rely on voxel-by-voxel correspondence and is robust towards noise. We show that these time-varying persistence diagrams can be clustered to find meaningful groupings between participants, and that they are also useful in studying within-subject brain state trajectories of subjects performing a particular task. Here, we apply both clustering and trajectory analysis techniques to a group of participants watching the movie 'Partly Cloudy'. We observe significant differences in both brain state trajectories and overall topological activity between adults and children watching the same movie.

Visual processing in the retina has been studied in great detail at all levels such that a comprehensive picture of the retina's cell types and the many neural circuits they form is emerging. However, the currently best performing models of retinal function are black-box CNN models which are agnostic to such biological knowledge. In particular, these models typically neglect the role of the many inhibitory circuits involving amacrine cells and the biophysical mechanisms underlying synaptic release. Here, we present a computational model of temporal processing in the inner retina, including inhibitory feedback circuits and realistic synaptic release mechanisms. Fit to the responses of bipolar cells, the model generalized well to new stimuli including natural movie sequences, performing on par with or better than a benchmark black-box model. In pharmacology experiments, the model replicated in silico the effect of blocking specific amacrine cell populations with high fidelity, indicating that it had learned key circuit functions. Also, more in depth comparisons showed that connectivity patterns learned by the model were well matched to connectivity patterns extracted from connectomics data. Thus, our model provides a biologically interpretable data-driven account of temporal processing in the inner retina, filling the gap between purely black-box and …

The search for biologically faithful synaptic plasticity rules has resulted in a large body of models. They are usually inspired by -- and fitted to -- experimental data, but they rarely produce neural dynamics that serve complex functions. These failures suggest that current plasticity models are still under-constrained by existing data. Here, we present an alternative approach that uses meta-learning to discover plausible synaptic plasticity rules. Instead of experimental data, the rules are constrained by the functions they implement and the structure they are meant to produce. Briefly, we parameterize synaptic plasticity rules by a Volterra expansion and then use supervised learning methods (gradient descent or evolutionary strategies) to minimize a problem-dependent loss function that quantifies how effectively a candidate plasticity rule transforms an initially random network into one with the desired function. We first validate our approach by re-discovering previously described plasticity rules, starting at the single-neuron level and ``Oja’s rule'', a simple Hebbian plasticity rule that captures the direction of most variability of inputs to a neuron (i.e., the first principal component). We expand the problem to the network level and ask the framework to find Oja’s rule together with an anti-Hebbian rule such that an initially random …

We study the task of replicating the functionality of black-box neural models, for which we only know the output class probabilities provided for a set of input images. We assume back-propagation through the black-box model is not possible and its training images are not available, e.g. the model could be exposed only through an API. In this context, we present a teacher-student framework that can distill the black-box (teacher) model into a student model with minimal accuracy loss. To generate useful data samples for training the student, our framework (i) learns to generate images on a proxy data set (with images and classes different from those used to train the black-box) and (ii) applies an evolutionary strategy to make sure that each generated data sample exhibits a high response for a specific class when given as input to the black box. Our framework is compared with several baseline and state-of-the-art methods on three benchmark data sets. The empirical evidence indicates that our model is superior to the considered baselines. Although our method does not back-propagate through the black-box network, it generally surpasses state-of-the-art methods that regard the teacher as a glass-box model. Our code is available at: https://github.com/antoniobarbalau/black-box-ripper.

Understanding of GAN training is still very limited. One major challenge is its non-convex-non-concave min-max objective, which may lead to sub-optimal local minima. In this work, we perform a global landscape analysis of the empirical loss of GANs. We prove that a class of separable-GAN, including the original JS-GAN, has exponentially many bad basins which are perceived as mode-collapse. We also study the relativistic pairing GAN (RpGAN) loss which couples the generated samples and the true samples. We prove that RpGAN has no bad basins. Experiments on synthetic data show that the predicted bad basin can indeed appear in training. We also perform experiments to support our theory that RpGAN has a better landscape than separable-GAN. For instance, we empirically show that RpGAN performs better than separable-GAN with relatively narrow neural nets. The code is available at \url{https://github.com/AilsaF/RS-GAN}.

Optimal Transport (OT) distances are now routinely used as loss functions in ML tasks. Yet, computing OT distances between arbitrary (i.e. not necessarily discrete) probability distributions remains an open problem. This paper introduces a new online estimator of entropy-regularized OT distances between two such arbitrary distributions. It uses streams of samples from both distributions to iteratively enrich a non-parametric representation of the transportation plan. Compared to the classic Sinkhorn algorithm, our method leverages new samples at each iteration, which enables a consistent estimation of the true regularized OT distance. We provide a theoretical analysis of the convergence of the online Sinkhorn algorithm, showing a nearly-1/n asymptotic sample complexity for the iterate sequence. We validate our method on synthetic 1-d to 10-d data and on real 3-d shape data.

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. [2016] provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses.

Our work is the first to address uniform stability of SGD on nonsmooth convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bound allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess …

A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothness - which shows how sensitive it is to changes of its input. Our goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. This leads to novel soft-max functions, each of which is optimal for a different application. The most commonly used soft-max function, called exponential mechanism, has optimal tradeoff between approximation measured in terms of expected additive approximation and smoothness measured with respect to Renyi Divergence. We introduce a soft-max function, called piece-wise linear soft-max, with optimal tradeoff between approximation, measured in terms of worst-case additive approximation and smoothness, measured with respect to l_q-norm. The worst-case approximation guarantee of the piece-wise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications Martins et al. 16, Laha et al. 18 and is not satisfied by the exponential mechanism. Moreover, the l_q-smoothness is suitable for applications in Mechanism Design and Game Theory where the piece-wise linear mechanism outperforms the exponential mechanism. Finally, we investigate another soft-max function, called power mechanism, …

Arguably, the two most popular accelerated or momentum-based optimization methods are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with a friction term. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyze the symplectic structure of both Nesterov and heavy ball, besides providing several new insights into these methods. Moreover, we propose a new algorithm based on a dissipative relativistic system that normalizes the momentum and may result in more stable/faster optimization. Importantly, such a method generalizes both Nesterov and heavy ball, each being recovered as distinct limiting cases, and has potential advantages at no additional cost.

Many learning algorithms, such as stochastic gradient descent, are affected by the order in which training examples are used. It is often observed that sampling the training examples without-replacement, also known as random reshuffling, causes learning algorithms to converge faster. We give a counterexample to the Operator Inequality of Noncommutative Arithmetic and Geometric Means, a longstanding conjecture that relates to the performance of random reshuffling in learning algorithms (Recht and Ré, "Toward a noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences," COLT 2012). We use this to give an example of a learning task and algorithm for which with-replacement random sampling actually outperforms random reshuffling.

Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk with the squared loss for linear models and kernel methods. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent methods. Our observation immediately yields excess risk guarantees for the path traced by the iterates of mirror descent in terms of offset complexities of certain function classes depending only on the choice of the mirror map, initialization point, step-size, and the number of iterations. We apply our theory to recover, in a rather clean and elegant manner via rather short proofs, some of the recent results in the implicit regularization literature, while also showing how to improve upon them in some settings.

Recently, there have been significant interests in studying the so-called "double-descent" of the generalization error of linear regression models under the overparameterized and overfitting regime, with the hope that such analysis may provide the first step towards understanding why overparameterized deep neural networks (DNN) still generalize well. However, to date most of these studies focused on the min L2-norm solution that overfits the data. In contrast, in this paper we study the overfitting solution that minimizes the L1-norm, which is known as Basis Pursuit (BP) in the compressed sensing literature. Under a sparse true linear regression model with p i.i.d. Gaussian features, we show that for a large range of p up to a limit that grows exponentially with the number of samples n, with high probability the model error of BP is upper bounded by a value that decreases with p. To the best of our knowledge, this is the first analytical result in the literature establishing the double-descent of overfitting BP for finite n and p. Further, our results reveal significant differences between the double-descent of BP and min L2-norm solutions. Specifically, the double-descent upper-bound of BP is independent of the signal strength, and for high SNR and …

We study problem-dependent rates, i.e., generalization errors that scale tightly with the variance or the effective loss at the "best hypothesis." Existing uniform convergence and localization frameworks, the most widely used tools to study this problem, often fail to simultaneously provide parameter localization and optimal dependence on the sample size. As a result, existing problem-dependent rates are often rather weak when the hypothesis class is "rich" and the worst-case bound of the loss is large. In this paper we propose a new framework based on a "uniform localized convergence" principle. We provide the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general "rich" classes; we also establish improved loss-dependent rate for standard empirical risk minimization.

We consider an underdetermined noisy linear regression model where the minimum-norm interpolating predictor is known to be consistent, and ask: can uniform convergence in a norm ball, or at least (following Nagarajan and Kolter) the subset of a norm ball that the algorithm selects on a typical input set, explain this success? We show that uniformly bounding the difference between empirical and population errors cannot show any learning in the norm ball, and cannot show consistency for any set, even one depending on the exact algorithm and distribution. But we argue we can explain the consistency of the minimal-norm interpolator with a slightly weaker, yet standard, notion: uniform convergence of zero-error predictors in a norm ball. We use this to bound the generalization error of low- (but not minimal-)norm interpolating predictors.

We consider the problem of robust and adaptive model predictive control (MPC) of a linear system, with unknown parameters that are learned along the way (adaptive), in a critical setting where failures must be prevented (robust). This problem has been studied from different perspectives by different communities. However, the existing theory deals only with the case of quadratic costs (the LQ problem), which limits applications to stabilisation and tracking tasks only. In order to handle more general (non-convex) costs that naturally arise in many practical problems, we carefully select and bring together several tools from different communities, namely non-asymptotic linear regression, recent results in interval prediction, and tree-based planning. Combining and adapting the theoretical guarantees at each layer is non trivial, and we provide the first end-to-end suboptimality analysis for this setting. Interestingly, our analysis naturally adapts to handle many models and combines with a data-driven robust model selection strategy, which enables to relax the modelling assumptions. Last, we strive to preserve tractability at any stage of the method, that we illustrate on two challenging simulated environments.

Curriculum reinforcement learning (CRL) improves the learning speed and stability of an agent by exposing it to a tailored series of tasks throughout learning. Despite empirical successes, an open question in CRL is how to automatically generate a curriculum for a given reinforcement learning (RL) agent, avoiding manual design. In this paper, we propose an answer by interpreting the curriculum generation as an inference problem, where distributions over tasks are progressively learned to approach the target task. This approach leads to an automatic curriculum generation, whose pace is controlled by the agent, with solid theoretical motivation and easily integrated with deep RL algorithms. In the conducted experiments, the curricula generated with the proposed algorithm significantly improve learning performance across several environments and deep RL algorithms, matching or outperforming state-of-the-art existing CRL algorithms.

Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far. We study KL regularization within an approximate value iteration scheme and show that it implicitly averages q-values. Leveraging this insight, we provide a very strong performance bound, the very first to combine two desirable aspects: a linear dependency to the horizon (instead of quadratic) and an error propagation term involving an averaging effect of the estimation errors (instead of an accumulation effect). We also study the more general case of an additional entropy regularizer. The resulting abstract scheme encompasses many existing RL algorithms. Some of our assumptions do not hold with neural networks, so we complement this theoretical analysis with an extensive empirical study.

We consider the problem of controlling a known linear dynamical system under stochastic noise, adversarially chosen costs, and bandit feedback. Unlike the full feedback setting where the entire cost function is revealed after each decision, here only the cost incurred by the learner is observed. We present a new and efficient algorithm that, for strongly convex and smooth costs, obtains regret that grows with the square root of the time horizon T. We also give extensions of this result to general convex, possibly non-smooth costs, and to non-stochastic system noise. A key component of our algorithm is a new technique for addressing bandit optimization of loss functions with memory.

The current dominant paradigm in sensorimotor control, whether imitation or reinforcement learning, is to train policies directly in raw action spaces such as torque, joint angle, or end-effector position. This forces the agent to make decision at each point in training, and hence, limits the scalability to continuous, high-dimensional, and long-horizon tasks. In contrast, research in classical robotics has, for a long time, exploited dynamical systems as a policy representation to learn robot behaviors via demonstrations. These techniques, however, lack the flexibility and generalizability provided by deep learning or deep reinforcement learning and have remained under-explored in such settings. In this work, we begin to close this gap and embed dynamics structure into deep neural network-based policies by reparameterizing action spaces with differential equations. We propose Neural Dynamic Policies (NPDs) that make predictions in trajectory distribution space as opposed to prior policy learning methods where action represents the raw control space. The embedded structure allows us to perform end-to-end policy learning under both reinforcement and imitation learning setups. We show that NDPs achieve better or comparable performance to state-of-the-art approaches on many robotic control tasks using both reward-based training and demonstrations. Project video and code are available at: https://shikharbahl.github.io/neural-dynamic-policies/.

Exploration is a key problem in reinforcement learning, since agents can only learn from data they acquire in the environment. With that in mind, maintaining a population of agents is an attractive method, as it allows data be collected with a diverse set of behaviors. This behavioral diversity is often boosted via multi-objective loss functions. However, those approaches typically leverage mean field updates based on pairwise distances, which makes them susceptible to cycling behaviors and increased redundancy. In addition, explicitly boosting diversity often has a detrimental impact on optimizing already fruitful behaviors for rewards. As such, the reward-diversity trade off typically relies on heuristics. Finally, such methods require behavioral representations, often handcrafted and domain specific. In this paper, we introduce an approach to optimize all members of a population simultaneously. Rather than using pairwise distance, we measure the volume of the entire population in a behavioral manifold, defined by task-agnostic behavioral embeddings. In addition, our algorithm Diversity via Determinants (DvD), adapts the degree of diversity during training using online learning techniques. We introduce both evolutionary and gradient-based instantiations of DvD and show they effectively improve exploration without reducing performance when better exploration is not required.

Adversarial Imitation Learning alternates between learning a discriminator -- which tells apart expert's demonstrations from generated ones -- and a generator's policy to produce trajectories that can fool this discriminator. This alternated optimization is known to be delicate in practice since it compounds unstable adversarial training with brittle and sample-inefficient reinforcement learning. We propose to remove the burden of the policy optimization steps by leveraging a novel discriminator formulation. Specifically, our discriminator is explicitly conditioned on two policies: the one from the previous generator's iteration and a learnable policy. When optimized, this discriminator directly learns the optimal generator's policy. Consequently, our discriminator's update solves the generator's optimization problem for free: learning a policy that imitates the expert does not require an additional optimization loop. This formulation effectively cuts by half the implementation and computational burden of Adversarial Imitation Learning algorithms by removing the Reinforcement Learning phase altogether. We show on a variety of tasks that our simpler approach is competitive to prevalent Imitation Learning methods.

Potential-based reward shaping provides an approach for designing good reward functions, with the purpose of speeding up learning. However, automatically finding potential functions for complex environments is a difficult problem (in fact, of the same difficulty as learning a value function from scratch). We propose a new framework for learning potential functions by leveraging ideas from graph representation learning. Our approach relies on Graph Convolutional Networks which we use as a key ingredient in combination with the probabilistic inference view of reinforcement learning. More precisely, we leverage Graph Convolutional Networks to perform message passing from rewarding states. The propagated messages can then be used as potential functions for reward shaping to accelerate learning. We verify empirically that our approach can achieve considerable improvements in both small and high-dimensional control problems.

Entropy regularization, smoothing of Q-values and neural network function approximator are key components of the state-of-the-art reinforcement learning (RL) algorithms, such as Soft Actor-Critic~\cite{haarnoja2018soft}. Despite the widespread use, the impact of these core techniques on the convergence of RL algorithms is not yet fully understood. In this work, we analyse these techniques from error propagation perspective using the approximate dynamic programming framework. In particular, our analysis shows that (1) value smoothing results in increased stability of the algorithm in exchange for slower convergence, (2) entropy regularization reduces overestimation errors at the cost of modifying the original problem, (3) we study a combination of these techniques that describes the Soft Actor-Critic algorithm.

In this work we consider partially observable environments with sparse rewards. We present a self-supervised representation learning method for image-based observations, which arranges embeddings respecting temporal distance of observations. This representation is empirically robust to stochasticity and suitable for novelty detection from the error of a predictive forward model. We consider episodic and life-long uncertainties to guide the exploration. We propose to estimate the missing information about the environment with the world model, which operates in the learned latent space. As a motivation of the method, we analyse the exploration problem in a tabular Partially Observable Labyrinth. We demonstrate the method on image-based hard exploration environments from the Atari benchmark and report significant improvement with respect to prior work. The source code of the method and all the experiments is available at https://github.com/htdt/lwm.

Recent advances in deep reinforcement learning (RL) have led to considerable progress in many 2-player zero-sum games, such as Go, Poker and Starcraft. The purely adversarial nature of such games allows for conceptually simple and principled application of RL methods. However real-world settings are many-agent, and agent interactions are complex mixtures of common-interest and competitive aspects. We consider Diplomacy, a 7-player board game designed to accentuate dilemmas resulting from many-agent interactions. It also features a large combinatorial action space and simultaneous moves, which are challenging for RL algorithms. We propose a simple yet effective approximate best response operator, designed to handle large combinatorial action spaces and simultaneous moves. We also introduce a family of policy iteration methods that approximate fictitious play. With these methods, we successfully apply RL to Diplomacy: we show that our agents convincingly outperform the previous state-of-the-art, and game theoretic equilibrium analysis shows that the new process yields consistent improvements.

A residual network may be regarded as a discretization of an ordinary differential equation (ODE) which, in the limit of time discretization, defines a continuous-depth network. Although important steps have been taken to realize the advantages of such continuous formulations, most current techniques assume identical layers. Indeed, existing works throw into relief the myriad difficulties of learning an infinite-dimensional parameter in a continuous-depth neural network. To this end, we introduce a shooting formulation which shifts the perspective from parameterizing a network layer-by-layer to parameterizing over optimal networks described only by a set of initial conditions. For scalability, we propose a novel particle-ensemble parameterization which fully specifies the optimal weight trajectory of the continuous-depth neural network. Our experiments show that our particle-ensemble shooting formulation can achieve competitive performance. Finally, though the current work is inspired by continuous-depth neural networks, the particle-ensemble shooting formulation also applies to discrete-time networks and may lead to a new fertile area of research in deep learning parameterization.

We describe a procedure for explaining neurons in deep representations by identifying compositional logical concepts that closely approximate neuron behavior. Compared to prior work that uses atomic labels as explanations, analyzing neurons compositionally allows us to more precisely and expressively characterize their behavior. We use this procedure to answer several questions on interpretability in models for vision and natural language processing. First, we examine the kinds of abstractions learned by neurons. In image classification, we find that many neurons learn highly abstract but semantically coherent visual concepts, while other polysemantic neurons detect multiple unrelated features; in natural language inference (NLI), neurons learn shallow lexical heuristics from dataset biases. Second, we see whether compositional explanations give us insight into model performance: vision neurons that detect human-interpretable concepts are positively correlated with task performance, while NLI neurons that fire for shallow heuristics are negatively correlated with task performance. Finally, we show how compositional explanations provide an accessible way for end users to produce simple "copy-paste" adversarial examples that change model behavior in predictable ways.

Ensembles of deep neural networks are known to achieve state-of-the-art performance in uncertainty estimation and lead to accuracy improvement. In this work, we focus on a classification problem and investigate the behavior of both non-calibrated and calibrated negative log-likelihood (CNLL) of a deep ensemble as a function of the ensemble size and the member network size. We indicate the conditions under which CNLL follows a power law w. r. t. ensemble size or member network size, and analyze the dynamics of the parameters of the discovered power laws. Our important practical finding is that one large network may perform worse than an ensemble of several medium-size networks with the same total number of parameters (we call this ensemble a memory split). Using the detected power law-like dependencies, we can predict (1) the possible gain from the ensembling of networks with given structure, (2) the optimal memory split given a memory budget, based on a relatively small number of trained networks.

We propose a general physics-based deep learning architecture for wave-based imaging problems. A key difficulty in imaging problems with a varying background wave speed is that the medium ``bends'' the waves differently depending on their position and direction. This space-bending geometry makes the equivariance to translations of convolutional networks an undesired inductive bias. We build an interpretable neural architecture inspired by Fourier integral operators (FIOs) which approximate the wave physics. FIOs model a wide range of imaging modalities, from seismology and radar to Doppler and ultrasound. We focus on learning the geometry of wave propagation captured by FIOs, which is implicit in the data, via a loss based on optimal transport. The proposed FIONet performs significantly better than the usual baselines on a number of imaging inverse problems, especially in out-of-distribution tests.

Modern neural networks are often regarded as complex black-box functions whose behavior is difficult to understand owing to their nonlinear dependence on the data and the nonconvexity in their loss landscapes. In this work, we show that these common perceptions can be completely false in the early phase of learning. In particular, we formally prove that, for a class of well-behaved input distributions, the early-time learning dynamics of a two-layer fully-connected neural network can be mimicked by training a simple linear model on the inputs. We additionally argue that this surprising simplicity can persist in networks with more layers and with convolutional architecture, which we verify empirically. Key to our analysis is to bound the spectral norm of the difference between the Neural Tangent Kernel (NTK) and an affine transform of the data kernel; however, unlike many previous results utilizing the NTK, we do not require the network to have disproportionately large width, and the network is allowed to escape the kernel regime later in training.

Exponential families are widely used in machine learning; they include many distributions in continuous and discrete domains (e.g., Gaussian, Dirichlet, Poisson, and categorical distributions via the softmax transformation). Distributions in each of these families have fixed support. In contrast, for finite domains, there has been recent work on sparse alternatives to softmax (e.g., sparsemax and alpha-entmax), which have varying support, being able to assign zero probability to irrelevant categories. These discrete sparse mappings have been used for improving interpretability of neural attention mechanisms. This paper expands that work in two directions: first, we extend alpha-entmax to continuous domains, revealing a link with Tsallis statistics and deformed exponential families. Second, we introduce continuous-domain attention mechanisms, deriving efficient gradient backpropagation algorithms for alpha in {1,2}. Experiments on attention-based text classification, machine translation, and visual question answering illustrate the use of continuous attention in 1D and 2D, showing that it allows attending to time intervals and compact regions.

Spiking neural networks (SNNs) are well suited for spatio-temporal learning and implementations on energy-efficient event-driven neuromorphic processors. However, existing SNN error backpropagation (BP) methods lack proper handling of spiking discontinuities and suffer from low performance compared with the BP methods for traditional artificial neural networks. In addition, a large number of time steps are typically required to achieve decent performance, leading to high latency and rendering spike-based computation unscalable to deep architectures. We present a novel Temporal Spike Sequence Learning Backpropagation (TSSL-BP) method for training deep SNNs, which breaks down error backpropagation across two types of inter-neuron and intra-neuron dependencies and leads to improved temporal learning precision. It captures inter-neuron dependencies through presynaptic firing times by considering the all-or-none characteristics of firing activities and captures intra-neuron dependencies by handling the internal evolution of each neuronal state in time. TSSL-BP efficiently trains deep SNNs within a much shortened temporal window of a few steps while improving the accuracy for various image classification datasets including CIFAR10.

In this paper, we show that although the minimizers of cross-entropy and related classification losses are off at infinity, network weights learned by gradient flow converge in direction, with an immediate corollary that network predictions, training errors, and the margin distribution also converge. This proof holds for deep homogeneous networks — a broad class of networks allowing for ReLU, max-pooling, linear, and convolutional layers — and we additionally provide empirical support not just close to the theory (e.g., the AlexNet), but also on non-homogeneous networks (e.g., the DenseNet). If the network further has locally Lipschitz gradients, we show that these gradients also converge in direction, and asymptotically align with the gradient flow path, with consequences on margin maximization, convergence of saliency maps, and a few other settings. Our analysis complements and is distinct from the well-known neural tangent and mean-field theories, and in particular makes no requirements on network width and initialization, instead merely requiring perfect classification accuracy. The proof proceeds by developing a theory of unbounded nonsmooth Kurdyka-Łojasiewicz inequalities for functions definable in an o-minimal structure, and is also applicable outside deep learning.

Neural ordinary differential equations are an attractive option for modelling temporal dynamics. However, a fundamental issue is that the solution to an ordinary differential equation is determined by its initial condition, and there is no mechanism for adjusting the trajectory based on subsequent observations. Here, we demonstrate how this may be resolved through the well-understood mathematics of \emph{controlled differential equations}. The resulting \emph{neural controlled differential equation} model is directly applicable to the general setting of partially-observed irregularly-sampled multivariate time series, and (unlike previous work on this problem) it may utilise memory-efficient adjoint-based backpropagation even across observations. We demonstrate that our model achieves state-of-the-art performance against similar (ODE or RNN based) models in empirical studies on a range of datasets. Finally we provide theoretical results demonstrating universal approximation, and that our model subsumes alternative ODE models.

In recent years, rapid progress has been made on the problem of single-channel sound separation using supervised training of deep neural networks. In such supervised approaches, a model is trained to predict the component sources from synthetic mixtures created by adding up isolated ground-truth sources. Reliance on this synthetic training data is problematic because good performance depends upon the degree of match between the training data and real-world audio, especially in terms of the acoustic conditions and distribution of sources. The acoustic properties can be challenging to accurately simulate, and the distribution of sound types may be hard to replicate. In this paper, we propose a completely unsupervised method, mixture invariant training (MixIT), that requires only single-channel acoustic mixtures. In MixIT, training examples are constructed by mixing together existing mixtures, and the model separates them into a variable number of latent sources, such that the separated sources can be remixed to approximate the original mixtures. We show that MixIT can achieve competitive performance compared to supervised methods on speech separation. Using MixIT in a semi-supervised learning setting enables unsupervised domain adaptation and learning from large amounts of real-world data without ground-truth source waveforms. In particular, we significantly improve reverberant speech …

It is known that learning deep neural-networks is computationally hard in the worst-case. In fact, the proofs of such hardness results show that even weakly learning deep networks is hard. In other words, no efficient algorithm can find a predictor that is slightly better than a random guess. However, we observe that on natural distributions of images, small patches of the input image are corre- lated to the target label, which implies that on such natural data, efficient weak learning is trivial. While in the distribution-free setting, the celebrated boosting results show that weak learning implies strong learning, in the distribution-specific setting this is not necessarily the case. We introduce a property of distributions, denoted “local correlation”, which requires that small patches of the input image and of intermediate layers of the target function are correlated to the target label. We empirically demonstrate that this property holds for the CIFAR and ImageNet data sets. The main technical results of the paper is proving that, for some classes of deep functions, weak learning implies efficient strong learning under the “local correlation” assumption.

We study reinforcement learning in continuous state and action spaces endowed with a metric. We provide a refined analysis of the algorithm of Sinclair, Banerjee, and Yu (2019) and show that its regret scales with the zooming dimension of the instance. This parameter, which originates in the bandit literature, captures the size of the subsets of near optimal actions and is always smaller than the covering dimension used in previous analyses. As such, our results are the first provably adaptive guarantees for reinforcement learning in metric spaces.

In this paper we provide a new efficient algorithm for approximately computing the profile maximum likelihood (PML) distribution, a prominent quantity in symmetric property estimation. We provide an algorithm which matches the previous best known efficient algorithms for computing approximate PML distributions and improves when the number of distinct observed frequencies in the given instance is small. We achieve this result by exploiting new sparsity structure in approximate PML distributions and providing a new matrix rounding algorithm, of independent interest. Leveraging this result, we obtain the first provable computationally efficient implementation of PseudoPML, a general framework for estimating a broad class of symmetric properties. Additionally, we obtain efficient PML-based estimators for distributions with small profile entropy, a natural instance-based complexity measure. Further, we provide a simpler and more practical PseudoPML implementation that matches the best-known theoretical guarantees of such an estimator and evaluate this method empirically.

Reinforcement learning has the potential to automate the acquisition of behavior in complex settings, but in order for it to be successfully deployed, a number of practical challenges must be addressed. First, in real world settings, when an agent attempts a tasks and fails, the environment must somehow "reset" so that the agent can attempt the task again. While easy in simulation, this could require considerable human effort in the real world, especially if the number of trials is very large. Second, real world learning is often limited by challenges in exploration, as complex, temporally extended behavior is often times difficult to acquire with random exploration. In this work, we show how a single method can allow an agent to acquire skills with minimal supervision while removing the need for resets. We do this by exploiting the insight that the need to reset" an agent to a broad set of initial states for a learning task provides a natural setting to learn a diverse set ofreset-skills." We propose a general-sum game formulation that naturally balances the objective of resetting and learning skills, and demonstrate that this approach improves performance on reset-free tasks, and additionally show that the skills we …

Unlabeled data learning has attracted considerable attention recently. However, it is still elusive to extract the expected high-level semantic feature with mere unsupervised learning. In the meantime, semi-supervised learning (SSL) demonstrates a promising future in leveraging few samples. In this paper, we combine both to propose an Unsupervised Semantic Aggregation and Deformable Template Matching (USADTM) framework for SSL, which strives to improve the classification performance with few labeled data and then reduce the cost in data annotating. Specifically, unsupervised semantic aggregation based on Triplet Mutual Information (T-MI) loss is explored to generate semantic labels for unlabeled data. Then the semantic labels are aligned to the actual class by the supervision of labeled data. Furthermore, a feature pool that stores the labeled samples is dynamically updated to assign proxy labels for unlabeled data, which are used as targets for cross-entropy minimization. Extensive experiments and analysis across four standard semi-supervised learning benchmarks validate that USADTM achieves top performance (e.g., 90.46% accuracy on CIFAR-10 with 40 labels and 95.20% accuracy with 250 labels). The code is released at https://github.com/taohan10200/USADTM.

In modern multilabel classification problems, each data instance belongs to a small number of classes among a large set of classes. In other words, these problems involve learning very sparse binary label vectors. Moreover, in the large-scale problems, the labels typically have certain (unknown) hierarchy. In this paper we exploit the sparsity of label vectors and the hierarchical structure to embed them in low-dimensional space using label groupings. Consequently, we solve the classification problem in a much lower dimensional space and then obtain labels in the original space using an appropriately defined lifting. Our method builds on the work of (Ubaru & Mazumdar, 2017), where the idea of group testing was also explored for multilabel classification. We first present a novel data-dependent grouping approach, where we use a group construction based on a low-rank Nonnegative Matrix Factorization (NMF) of the label matrix of training instances. The construction also allows us, using recent results, to develop a fast prediction algorithm that has a \emph{logarithmic runtime in the number of labels}. We then present a hierarchical partitioning approach that exploits the label hierarchy in large-scale problems to divide the large label space into smaller sub-problems, which can then be solved independently via …

We introduce Transductive Infomation Maximization (TIM) for few-shot learning. Our method maximizes the mutual information between the query features and their label predictions for a given few-shot task, in conjunction with a supervision loss based on the support set. Furthermore, we propose a new alternating-direction solver for our mutual-information loss, which substantially speeds up transductive inference convergence over gradient-based optimization, while yielding similar accuracy. TIM inference is modular: it can be used on top of any base-training feature extractor. Following standard transductive few-shot settings, our comprehensive experiments demonstrate that TIM outperforms state-of-the-art methods significantly across various datasets and networks, while used on top of a fixed feature extractor trained with simple cross-entropy on the base classes, without resorting to complex meta-learning schemes. It consistently brings between 2% and 5% improvement in accuracy over the best performing method, not only on all the well-established few-shot benchmarks but also on more challenging scenarios, with domain shifts and larger numbers of classes.

Meta-learning algorithms aim to learn two components: a model that predicts targets for a task, and a base learner that updates that model when given examples from a new task. This additional level of learning can be powerful, but it also creates another potential source of overfitting, since we can now overfit in either the model or the base learner. We describe both of these forms of meta-learning overfitting, and demonstrate that they appear experimentally in common meta-learning benchmarks. We introduce an information-theoretic framework of meta-augmentation, whereby adding randomness discourages the base learner and model from learning trivial solutions that do not generalize to new tasks. We demonstrate that meta-augmentation produces large complementary benefits to recently proposed meta-regularization techniques.

Contrastive learning between multiple views of the data has recently achieved state of the art performance in the field of self-supervised representation learning. Despite its success, the influence of different view choices has been less studied. In this paper, we use theoretical and empirical analysis to better understand the importance of view selection, and argue that we should reduce the mutual information (MI) between views while keeping task-relevant information intact. To verify this hypothesis, we devise unsupervised and semi-supervised frameworks that learn effective views by aiming to reduce their MI. We also consider data augmentation as a way to reduce MI, and show that increasing data augmentation indeed leads to decreasing MI and improves downstream classification accuracy. As a by-product, we achieve a new state-of-the-art accuracy on unsupervised pre-training for ImageNet classification (73% top-1 linear readout with a ResNet-50).

In self-supervised learning, a system is tasked with achieving a surrogate objective by defining alternative targets on a set of unlabeled data. The aim is to build useful representations that can be used in downstream tasks, without costly manual annotation. In this work, we propose a novel self-supervised formulation of relational reasoning that allows a learner to bootstrap a signal from information implicit in unlabeled data. Training a relation head to discriminate how entities relate to themselves (intra-reasoning) and other entities (inter-reasoning), results in rich and descriptive representations in the underlying neural network backbone, which can be used in downstream tasks such as classification and image retrieval. We evaluate the proposed method following a rigorous experimental procedure, using standard datasets, protocols, and backbones. Self-supervised relational reasoning outperforms the best competitor in all conditions by an average 14% in accuracy, and the most recent state-of-the-art model by 3%. We link the effectiveness of the method to the maximization of a Bernoulli log-likelihood, which can be considered as a proxy for maximizing the mutual information, resulting in a more efficient objective with respect to the commonly used contrastive losses.

Existing semi-supervised learning (SSL) algorithms use a single weight to balance the loss of labeled and unlabeled examples, i.e., all unlabeled examples are equally weighted. But not all unlabeled data are equal. In this paper we study how to use a different weight for “every” unlabeled example. Manual tuning of all those weights -- as done in prior work -- is no longer possible. Instead, we adjust those weights via an algorithm based on the influence function, a measure of a model's dependency on one training example. To make the approach efficient, we propose a fast and effective approximation of the influence function. We demonstrate that this technique outperforms state-of-the-art methods on semi-supervised image and language classification tasks.

While semi-supervised learning (SSL) has proven to be a promising way for leveraging unlabeled data when labeled data is scarce, the existing SSL algorithms typically assume that training class distributions are balanced. However, these SSL algorithms trained under imbalanced class distributions can severely suffer when generalizing to a balanced testing criterion, since they utilize biased pseudo-labels of unlabeled data toward majority classes. To alleviate this issue, we formulate a convex optimization problem to softly refine the pseudo-labels generated from the biased model, and develop a simple algorithm, named Distribution Aligning Refinery of Pseudo-label (DARP) that solves it provably and efficiently. Under various class imbalanced semi-supervised scenarios, we demonstrate the effectiveness of DARP and its compatibility with state-of-the-art SSL schemes.

Contrastive learning applied to self-supervised representation learning has seen a resurgence in recent years, leading to state of the art performance in the unsupervised training of deep image models. Modern batch contrastive approaches subsume or significantly outperform traditional contrastive losses such as triplet, max-margin and the N-pairs loss. In this work, we extend the self-supervised batch contrastive approach to the fully-supervised setting, allowing us to effectively leverage label information. Clusters of points belonging to the same class are pulled together in embedding space, while simultaneously pushing apart clusters of samples from different classes. We analyze two possible versions of the supervised contrastive (SupCon) loss, identifying the best-performing formulation of the loss. On ResNet-200, we achieve top-1 accuracy of 81.4% on the ImageNet dataset, which is 0.8% above the best number reported for this architecture. We show consistent outperformance over cross-entropy on other datasets and two ResNet variants. The loss shows benefits for robustness to natural corruptions, and is more stable to hyperparameter settings such as optimizers and data augmentations. In reduced data settings, it outperforms cross-entropy significantly. Our loss function is simple to implement and reference TensorFlow code is released at https://t.ly/supcon.

A good teacher can adjust a curriculum based on students' learning history. By analogy, in this paper, we study the dynamics of a deep neural network's (DNN) performance on individual samples during its learning process. The observed properties allow us to develop an adaptive curriculum that leads to faster learning of more accurate models. We introduce dynamic instance hardness (DIH), the exponential moving average of a sample's instantaneous hardness (e.g., a loss, or a change in output) over the training history. A low DIH indicates that a model retains knowledge about a sample over time. For DNNs, we find that a sample's DIH early in training predicts its DIH in later stages. Hence, we can train a model using samples mostly with higher DIH and safely deprioritize those with lower DIH. This motivates a DIH guided curriculum learning (DIHCL) procedure. Compared to existing CL methods: (1) DIH is more stable over time than using only instantaneous hardness, which is noisy due to stochastic training and DNN's non-smoothness; (2) DIHCL is computationally inexpensive since it uses only a byproduct of back-propagation and thus does not require extra inference. On 11 datasets, DIHCL significantly outperforms random mini-batch SGD and recent CL methods …

Curriculum learning is a technique to improve a model performance and generalization based on the idea that easy samples should be presented before difficult ones during training. While it is generally complex to estimate a priori the difficulty of a given sample, recent works have shown that curriculum learning can be formulated dynamically in a self-supervised manner. The key idea is to somehow estimate the importance (or weight) of each sample directly during training based on the observation that easy and hard samples behave differently and can therefore be separated. However, these approaches are usually limited to a specific task (e.g., classification) and require extra data annotations, layers or parameters as well as a dedicated training procedure. We propose instead a simple and generic method that can be applied to a variety of losses and tasks without any change in the learning procedure. It consists in appending a novel loss function on top of any existing task loss, hence its name: the SuperLoss. Its main effect is to automatically downweight the contribution of samples with a large loss, i.e. hard samples, effectively mimicking the core principle of curriculum learning. As a side effect, we show that our loss prevents the …

Neuroimaging studies produce gigabytes of spatio-temporal data for a small number of participants and stimuli. Recent work increasingly suggests that the common practice of averaging across participants and stimuli leaves out systematic and meaningful information. We propose Neural Topographic Factor Analysis (NTFA), a probabilistic factor analysis model that infers embeddings for participants and stimuli. These embeddings allow us to reason about differences between participants and stimuli as signal rather than noise. We evaluate NTFA on data from an in-house pilot experiment, as well as two publicly available datasets. We demonstrate that inferring representations for participants and stimuli improves predictive generalization to unseen data when compared to previous topographic methods. We also demonstrate that the inferred latent factor representations are useful for downstream tasks such as multivoxel pattern analysis and functional connectivity.

Within months of birth, children develop meaningful expectations about the world around them. How much of this early knowledge can be explained through generic learning mechanisms applied to sensory data, and how much of it requires more substantive innate inductive biases? Addressing this fundamental question in its full generality is currently infeasible, but we can hope to make real progress in more narrowly defined domains, such as the development of high-level visual categories, thanks to improvements in data collecting technology and recent progress in deep learning. In this paper, our goal is precisely to achieve such progress by utilizing modern self-supervised deep learning methods and a recent longitudinal, egocentric video dataset recorded from the perspective of three young children (Sullivan et al., 2020). Our results demonstrate the emergence of powerful, high-level visual representations from developmentally realistic natural videos using generic self-supervised learning objectives.

Learning from indirect supervision signals is important in real-world AI applications when, often, gold labels are missing or too costly. In this paper, we develop a unified theoretical framework for multi-class classification when the supervision is provided by a variable that contains nonzero mutual information with the gold label. The nature of this problem is determined by (i) the transition probability from the gold labels to the indirect supervision variables and (ii) the learner's prior knowledge about the transition. Our framework relaxes assumptions made in the literature, and supports learning with unknown, non-invertible and instance-dependent transitions. Our theory introduces a novel concept called \emph{separation}, which characterizes the learnability and generalization bounds. We also demonstrate the application of our framework via concrete novel results in a variety of learning scenarios such as learning with superset annotations and joint supervision signals.

Learning from label proportions (LLP) is a weakly supervised setting for classification in which unlabeled training instances are grouped into bags, and each bag is annotated with the proportion of each class occurring in that bag. Prior work on LLP has yet to establish a consistent learning procedure, nor does there exist a theoretically justified, general purpose training criterion. In this work we address these two issues by posing LLP in terms of mutual contamination models (MCMs), which have recently been applied successfully to study various other weak supervision settings. In the process, we establish several novel technical results for MCMs, including unbiased losses and generalization error bounds under non-iid sampling plans. We also point out the limitations of a common experimental setting for LLP, and propose a new one based on our MCM framework.

The ability to record activities from hundreds of neurons simultaneously in the brain has placed an increasing demand for developing appropriate statistical techniques to analyze such data. Recently, deep generative models have been proposed to fit neural population responses. While these methods are flexible and expressive, the downside is that they can be difficult to interpret and identify. To address this problem, we propose a method that integrates key ingredients from latent models and traditional neural encoding models. Our method, pi-VAE, is inspired by recent progress on identifiable variational auto-encoder, which we adapt to make appropriate for neuroscience applications. Specifically, we propose to construct latent variable models of neural activity while simultaneously modeling the relation between the latent and task variables (non-neural variables, e.g. sensory, motor, and other externally observable states). The incorporation of task variables results in models that are not only more constrained, but also show qualitative improvements in interpretability and identifiability. We validate pi-VAE using synthetic data, and apply it to analyze neurophysiological datasets from rat hippocampus and macaque motor cortex. We demonstrate that pi-VAE not only fits the data better, but also provides unexpected novel insights into the structure of the neural codes.

Neural datasets often contain measurements of neural activity across multiple trials of a repeated stimulus or behavior. An important problem in the analysis of such datasets is to characterize systematic aspects of neural activity that carry information about the repeated stimulus or behavior of interest, which can be considered signal'', and to separate them from the trial-to-trial fluctuations in activity that are not time-locked to the stimulus, which for purposes of such analyses can be considerednoise''. Gaussian Process factor models provide a powerful tool for identifying shared structure in high-dimensional neural data. However, they have not yet been adapted to the problem of characterizing signal and noise in multi-trial datasets. Here we address this shortcoming by proposing ``signal-noise'' Poisson-spiking Gaussian Process Factor Analysis (SNP-GPFA), a flexible latent variable model that resolves signal and noise latent structure in neural population spiking activity. To learn the parameters of our model, we introduce a Fourier-domain black box variational inference method that quickly identifies smooth latent structure. The resulting model reliably uncovers latent signal and trial-to-trial noise-related fluctuations in large-scale recordings. We use this model to show that in monkey V1, noise fluctuations perturb neural activity within a subspace orthogonal to signal activity, …

The brain takes uncertainty intrinsic to our world into account. For example, associating spatial locations with rewards requires to predict not only expected reward at new spatial locations but also its uncertainty to avoid catastrophic events and forage safely. A powerful and flexible framework for nonlinear regression that takes uncertainty into account in a principled Bayesian manner is Gaussian process (GP) regression. Here I propose that the brain implements GP regression and present neural networks (NNs) for it. First layer neurons, e.g.\ hippocampal place cells, have tuning curves that correspond to evaluations of the GP kernel. Output neurons explicitly and distinctively encode predictive mean and variance, as observed in orbitofrontal cortex (OFC) for the case of reward prediction. Because the weights of a NN implementing exact GP regression do not arise with biological plasticity rules, I present approximations to obtain local (anti-)Hebbian synaptic learning rules. The resulting neuronal network approximates the full GP well compared to popular sparse GP approximations and achieves comparable predictive performance.

Neural Coding analyses often reflect an assumption that neural populations respond uniquely and consistently to particular stimuli. For example, analyses of spatial remapping in hippocampal populations often assume that each environment has one unique representation and that remapping occurs over long time scales as an animal traverses between distinct environments. However, as neuroscience experiments begin to explore more naturalistic tasks and stimuli, and reflect more ambiguity in neural representations, methods for analyzing population neural codes must adapt to reflect these features. In this paper, we develop a new state-space modeling framework to address two important issues related to remapping. First, neurons may exhibit significant trial-to-trial or moment-to-moment variability in the firing patterns used to represent a particular environment or stimulus. Second, in ambiguous environments and tasks that involve cognitive uncertainty, neural populations may rapidly fluctuate between multiple representations. The state-space model addresses these two issues by integrating an observation model, which allows for multiple representations of the same stimulus or environment, with a state model, which characterizes the moment-by-moment probability of a shift in the neural representation. These models allow us to compute instantaneous estimates of the stimulus or environment currently represented by the population. We demonstrate the application of …

Neurons mainly communicate through spikes, and much effort has been spent to understand how the dynamics of spiking neural networks (SNNs) relates to their connectivity. Meanwhile, most major advances in machine learning have been made with simpler, rate-based networks, with SNNs only recently showing competitive results, largely thanks to transferring insights from rate to spiking networks. However, it is still an open question exactly which computations SNNs perform. Recently, the time-averaged firing rates of several SNNs were shown to yield the solutions to convex optimization problems. Here we turn these findings around and show that virtually all inhibition-dominated SNNs can be understood through the lens of convex optimization, with network connectivity, timescales, and firing thresholds being intricately linked to the parameters of underlying convex optimization problems. This approach yields new, geometric insights into the computations performed by spiking networks. In particular, we establish a class of SNNs whose instantaneous output provides a solution to linear or quadratic programming problems, and we thereby reveal their input-output mapping. Using these insights, we derive local, supervised learning rules that can approximate given convex input-output functions, and we show that the resulting networks are consistent with many features from biological networks, such as low …

Recent technological advances in systems neuroscience have led to a shift away from using simple tasks, with low-dimensional, well-controlled stimuli, towards trying to understand neural activity during naturalistic behavior. However, with the increase in number and complexity of task-relevant features, standard analyses such as estimating tuning functions become challenging. Here, we use a Poisson generalized additive model (P-GAM) with spline nonlinearities and an exponential link function to map a large number of task variables (input stimuli, behavioral outputs, or activity of other neurons, modeled as discrete events or continuous variables) into spike counts. We develop efficient procedures for parameter learning by optimizing a generalized cross-validation score and infer marginal confidence bounds for the contribution of each feature to neural responses. This allows us to robustly identify a minimal set of task features that each neuron is responsive to, circumventing computationally demanding model comparison. We show that our estimation procedure outperforms traditional regularized GLMs in terms of both fit quality and computing time. When applied to neural recordings from monkeys performing a virtual reality spatial navigation task, P-GAM reveals mixed selectivity and preferential coupling between neurons with similar tuning.

Generalized linear models are one of the most efficient paradigms for predicting the correlated stochastic activity of neuronal networks in response to external stimuli, with applications in many brain areas. However, when dealing with complex stimuli, the inferred coupling parameters often do not generalise across different stimulus statistics, leading to degraded performance and blowup instabilities. Here, we develop a two-step inference strategy that allows us to train robust generalised linear models of interacting neurons, by explicitly separating the effects of correlations in the stimulus from network interactions in each training step. Applying this approach to the responses of retinal ganglion cells to complex visual stimuli, we show that, compared to classical methods, the models trained in this way exhibit improved performance, are more stable, yield robust interaction networks, and generalise well across complex visual statistics. The method can be extended to deep convolutional neural networks, leading to models with high predictive accuracy for both the neuron firing rates and their correlations.

Neural-Matrix style, high-density electrode arrays for brain-machine interfaces (BMIs) and neuroscientific research require the use of multiplexing: Each recording channel can be routed to one of several electrode sites on the array. This capability allows the user to flexibly distribute recording channels to the locations where the most desirable neural signals can be resolved. For example, in the Neuropixel probe, 960 electrodes can be addressed by 384 recording channels. However, currently no adaptive methods exist to use recorded neural data to optimize/customize the electrode selections per recording context. Here, we present an algorithm called classification-based selection (CBS) that optimizes the joint electrode selections for all recording channels so as to maximize isolation quality of detected neurons. We show, in experiments using Neuropixels in non-human primates, that this algorithm yields a similar number of isolated neurons as would be obtained if all electrodes were recorded simultaneously. Neuron counts were 41-85% improved over previously published electrode selection strategies. The neurons isolated from electrodes selected by CBS were a 73% match, by spike timing, to the complete set of recordable neurons around the probe. The electrodes selected by CBS exhibited higher average per-recording-channel signal-to-noise ratio. CBS, and selection optimization in general, could play …

A common goal in the analysis of neural data is to compress large population recordings into sets of interpretable, low-dimensional latent trajectories. This problem can be approached using Gaussian process (GP)-based methods which provide uncertainty quantification and principled model selection. However, standard GP priors do not distinguish between underlying dynamical processes and other forms of temporal autocorrelation. Here, we propose a new family of “dynamical” priors over trajectories, in the form of GP covariance functions that express a property shared by most dynamical systems: temporal non-reversibility. Non-reversibility is a universal signature of autonomous dynamical systems whose state trajectories follow consistent flow fields, such that any observed trajectory could not occur in reverse. Our new multi-output GP kernels can be used as drop-in replacements for standard kernels in multivariate regression, but also in latent variable models such as Gaussian process factor analysis (GPFA). We therefore introduce GPFADS (Gaussian Process Factor Analysis with Dynamical Structure), which models single-trial neural population activity using low-dimensional, non-reversible latent processes. Unlike previously proposed non-reversible multi-output kernels, ours admits a Kronecker factorization enabling fast and memory-efficient learning and inference. We apply GPFADS to synthetic data and show that it correctly recovers ground truth phase portraits. GPFADS also …

Excitation-inhibition balance is ubiquitously observed in the cortex. Recent studies suggest an intriguing link between balance on fast timescales, tight balance, and efficient information coding with spikes. We further this connection by taking a principled approach to optimal balanced networks of excitatory (E) and inhibitory(I) neurons. By deriving E-I spiking neural networks from greedy spike-based optimizations of constrained minimax objectives, we show that tight balance arises from correcting for deviations from the minimax optima. We predict specific neuron firing rates in the networks by solving the minimax problems, going beyond statistical theories of balanced networks. We design minimax objectives for reconstruction of an input signal, associative memory, and storage of manifold attractors, and derive from them E-I networks that perform the computation. Overall, we present a novel normative modeling approach for spiking E-I networks, going beyond the widely-used energy-minimizing networks that violate Dale’s law. Our networks can be used to model cortical circuits and computations.

Biological neural networks face a formidable task: performing reliable computations in the face of intrinsic stochasticity in individual neurons, imprecisely specified synaptic connectivity, and nonnegligible delays in synaptic transmission. A common approach to combatting such biological heterogeneity involves averaging over large redundant networks of N neurons resulting in coding errors that decrease classically as the square root of N. Recent work demonstrated a novel mechanism whereby recurrent spiking networks could efficiently encode dynamic stimuli achieving a superclassical scaling in which coding errors decrease as 1/N. This specific mechanism involved two key ideas: predictive coding, and a tight balance, or cancellation between strong feedforward inputs and strong recurrent feedback. However, the theoretical principles governing the efficacy of balanced predictive coding and its robustness to noise, synaptic weight heterogeneity and communication delays remain poorly understood. To discover such principles, we introduce an analytically tractable model of balanced predictive coding, in which the degree of balance and the degree of weight disorder can be dissociated unlike in previous balanced network models, and we develop a mean-field theory of coding accuracy. Overall, our work provides and solves a general theoretical framework for dissecting the differential contributions neural noise, synaptic disorder, chaos, synaptic delays, and …

A common problem in neuroscience is to elucidate the collective neural representations of behaviorally important variables such as head direction, spatial location, upcoming movements, or mental spatial transformations. Often, these latent variables are internal constructs not directly accessible to the experimenter. Here, we propose a new probabilistic latent variable model to simultaneously identify the latent state and the way each neuron contributes to its representation in an unsupervised way. In contrast to previous models which assume Euclidean latent spaces, we embrace the fact that latent states often belong to symmetric manifolds such as spheres, tori, or rotation groups of various dimensions. We therefore propose the manifold Gaussian process latent variable model (mGPLVM), where neural responses arise from (i) a shared latent variable living on a specific manifold, and (ii) a set of non-parametric tuning curves determining how each neuron contributes to the representation. Cross-validated comparisons of models with different topologies can be used to distinguish between candidate manifolds, and variational inference enables quantification of uncertainty. We demonstrate the validity of the approach on several synthetic datasets, as well as on calcium recordings from the ellipsoid body of Drosophila melanogaster and extracellular recordings from the mouse anterodorsal thalamic nucleus. These circuits …

One of the primary goals of systems neuroscience is to relate the structure of neural circuits to their function, yet patterns of connectivity are difficult to establish when recording from large populations in behaving organisms. Many previous approaches have attempted to estimate functional connectivity between neurons using statistical modeling of observational data, but these approaches rely heavily on parametric assumptions and are purely correlational. Recently, however, holographic photostimulation techniques have made it possible to precisely target selected ensembles of neurons, offering the possibility of establishing direct causal links. A naive method for inferring functional connections is to stimulate each individual neuron multiple times and observe the responses of cells in the local network, but this approach scales poorly with the number of neurons. Here, we propose a method based on noisy group testing that drastically increases the efficiency of this process in sparse networks. By stimulating small ensembles of neurons, we show that it is possible to recover binarized network connectivity with a number of tests that grows only logarithmically with population size under minimal statistical assumptions. Moreover, we prove that our approach, which reduces to an efficiently solvable convex optimization problem, can be related to Variational Bayesian inference on …

Modern recording techniques can generate large-scale measurements of multiple neural populations over extended time periods. However, it remains a challenge to model non-stationary interactions between high-dimensional populations of neurons. To tackle this challenge, we develop recurrent switching linear dynamical systems models for multiple populations. Here, each high-dimensional neural population is represented by a unique set of latent variables, which evolve dynamically in time. Populations interact with each other through this low-dimensional space. We allow the nature of these interactions to change over time by using a discrete set of dynamical states. Additionally, we parameterize these discrete state transition rules to capture which neural populations are responsible for switching between interaction states. To fit the model, we use variational expectation-maximization with a structured mean-field approximation. After validating the model on simulations, we apply it to two different neural datasets: spiking activity from motor areas in a non-human primate, and calcium imaging from neurons in the nematode \textit{C. elegans}. In both datasets, the model reveals behaviorally-relevant discrete states with unique inter-population interactions and different populations that predict transitioning between these states.

The standard approach to fitting an autoregressive spike train model is to maximize the likelihood for one-step prediction. This maximum likelihood estimation (MLE) often leads to models that perform poorly when generating samples recursively for more than one time step. Moreover, the generated spike trains can fail to capture important features of the data and even show diverging firing rates. To alleviate this, we propose to directly minimize the divergence between neural recorded and model generated spike trains using spike train kernels. We develop a method that stochastically optimizes the maximum mean discrepancy induced by the kernel. Experiments performed on both real and synthetic neural data validate the proposed approach, showing that it leads to well-behaving models. Using different combinations of spike train kernels, we show that we can control the trade-off between different features which is critical for dealing with model-mismatch.

We introduce manifold-learning flows (ℳ-flows), a new class of generative models that simultaneously learn the data manifold as well as a tractable probability density on that manifold. Combining aspects of normalizing flows, GANs, autoencoders, and energy-based models, they have the potential to represent data sets with a manifold structure more faithfully and provide handles on dimensionality reduction, denoising, and out-of-distribution detection. We argue why such models should not be trained by maximum likelihood alone and present a new training algorithm that separates manifold and density updates. In a range of experiments we demonstrate how ℳ-flows learn the data manifold and allow for better inference than standard flows in the ambient data space.

Learning from different data types is a long-standing goal in machine learning research, as multiple information sources co-occur when describing natural phenomena. However, existing generative models that approximate a multimodal ELBO rely on difficult or inefficient training schemes to learn a joint distribution and the dependencies between modalities. In this work, we propose a novel, efficient objective function that utilizes the Jensen-Shannon divergence for multiple distributions. It simultaneously approximates the unimodal and joint multimodal posteriors directly via a dynamic prior. In addition, we theoretically prove that the new multimodal JS-divergence (mmJSD) objective optimizes an ELBO. In extensive experiments, we demonstrate the advantage of the proposed mmJSD model compared to previous work in unsupervised, generative learning tasks.

Normalizing flows and variational autoencoders are powerful generative models that can represent complicated density functions. However, they both impose constraints on the models: Normalizing flows use bijective transformations to model densities whereas VAEs learn stochastic transformations that are non-invertible and thus typically do not provide tractable estimates of the marginal likelihood. In this paper, we introduce SurVAE Flows: A modular framework of composable transformations that encompasses VAEs and normalizing flows. SurVAE Flows bridge the gap between normalizing flows and VAEs with surjective transformations, wherein the transformations are deterministic in one direction -- thereby allowing exact likelihood computation, and stochastic in the reverse direction -- hence providing a lower bound on the corresponding likelihood. We show that several recently proposed methods, including dequantization and augmented normalizing flows, can be expressed as SurVAE Flows. Finally, we introduce common operations such as the max value, the absolute value, sorting and stochastic permutation as composable layers in SurVAE Flows.

Continuous-time event data are common in applications such as individual behavior data, financial transactions, and medical health records. Modeling such data can be very challenging, in particular for applications with many different types of events,since it requires a model to predict the event types as well as the time of occurrence. Recurrent neural networks that parameterize time-varying intensity functions are the current state-of-the-art for predictive modeling with such data. These models typically assume that all event sequences come from the same data distribution. However, in many applications event sequences are generated by different sources,or users, and their characteristics can be very different. In this paper, we extend the broad class of neural marked point process models to mixtures of latent embeddings,where each mixture component models the characteristic traits of a given user. Our approach relies on augmenting these models with a latent variable that encodes user characteristics, represented by a mixture model over user behavior that is trained via amortized variational inference. We evaluate our methods on four large real-world datasets and demonstrate systematic improvements from our approach over existing work for a variety of predictive metrics such as log-likelihood, next event ranking, and source-of-sequence identification.

Undirected graphical models are compact representations of joint probability distributions over random variables. To solve inference tasks of interest, graphical models of arbitrary topology can be trained using empirical risk minimization. However, to solve inference tasks that were not seen during training, these models (EGMs) often need to be re-trained. Instead, we propose an inference-agnostic adversarial training framework which produces an infinitely-large ensemble of graphical models (AGMs). The ensemble is optimized to generate data within the GAN framework, and inference is performed using a finite subset of these models. AGMs perform comparably with EGMs on inference tasks that the latter were specifically optimized for. Most importantly, AGMs show significantly better generalization to unseen inference tasks compared to EGMs, as well as deep neural architectures like GibbsNet and VAEAC which allow arbitrary conditioning. Finally, AGMs allow fast data sampling, competitive with Gibbs sampling from EGMs.

Despite progress in training neural networks for lossy image compression, current approaches fail to maintain both perceptual quality and abstract features at very low bitrates. Encouraged by recent success in learning discrete representations with Vector Quantized Variational Autoencoders (VQ-VAEs), we motivate the use of a hierarchy of VQ-VAEs to attain high factors of compression. We show that the combination of stochastic quantization and hierarchical latent structure aids likelihood-based image compression. This leads us to introduce a novel objective for training hierarchical VQ-VAEs. Our resulting scheme produces a Markovian series of latent variables that reconstruct images of high-perceptual quality which retain semantically meaningful features. We provide qualitative and quantitative evaluations on the CelebA and MNIST datasets.

Normalizing flows have shown great promise for modelling flexible probability distributions in a computationally tractable way. However, whilst data is often naturally described on Riemannian manifolds such as spheres, torii, and hyperbolic spaces, most normalizing flows implicitly assume a flat geometry, making them either misspecified or ill-suited in these situations. To overcome this problem, we introduce Riemannian continuous normalizing flows, a model which admits the parametrization of flexible probability measures on smooth manifolds by defining flows as the solution to ordinary differential equations. We show that this approach can lead to substantial improvements on both synthetic and real-world data when compared to standard flows or previously introduced projected flows.

Several machine learning applications involve the optimization of higher-order derivatives (e.g., gradients of gradients) during training, which can be expensive with respect to memory and computation even with automatic differentiation. As a typical example in generative modeling, score matching~(SM) involves the optimization of the trace of a Hessian. To improve computing efficiency, we rewrite the SM objective and its variants in terms of directional derivatives, and present a generic strategy to efficiently approximate any-order directional derivative with finite difference~(FD). Our approximation only involves function evaluations, which can be executed in parallel, and no gradient computations. Thus, it reduces the total computational cost while also improving numerical stability. We provide two instantiations by reformulating variants of SM objectives into the FD forms. Empirically, we demonstrate that our methods produce results comparable to the gradient-based counterparts while being much more computationally efficient.

We propose an energy-based model (EBM) in the latent space of a generator model, so that the EBM serves as a prior model that stands on the top-down network of the generator model. Both the latent space EBM and the top-down network can be learned jointly by maximum likelihood, which involves short-run MCMC sampling from both the prior and posterior distributions of the latent vector. Due to the low dimensionality of the latent space and the expressiveness of the top-down network, a simple EBM in latent space can capture regularities in the data effectively, and MCMC sampling in latent space is efficient and mixes well. We show that the learned model exhibits strong performances in terms of image and text generation and anomaly detection. The one-page code can be found in supplementary materials.

Achieving the full promise of the Thermodynamic Variational Objective (TVO), a recently proposed variational inference objective that lower-bounds the log evidence via one-dimensional Riemann integration, requires choosing a ``schedule'' of sorted discretization points. This paper introduces a bespoke Gaussian process bandit optimization method for automatically choosing these points. Our approach not only automates their one-time selection, but also dynamically adapts their positions over the course of optimization, leading to improved model learning and inference. We provide theoretical guarantees that our bandit optimization converges to the regret-minimizing choice of integration points. Empirical validation of our algorithm is provided in terms of improved learning and inference in Variational Autoencoders and sigmoid belief networks.

Deep generative models often perform poorly in real-world applications due to the heterogeneity of natural data sets. Heterogeneity arises from data containing different types of features (categorical, ordinal, continuous, etc.) and features of the same type having different marginal distributions. We propose an extension of variational autoencoders (VAEs) called VAEM to handle such heterogeneous data. VAEM is a deep generative model that is trained in a two stage manner, such that the first stage provides a more uniform representation of the data to the second stage, thereby sidestepping the problems caused by heterogeneous data. We provide extensions of VAEM to handle partially observed data, and demonstrate its performance in data generation, missing data prediction and sequential feature selection tasks. Our results show that VAEM broadens the range of real-world applications where deep generative models can be successfully deployed.

The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) – an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

Reconciling symbolic and distributed representations is a crucial challenge that can potentially resolve the limitations of current deep learning. Remarkable advances in this direction have been achieved recently via generative object-centric representation models. While learning a recognition model that infers object-centric symbolic representations like bounding boxes from raw images in an unsupervised way, no such model can provide another important ability of a generative model, i.e., generating (sampling) according to the structure of learned world density. In this paper, we propose Generative Neurosymbolic Machines, a generative model that combines the benefits of distributed and symbolic representations to support both structured representations of symbolic components and density-based generation. These two crucial properties are achieved by a two-layer latent hierarchy with the global distributed latent for flexible density modeling and the structured symbolic latent map. To increase the model flexibility in this hierarchical structure, we also propose the StructDRAW prior. In experiments, we show that the proposed model significantly outperforms the previous structured representation models as well as the state-of-the-art non-structured generative models in terms of both structure accuracy and image generation quality.

Temporal point process (TPP) models combined with recurrent neural networks provide a powerful framework for modeling continuous-time event data. While such models are flexible, they are inherently sequential and therefore cannot benefit from the parallelism of modern hardware. By exploiting the recent developments in the field of normalizing flows, we design TriTPP - a new class of non-recurrent TPP models, where both sampling and likelihood computation can be done in parallel. TriTPP matches the flexibility of RNN-based methods but permits several orders of magnitude faster sampling. This enables us to use the new model for variational inference in continuous-time discrete-state systems. We demonstrate the advantages of the proposed framework on synthetic and real-world datasets.

This work addresses efficient inference and learning in switching Gaussian linear dynamical systems using a Rao-Blackwellised particle filter and a corresponding Monte Carlo objective. To improve the forecasting capabilities, we extend this classical model by conditionally linear state-to-switch dynamics, while leaving the partial tractability of the conditional Gaussian linear part intact. Furthermore, we use an auxiliary variable approach with a decoder-type neural network that allows for more complex non-linear emission models and multivariate observations. We propose a Monte Carlo objective that leverages the conditional linearity by computing the corresponding conditional expectations in closed-form and a suitable proposal distribution that is factorised similarly to the optimal proposal distribution. We evaluate our approach on several popular time series forecasting datasets as well as image streams of simulated physical systems. Our results show improved forecasting performance compared to other deep state-space model approaches.

The current dominant paradigm in sensorimotor control, whether imitation or reinforcement learning, is to train policies directly in raw action spaces such as torque, joint angle, or end-effector position. This forces the agent to make decision at each point in training, and hence, limits the scalability to continuous, high-dimensional, and long-horizon tasks. In contrast, research in classical robotics has, for a long time, exploited dynamical systems as a policy representation to learn robot behaviors via demonstrations. These techniques, however, lack the flexibility and generalizability provided by deep learning or deep reinforcement learning and have remained under-explored in such settings. In this work, we begin to close this gap and embed dynamics structure into deep neural network-based policies by reparameterizing action spaces with differential equations. We propose Neural Dynamic Policies (NPDs) that make predictions in trajectory distribution space as opposed to prior policy learning methods where action represents the raw control space. The embedded structure allows us to perform end-to-end policy learning under both reinforcement and imitation learning setups. We show that NDPs achieve better or comparable performance to state-of-the-art approaches on many robotic control tasks using both reward-based training and demonstrations. Project video and code are available at: https://shikharbahl.github.io/neural-dynamic-policies/.

Offline reinforcement learning (RL), also known as batch RL, offers the prospect of policy optimization from large pre-recorded datasets without online environment interaction. It addresses challenges with regard to the cost of data collection and safety, both of which are particularly pertinent to real-world applications of RL. Unfortunately, most off-policy algorithms perform poorly when learning from a fixed dataset. In this paper, we propose a novel offline RL algorithm to learn policies from data using a form of critic-regularized regression (CRR). We find that CRR performs surprisingly well and scales to tasks with high-dimensional state and action spaces -- outperforming several state-of-the-art offline RL algorithms by a significant margin on a wide range of benchmark tasks.

Learning from Observations (LfO) is a practical reinforcement learning scenario from which many applications can benefit through the reuse of incomplete resources. Compared to conventional imitation learning (IL), LfO is more challenging because of the lack of expert action guidance. In both conventional IL and LfO, distribution matching is at the heart of their foundation. Traditional distribution matching approaches are sample-costly which depend on on-policy transitions for policy learning. Towards sample-efficiency, some off-policy solutions have been proposed, which, however, either lack comprehensive theoretical justifications or depend on the guidance of expert actions. In this work, we propose a sample-efficient LfO approach which enables off-policy optimization in a principled manner. To further accelerate the learning procedure, we regulate the policy update with an inverse action model, which assists distribution matching from the perspective of mode-covering. Extensive empirical results on challenging locomotion tasks indicate that our approach is comparable with state-of-the-art in terms of both sample-efficiency and asymptotic performance.

Popular Maximum Entropy Inverse Reinforcement Learning approaches require the computation of expected state visitation frequencies for the optimal policy under an estimate of the reward function. This usually requires intermediate value estimation in the inner loop of the algorithm, slowing down convergence considerably. In this work, we introduce a novel class of algorithms that only needs to solve the MDP underlying the demonstrated behavior once to recover the expert policy. This is possible through a formulation that exploits a probabilistic behavior assumption for the demonstrations within the structure of Q-learning. We propose Inverse Action-value Iteration which is able to fully recover an underlying reward of an external agent in closed-form analytically. We further provide an accompanying class of sampling-based variants which do not depend on a model of the environment. We show how to extend this class of algorithms to continuous state-spaces via function approximation and how to estimate a corresponding action-value function, leading to a policy as close as possible to the policy of the external agent, while optionally satisfying a list of predefined hard constraints. We evaluate the resulting algorithms called Inverse Action-value Iteration, Inverse Q-learning and Deep Inverse Q-learning on the Objectworld benchmark, showing a speedup of …

Value estimation is a critical component of the reinforcement learning (RL) paradigm. The question of how to effectively learn value predictors from data is one of the major problems studied by the RL community, and different approaches exploit structure in the problem domain in different ways. Model learning can make use of the rich transition structure present in sequences of observations, but this approach is usually not sensitive to the reward function. In contrast, model-free methods directly leverage the quantity of interest from the future, but receive a potentially weak scalar signal (an estimate of the return). We develop an approach for representation learning in RL that sits in between these two extremes: we propose to learn what to model in a way that can directly help value prediction. To this end, we determine which features of the future trajectory provide useful information to predict the associated return. This provides tractable prediction targets that are directly relevant for a task, and can thus accelerate learning the value function. The idea can be understood as reasoning, in hindsight, about which aspects of the future observations could help past value prediction. We show how this can help dramatically even in simple policy …

Exploration is a key problem in reinforcement learning, since agents can only learn from data they acquire in the environment. With that in mind, maintaining a population of agents is an attractive method, as it allows data be collected with a diverse set of behaviors. This behavioral diversity is often boosted via multi-objective loss functions. However, those approaches typically leverage mean field updates based on pairwise distances, which makes them susceptible to cycling behaviors and increased redundancy. In addition, explicitly boosting diversity often has a detrimental impact on optimizing already fruitful behaviors for rewards. As such, the reward-diversity trade off typically relies on heuristics. Finally, such methods require behavioral representations, often handcrafted and domain specific. In this paper, we introduce an approach to optimize all members of a population simultaneously. Rather than using pairwise distance, we measure the volume of the entire population in a behavioral manifold, defined by task-agnostic behavioral embeddings. In addition, our algorithm Diversity via Determinants (DvD), adapts the degree of diversity during training using online learning techniques. We introduce both evolutionary and gradient-based instantiations of DvD and show they effectively improve exploration without reducing performance when better exploration is not required.

Adversarial Imitation Learning alternates between learning a discriminator -- which tells apart expert's demonstrations from generated ones -- and a generator's policy to produce trajectories that can fool this discriminator. This alternated optimization is known to be delicate in practice since it compounds unstable adversarial training with brittle and sample-inefficient reinforcement learning. We propose to remove the burden of the policy optimization steps by leveraging a novel discriminator formulation. Specifically, our discriminator is explicitly conditioned on two policies: the one from the previous generator's iteration and a learnable policy. When optimized, this discriminator directly learns the optimal generator's policy. Consequently, our discriminator's update solves the generator's optimization problem for free: learning a policy that imitates the expert does not require an additional optimization loop. This formulation effectively cuts by half the implementation and computational burden of Adversarial Imitation Learning algorithms by removing the Reinforcement Learning phase altogether. We show on a variety of tasks that our simpler approach is competitive to prevalent Imitation Learning methods.

Potential-based reward shaping provides an approach for designing good reward functions, with the purpose of speeding up learning. However, automatically finding potential functions for complex environments is a difficult problem (in fact, of the same difficulty as learning a value function from scratch). We propose a new framework for learning potential functions by leveraging ideas from graph representation learning. Our approach relies on Graph Convolutional Networks which we use as a key ingredient in combination with the probabilistic inference view of reinforcement learning. More precisely, we leverage Graph Convolutional Networks to perform message passing from rewarding states. The propagated messages can then be used as potential functions for reward shaping to accelerate learning. We verify empirically that our approach can achieve considerable improvements in both small and high-dimensional control problems.

Learning with sparse rewards remains a significant challenge in reinforcement learning (RL), especially when the aim is to train a policy capable of achieving multiple different goals. To date, the most successful approaches for dealing with multi-goal, sparse reward environments have been model-free RL algorithms. In this work we propose PlanGAN, a model-based algorithm specifically designed for solving multi-goal tasks in environments with sparse rewards. Our method builds on the fact that any trajectory of experience collected by an agent contains useful information about how to achieve the goals observed during that trajectory. We use this to train an ensemble of conditional generative models (GANs) to generate plausible trajectories that lead the agent from its current state towards a specified goal. We then combine these imagined trajectories into a novel planning algorithm in order to achieve the desired goal as efficiently as possible. The performance of PlanGAN has been tested on a number of robotic navigation/manipulation tasks in comparison with a range of model-free reinforcement learning baselines, including Hindsight Experience Replay. Our studies indicate that PlanGAN can achieve comparable performance whilst being around 4-8 times more sample efficient.

Curriculum reinforcement learning (CRL) improves the learning speed and stability of an agent by exposing it to a tailored series of tasks throughout learning. Despite empirical successes, an open question in CRL is how to automatically generate a curriculum for a given reinforcement learning (RL) agent, avoiding manual design. In this paper, we propose an answer by interpreting the curriculum generation as an inference problem, where distributions over tasks are progressively learned to approach the target task. This approach leads to an automatic curriculum generation, whose pace is controlled by the agent, with solid theoretical motivation and easily integrated with deep RL algorithms. In the conducted experiments, the curricula generated with the proposed algorithm significantly improve learning performance across several environments and deep RL algorithms, matching or outperforming state-of-the-art existing CRL algorithms.

Reinforcement learning with sparse rewards is challenging because an agent can rarely obtain non-zero rewards and hence, gradient-based optimization of parameterized policies can be incremental and slow. Recent work demonstrated that using a memory buffer of previous successful trajectories can result in more effective policies. However, existing methods may overly exploit past successful experiences, which can encourage the agent to adopt sub-optimal and myopic behaviors. In this work, instead of focusing on good experiences with limited diversity, we propose to learn a trajectory-conditioned policy to follow and expand diverse past trajectories from a memory buffer. Our method allows the agent to reach diverse regions in the state space and improve upon the past trajectories to reach new states. We empirically show that our approach significantly outperforms count-based exploration methods (parametric approach) and self-imitation learning (parametric approach with non-parametric memory) on various complex tasks with local optima. In particular, without using expert demonstrations or resetting to arbitrary states, we achieve the state-of-the-art scores under five billion number of frames, on challenging Atari games such as Montezuma’s Revenge and Pitfall.

Reward shaping is an effective technique for incorporating domain knowledge into reinforcement learning (RL). Existing approaches such as potential-based reward shaping normally make full use of a given shaping reward function. However, since the transformation of human knowledge into numeric reward values is often imperfect due to reasons such as human cognitive bias, completely utilizing the shaping reward function may fail to improve the performance of RL algorithms. In this paper, we consider the problem of adaptively utilizing a given shaping reward function. We formulate the utilization of shaping rewards as a bi-level optimization problem, where the lower level is to optimize policy using the shaping rewards and the upper level is to optimize a parameterized shaping weight function for true reward maximization. We formally derive the gradient of the expected true reward with respect to the shaping weight function parameters and accordingly propose three learning algorithms based on different assumptions. Experiments in sparse-reward cartpole and MuJoCo environments show that our algorithms can fully exploit beneficial shaping rewards, and meanwhile ignore unbeneficial shaping rewards or even transform them into beneficial ones.

Imitation learning (IL) aims to learn a policy from expert demonstrations that minimizes the discrepancy between the learner and expert behaviors. Various imitation learning algorithms have been proposed with different pre-determined divergences to quantify the discrepancy. This naturally gives rise to the following question: Given a set of expert demonstrations, which divergence can recover the expert policy more accurately with higher data efficiency? In this work, we propose f-GAIL – a new generative adversarial imitation learning model – that automatically learns a discrepancy measure from the f-divergence family as well as a policy capable of producing expert-like behaviors. Compared with IL baselines with various predefined divergence measures, f-GAIL learns better policies with higher data efficiency in six physics-based control tasks.

Consider learning a policy purely on the basis of demonstrated behavior---that is, with no access to reinforcement signals, no knowledge of transition dynamics, and no further interaction with the environment. This strictly batch imitation learning problem arises wherever live experimentation is costly, such as in healthcare. One solution is simply to retrofit existing algorithms for apprenticeship learning to work in the offline setting. But such an approach leans heavily on off-policy evaluation or offline model estimation, and can be indirect and inefficient. We argue that a good solution should be able to explicitly parameterize a policy (i.e. respecting action conditionals), implicitly learn from rollout dynamics (i.e. leveraging state marginals), and---crucially---operate in an entirely offline fashion. To address this challenge, we propose a novel technique by energy-based distribution matching (EDM): By identifying parameterizations of the (discriminative) model of a policy with the (generative) energy function for state distributions, EDM yields a simple but effective solution that equivalently minimizes a divergence between the occupancy measure for the demonstrator and a model thereof for the imitator. Through experiments with application to control and healthcare settings, we illustrate consistent performance gains over existing algorithms for strictly batch imitation learning.

Deep reinforcement learning includes a broad family of algorithms that parameterise an internal representation, such as a value function or policy, by a deep neural network. Each algorithm optimises its parameters with respect to an objective, such as Q-learning or policy gradient, that defines its semantics. In this work, we propose an algorithm based on meta-gradient descent that discovers its own objective, flexibly parameterised by a deep neural network, solely from interactive experience with its environment. Over time, this allows the agent to learn how to learn increasingly effectively. Furthermore, because the objective is discovered online, it can adapt to changes over time. We demonstrate that the algorithm discovers how to address several important issues in RL, such as bootstrapping, non-stationarity, and off-policy learning. On the Atari Learning Environment, the meta-gradient algorithm adapts over time to learn with greater efficiency, eventually outperforming the median score of a strong actor-critic baseline.

Inverse Reinforcement Learning addresses the problem of inferring an expert's reward function from demonstrations. However, in many applications, we not only have access to the expert's near-optimal behaviour, but we also observe part of her learning process. In this paper, we propose a new algorithm for this setting, in which the goal is to recover the reward function being optimized by an agent, given a sequence of policies produced during learning. Our approach is based on the assumption that the observed agent is updating her policy parameters along the gradient direction. Then we extend our method to deal with the more realistic scenario where we only have access to a dataset of learning trajectories. For both settings, we provide theoretical insights into our algorithms' performance. Finally, we evaluate the approach in a simulated GridWorld environment and on the MuJoCo environments, comparing it with the state-of-the-art baseline.

In online convex optimization (OCO), Lipschitz continuity of the functions is commonly assumed in order to obtain sublinear regret. Moreover, many algorithms have only logarithmic regret when these functions are also strongly convex. Recently, researchers from convex optimization proposed the notions of relative Lipschitz continuity'' andrelative strong convexity''. Both of the notions are generalizations of their classical counterparts. It has been shown that subgradient methods in the relative setting have performance analogous to their performance in the classical setting.

In this work, we consider OCO for relative Lipschitz and relative strongly convex functions. We extend the known regret bounds for classical OCO algorithms to the relative setting. Specifically, we show regret bounds for the follow the regularized leader algorithms and a variant of online mirror descent. Due to the generality of these methods, these results yield regret bounds for a wide variety of OCO algorithms. Furthermore, we further extend the results to algorithms with extra regularization such as regularized dual averaging.

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. [2016] provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses.

Our work is the first to address uniform stability of SGD on nonsmooth convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bound allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess …

Conditional stochastic optimization covers a variety of applications ranging from invariant learning and causal inference to meta-learning. However, constructing unbiased gradient estimators for such problems is challenging due to the composition structure. As an alternative, we propose a biased stochastic gradient descent (BSGD) algorithm and study the bias-variance tradeoff under different structural assumptions. We establish the sample complexities of BSGD for strongly convex, convex, and weakly convex objectives under smooth and non-smooth conditions. Our lower bound analysis shows that the sample complexities of BSGD cannot be improved for general convex objectives and nonconvex objectives except for smooth nonconvex objectives with Lipschitz continuous gradient estimator. For this special setting, we propose an accelerated algorithm called biased SpiderBoost (BSpiderBoost) that matches the lower bound complexity. We further conduct numerical experiments on invariant logistic regression and model-agnostic meta-learning to illustrate the performance of BSGD and BSpiderBoost.

Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020). As a special case of this framework, we propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast with a rate which not only is dimension-free, but also has no dependence on the target distribution. We give an application of this result to the problem of sampling from the uniform distribution on a convex body using a strategy inspired by interior-point methods. Our general approach follows the recent trend of linking sampling and optimization and highlights the role of the chi-squared divergence. In particular, it yields new results on the convergence of the vanilla Langevin diffusion in Wasserstein distance.

We address the problem of zero-order optimization of a strongly convex function. The goal is to find the minimizer of the function by a sequential exploration of its function values, under measurement noise. We study the impact of higher order smoothness properties of the function on the optimization error and on the online regret. To solve this problem we consider a randomized approximation of the projected gradient descent algorithm. The gradient is estimated by a randomized procedure involving two function evaluations and a smoothing kernel. We derive upper bounds for this algorithm both in the constrained and unconstrained settings and prove minimax lower bounds for any sequential search method. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters. Based on this algorithm, we also propose an estimator of the minimum value of the function achieving almost sharp oracle behavior. We compare our results with the state-of-the-art, highlighting a number of key improvements.

Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk with the squared loss for linear models and kernel methods. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent methods. Our observation immediately yields excess risk guarantees for the path traced by the iterates of mirror descent in terms of offset complexities of certain function classes depending only on the choice of the mirror map, initialization point, step-size, and the number of iterations. We apply our theory to recover, in a rather clean and elegant manner via rather short proofs, some of the recent results in the implicit regularization literature, while also showing how to improve upon them in some settings.

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between \emph{one-dimensional random projections} of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.

The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios.

We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of Stochastic Convex Optimization. As a first step, we provide a simple construction that rules out the existence of a \emph{distribution-independent} implicit regularizer that governs the generalization ability of SGD. We then demonstrate a learning problem that rules out a very general class of \emph{distribution-dependent} implicit regularizers from explaining generalization, which includes strongly convex regularizers as well as non-degenerate norm-based regularizations. Certain aspects of our constructions point out to significant difficulties in providing a comprehensive explanation of an algorithm's generalization performance by solely arguing about its implicit regularization properties.

We study problem-dependent rates, i.e., generalization errors that scale tightly with the variance or the effective loss at the "best hypothesis." Existing uniform convergence and localization frameworks, the most widely used tools to study this problem, often fail to simultaneously provide parameter localization and optimal dependence on the sample size. As a result, existing problem-dependent rates are often rather weak when the hypothesis class is "rich" and the worst-case bound of the loss is large. In this paper we propose a new framework based on a "uniform localized convergence" principle. We provide the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general "rich" classes; we also establish improved loss-dependent rate for standard empirical risk minimization.

We consider an underdetermined noisy linear regression model where the minimum-norm interpolating predictor is known to be consistent, and ask: can uniform convergence in a norm ball, or at least (following Nagarajan and Kolter) the subset of a norm ball that the algorithm selects on a typical input set, explain this success? We show that uniformly bounding the difference between empirical and population errors cannot show any learning in the norm ball, and cannot show consistency for any set, even one depending on the exact algorithm and distribution. But we argue we can explain the consistency of the minimal-norm interpolator with a slightly weaker, yet standard, notion: uniform convergence of zero-error predictors in a norm ball. We use this to bound the generalization error of low- (but not minimal-)norm interpolating predictors.

Exponential bounds on the estimation error are given for the plug-in estimator of weighted areas under the ROC curve. The bounds hold for single score functions and uniformly over classes of functions, whose complexity can be controlled by Gaussian or Rademacher averages. The results justify learning algorithms which select score functions to maximize the empirical partial area under the curve (pAUC). They also illustrate the use of some recent advances in the theory of nonlinear empirical processes.

Neural Ordinary Differential Equations (NODEs) are a new class of models that transform data continuously through infinite-depth architectures. The continuous nature of NODEs has made them particularly suitable for learning the dynamics of complex physical systems. While previous work has mostly been focused on first order ODEs, the dynamics of many systems, especially in classical physics, are governed by second order laws. In this work, we consider Second Order Neural ODEs (SONODEs). We show how the adjoint sensitivity method can be extended to SONODEs and prove that the optimisation of a first order coupled ODE is equivalent and computationally more efficient. Furthermore, we extend the theoretical understanding of the broader class of Augmented NODEs (ANODEs) by showing they can also learn higher order dynamics with a minimal number of augmented dimensions, but at the cost of interpretability. This indicates that the advantages of ANODEs go beyond the extra space offered by the augmented dimensions, as originally thought. Finally, we compare SONODEs and ANODEs on synthetic and real dynamical systems and demonstrate that the inductive biases of the former generally result in faster training and better performance.

Learning object-centric representations of complex scenes is a promising step towards enabling efficient abstract reasoning from low-level perceptual features. Yet, most deep learning approaches learn distributed representations that do not capture the compositional properties of natural scenes. In this paper, we present the Slot Attention module, an architectural component that interfaces with perceptual representations such as the output of a convolutional neural network and produces a set of task-dependent abstract representations which we call slots. These slots are exchangeable and can bind to any object in the input by specializing through a competitive procedure over multiple rounds of attention. We empirically demonstrate that Slot Attention can extract object-centric representations that enable generalization to unseen compositions when trained on unsupervised object discovery and supervised property prediction tasks.

Graph neural networks (GNNs) have emerged as a powerful tool for learning software engineering tasks including code completion, bug finding, and program repair. They benefit from leveraging program structure like control flow graphs, but they are not well-suited to tasks like program execution that require far more sequential reasoning steps than number of GNN propagation steps. Recurrent neural networks (RNNs), on the other hand, are well-suited to long sequential chains of reasoning, but they do not naturally incorporate program structure and generally perform worse on the above tasks. Our aim is to achieve the best of both worlds, and we do so by introducing a novel GNN architecture, the Instruction Pointer Attention Graph Neural Networks (IPA-GNN), which achieves improved systematic generalization on the task of learning to execute programs using control flow graphs. The model arises by considering RNNs operating on program traces with branch decisions as latent variables. The IPA-GNN can be seen either as a continuous relaxation of the RNN model or as a GNN variant more tailored to execution. To test the models, we propose evaluating systematic generalization on learning to execute using control flow graphs, which tests sequential reasoning and use of program structure. More practically, …

Probabilistic forecasting consists in predicting a distribution of possible future outcomes. In this paper, we address this problem for non-stationary time series, which is very challenging yet crucially important. We introduce the STRIPE model for representing structured diversity based on shape and time features, ensuring both probable predictions while being sharp and accurate. STRIPE is agnostic to the forecasting model, and we equip it with a diversification mechanism relying on determinantal point processes (DPP). We introduce two DPP kernels for modelling diverse trajectories in terms of shape and time, which are both differentiable and proved to be positive semi-definite. To have an explicit control on the diversity structure, we also design an iterative sampling mechanism to disentangle shape and time representations in the latent space. Experiments carried out on synthetic datasets show that STRIPE significantly outperforms baseline methods for representing diversity, while maintaining accuracy of the forecasting model. We also highlight the relevance of the iterative sampling scheme and the importance to use different criteria for measuring quality and diversity. Finally, experiments on real datasets illustrate that STRIPE is able to outperform state-of-the-art probabilistic forecasting approaches in the best sample prediction.

A residual network may be regarded as a discretization of an ordinary differential equation (ODE) which, in the limit of time discretization, defines a continuous-depth network. Although important steps have been taken to realize the advantages of such continuous formulations, most current techniques assume identical layers. Indeed, existing works throw into relief the myriad difficulties of learning an infinite-dimensional parameter in a continuous-depth neural network. To this end, we introduce a shooting formulation which shifts the perspective from parameterizing a network layer-by-layer to parameterizing over optimal networks described only by a set of initial conditions. For scalability, we propose a novel particle-ensemble parameterization which fully specifies the optimal weight trajectory of the continuous-depth neural network. Our experiments show that our particle-ensemble shooting formulation can achieve competitive performance. Finally, though the current work is inspired by continuous-depth neural networks, the particle-ensemble shooting formulation also applies to discrete-time networks and may lead to a new fertile area of research in deep learning parameterization.

A finite-state machine (FSM) is a computation model to process binary strings in sequential circuits. Hence, a single-input linear FSM is conventionally used to implement complex single-input functions , such as tanh and exponentiation functions, in stochastic computing (SC) domain where continuous values are represented by sequences of random bits. In this paper, we introduce a method that can train a multi-layer FSM-based network where FSMs are connected to every FSM in the previous and the next layer. We show that the proposed FSM-based network can synthesize multi-input complex functions such as 2D Gabor filters and can perform non-sequential tasks such as image classifications on stochastic streams with no multiplication since FSMs are implemented by look-up tables only. Inspired by the capability of FSMs in processing binary streams, we then propose an FSM-based model that can process time series data when performing temporal tasks such as character-level language modeling. Unlike long short-term memories (LSTMs) that unroll the network for each input time step and perform back-propagation on the unrolled network, our FSM-based model requires to backpropagate gradients only for the current input time step while it is still capable of learning long-term dependencies. Therefore, our FSM-based model can learn extremely …

We consider the task of solving generic inverse problems, where one wishes to determine the hidden parameters of a natural system that will give rise to a particular set of measurements. Recently many new approaches based upon deep learning have arisen, generating promising results. We conceptualize these models as different schemes for efficiently, but randomly, exploring the space of possible inverse solutions. As a result, the accuracy of each approach should be evaluated as a function of time rather than a single estimated solution, as is often done now. Using this metric, we compare several state-of-the-art inverse modeling approaches on four benchmark tasks: two existing tasks, a new 2-dimensional sinusoid task, and a challenging modern task of meta-material design. Finally, inspired by our conception of the inverse problem, we explore a simple solution that uses a deep neural network as a surrogate (i.e., approximation) for the forward model, and then uses backpropagation with respect to the model input to search for good inverse solutions. Variations of this approach - which we term the neural adjoint (NA) - have been explored recently on specific problems, and here we evaluate it comprehensively on our benchmark. We find that the addition of a …

Many real-world systems, such as moving planets, can be considered as multi-agent dynamic systems, where objects interact with each other and co-evolve along with the time. Such dynamics is usually difficult to capture, and understanding and predicting the dynamics based on observed trajectories of objects become a critical research problem in many domains. Most existing algorithms, however, assume the observations are regularly sampled and all the objects can be fully observed at each sampling time, which is impractical for many applications. In this paper, we pro-pose to learn system dynamics from irregularly-sampled and partial observations with underlying graph structure for the first time. To tackle the above challenge, we present LG-ODE, a latent ordinary differential equation generative model for modeling multi-agent dynamic system with known graph structure. It can simultaneously learn the embedding of high dimensional trajectories and infer continuous latent system dynamics. Our model employs a novel encoder parameterized by a graph neural network that can infer initial states in an unsupervised way from irregularly-sampled partial observations of structural objects and utilizes neuralODE to infer arbitrarily complex continuous-time latent dynamics. Experiments on motion capture, spring system, and charged particle datasets demonstrate the effectiveness of our approach.

Transformers have been proven a successful model for a variety of tasks in sequence modeling. However, computing the attention matrix, which is their key component, has quadratic complexity with respect to the sequence length, thus making them prohibitively expensive for large sequences. To address this, we propose clustered attention, which instead of computing the attention for every query, groups queries into clusters and computes attention just for the centroids. To further improve this approximation, we use the computed clusters to identify the keys with the highest attention per query and compute the exact key/query dot products. This results in a model with linear complexity with respect to the sequence length for a fixed number of clusters. We evaluate our approach on two automatic speech recognition datasets and show that our model consistently outperforms vanilla transformers for a given computational budget. Finally, we demonstrate that our model can approximate arbitrarily complex attention distributions with a minimal number of clusters by approximating a pretrained BERT model on GLUE and SQuAD benchmarks with only 25 clusters and no loss in performance.

Exponential families are widely used in machine learning; they include many distributions in continuous and discrete domains (e.g., Gaussian, Dirichlet, Poisson, and categorical distributions via the softmax transformation). Distributions in each of these families have fixed support. In contrast, for finite domains, there has been recent work on sparse alternatives to softmax (e.g., sparsemax and alpha-entmax), which have varying support, being able to assign zero probability to irrelevant categories. These discrete sparse mappings have been used for improving interpretability of neural attention mechanisms. This paper expands that work in two directions: first, we extend alpha-entmax to continuous domains, revealing a link with Tsallis statistics and deformed exponential families. Second, we introduce continuous-domain attention mechanisms, deriving efficient gradient backpropagation algorithms for alpha in {1,2}. Experiments on attention-based text classification, machine translation, and visual question answering illustrate the use of continuous attention in 1D and 2D, showing that it allows attending to time intervals and compact regions.

Normalizing flows transform a simple base distribution into a complex target distribution and have proved to be powerful models for data generation and density estimation. In this work, we propose a novel type of normalizing flow driven by a differential deformation of the continuous-time Wiener process. As a result, we obtain a rich time series model whose observable process inherits many of the appealing properties of its base process, such as efficient computation of likelihoods and marginals. Furthermore, our continuous treatment provides a natural framework for irregular time series with an independent arrival process, including straightforward interpolation. We illustrate the desirable properties of the proposed model on popular stochastic processes and demonstrate its superior flexibility to variational RNN and latent ODE baselines in a series of experiments on synthetic and real-world data.

We present high quality image synthesis results using diffusion probabilistic models, a class of latent variable models inspired by considerations from nonequilibrium thermodynamics. Our best results are obtained by training on a weighted variational bound designed according to a novel connection between diffusion probabilistic models and denoising score matching with Langevin dynamics, and our models naturally admit a progressive lossy decompression scheme that can be interpreted as a generalization of autoregressive decoding. On the unconditional CIFAR10 dataset, we obtain an Inception score of 9.46 and a state-of-the-art FID score of 3.17. On 256x256 LSUN, we obtain sample quality similar to ProgressiveGAN.

Attention and self-attention mechanisms, are now central to state-of-the-art deep learning on sequential tasks. However, most recent progress hinges on heuristic approaches with limited understanding of attention's role in model optimization and computation, and rely on considerable memory and computational resources that scale poorly. In this work, we present a formal analysis of how self-attention affects gradient propagation in recurrent networks, and prove that it mitigates the problem of vanishing gradients when trying to capture long-term dependencies by establishing concrete bounds for gradient norms. Building on these results, we propose a relevancy screening mechanism, inspired by the cognitive process of memory consolidation, that allows for a scalable use of sparse self-attention with recurrence. While providing guarantees to avoid vanishing gradients, we use simple numerical experiments to demonstrate the tradeoffs in performance and computational resources by efficiently balancing attention and recurrence. Based on our results, we propose a concrete direction of research to improve scalability of attentive networks.

Neural ordinary differential equations are an attractive option for modelling temporal dynamics. However, a fundamental issue is that the solution to an ordinary differential equation is determined by its initial condition, and there is no mechanism for adjusting the trajectory based on subsequent observations. Here, we demonstrate how this may be resolved through the well-understood mathematics of \emph{controlled differential equations}. The resulting \emph{neural controlled differential equation} model is directly applicable to the general setting of partially-observed irregularly-sampled multivariate time series, and (unlike previous work on this problem) it may utilise memory-efficient adjoint-based backpropagation even across observations. We demonstrate that our model achieves state-of-the-art performance against similar (ODE or RNN based) models in empirical studies on a range of datasets. Finally we provide theoretical results demonstrating universal approximation, and that our model subsumes alternative ODE models.

Combinatorial optimization (CO) problems are notoriously challenging for neural networks, especially in the absence of labeled instances. This work proposes an unsupervised learning framework for CO problems on graphs that can provide integral solutions of certified quality. Inspired by Erdos' probabilistic method, we use a neural network to parametrize a probability distribution over sets. Crucially, we show that when the network is optimized w.r.t. a suitably chosen loss, the learned distribution contains, with controlled probability, a low-cost integral solution that obeys the constraints of the combinatorial problem. The probabilistic proof of existence is then derandomized to decode the desired solutions. We demonstrate the efficacy of this approach to obtain valid
solutions to the maximum clique problem and to perform local graph clustering. Our method achieves competitive results on both real datasets and synthetic hard instances.

Existing Graph Neural Network (GNN) methods that learn inductive unsupervised graph representations focus on learning node and edge representations by predicting observed edges in the graph. Although such approaches have shown advances in downstream node classification tasks, they are ineffective in jointly representing larger k-node sets, k{>}2. We propose MHM-GNN, an inductive unsupervised graph representation approach that combines joint k-node representations with energy-based models (hypergraph Markov networks) and GNNs. To address the intractability of the loss that arises from this combination, we endow our optimization with a loss upper bound using a finite-sample unbiased Markov Chain Monte Carlo estimator. Our experiments show that the unsupervised joint k-node representations of MHM-GNN produce better unsupervised representations than existing approaches from the literature.

Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the actual task the GNN is solving. In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs. Here we introduce Pointer Graph Networks (PGNs) which augment sets or graphs with additional inferred edges for improved model generalisation ability. PGNs allow each node to dynamically point to another node, followed by message passing over these pointers. The sparsity of this adaptable graph structure makes learning tractable while still being sufficiently expressive to simulate complex algorithms. Critically, the pointing mechanism is directly supervised to model long-term sequences of operations on classical data structures, incorporating useful structural inductive biases from theoretical computer science. Qualitatively, we demonstrate that PGNs can learn parallelisable variants of pointer-based data structures, namely disjoint set unions and link/cut trees. PGNs generalise out-of-distribution to 5x larger test inputs on dynamic graph connectivity tasks, outperforming unrestricted GNNs and Deep Sets.

The ability to detect and count certain substructures in graphs is important for solving many tasks on graph-structured data, especially in the contexts of computational chemistry and biology as well as social network analysis. Inspired by this, we propose to study the expressive power of graph neural networks (GNNs) via their ability to count attributed graph substructures, extending recent works that examine their power in graph isomorphism testing and function approximation. We distinguish between two types of substructure counting: induced-subgraph-count and subgraph-count, and establish both positive and negative answers for popular GNN architectures. Specifically, we prove that Message Passing Neural Networks (MPNNs), 2-Weisfeiler-Lehman (2-WL) and 2-Invariant Graph Networks (2-IGNs) cannot perform induced-subgraph-count of substructures consisting of 3 or more nodes, while they can perform subgraph-count of star-shaped substructures. As an intermediary step, we prove that 2-WL and 2-IGNs are equivalent in distinguishing non-isomorphic graphs, partly answering an open problem raised in Maron et al. (2019). We also prove positive results for k-WL and k-IGNs as well as negative results for k-WL with a finite number of iterations. We then conduct experiments that support the theoretical results for MPNNs and 2-IGNs. Moreover, motivated by substructure counting and inspired by Murphy et …

Learning representations of sets of nodes in a graph is crucial for applications ranging from node-role discovery to link prediction and molecule classification. Graph Neural Networks (GNNs) have achieved great success in graph representation learning. However, expressive power of GNNs is limited by the 1-Weisfeiler-Lehman (WL) test and thus GNNs generate identical representations for graph substructures that may in fact be very different. More powerful GNNs, proposed recently by mimicking higher-order-WL tests, only focus on representing entire graphs and they are computationally inefficient as they cannot utilize sparsity of the underlying graph. Here we propose and mathematically analyze a general class of structure-related features, termed Distance Encoding (DE). DE assists GNNs in representing any set of nodes, while providing strictly more expressive power than the 1-WL test. DE captures the distance between the node set whose representation is to be learned and each node in the graph. To capture the distance DE can apply various graph-distance measures such as shortest path distance or generalized PageRank scores. We propose two ways for GNNs to use DEs (1) as extra node features, and (2) as controllers of message aggregation in GNNs. Both approaches can utilize the sparse structure of the underlying graph, …

We study the problem of semi-supervised learning on graphs, for which graph neural networks (GNNs) have been extensively explored. However, most existing GNNs inherently suffer from the limitations of over-smoothing, non-robustness, and weak-generalization when labeled nodes are scarce. In this paper, we propose a simple yet effective framework—GRAPH RANDOM NEURAL NETWORKS (GRAND)—to address these issues. In GRAND, we first design a random propagation strategy to perform graph data augmentation. Then we leverage consistency regularization to optimize the prediction consistency of unlabeled nodes across different data augmentations. Extensive experiments on graph benchmark datasets suggest that GRAND significantly outperforms state-of- the-art GNN baselines on semi-supervised node classification. Finally, we show that GRAND mitigates the issues of over-smoothing and non-robustness, exhibiting better generalization behavior than existing GNNs. The source code of GRAND is publicly available at https://github.com/Grand20/grand.

Generalizable, transferrable, and robust representation learning on graph-structured data remains a challenge for current graph neural networks (GNNs). Unlike what has been developed for convolutional neural networks (CNNs) for image data, self-supervised learning and pre-training are less explored for GNNs. In this paper, we propose a graph contrastive learning (GraphCL) framework for learning unsupervised representations of graph data. We first design four types of graph augmentations to incorporate various priors. We then systematically study the impact of various combinations of graph augmentations on multiple datasets, in four different settings: semi-supervised, unsupervised, and transfer learning as well as adversarial attacks. The results show that, even without tuning augmentation extents nor using sophisticated GNN architectures, our GraphCL framework can produce graph representations of similar or better generalizability, transferrability, and robustness compared to state-of-the-art methods. We also investigate the impact of parameterized graph augmentation extents and patterns, and observe further performance gains in preliminary experiments. Our codes are available at https://github.com/Shen-Lab/GraphCL.

Graph convolutional networks (GCNs) have shown promising results in processing graph data by extracting structure-aware features. This gave rise to extensive work in geometric deep learning, focusing on designing network architectures that ensure neuron activations conform to regularity patterns within the input graph. However, in most cases the graph structure is only accounted for by considering the similarity of activations between adjacent nodes, which limits the capabilities of such methods to discriminate between nodes in a graph. Here, we propose to augment conventional GCNs with geometric scattering transforms and residual convolutions. The former enables band-pass filtering of graph signals, thus alleviating the so-called oversmoothing often encountered in GCNs, while the latter is introduced to clear the resulting features of high-frequency noise. We establish the advantages of the presented Scattering GCN with both theoretical results establishing the complementary benefits of scattering and GCN features, as well as experimental results showing the benefits of our method compared to leading graph neural networks for semi-supervised node classification, including the recently proposed GAT network that typically alleviates oversmoothing using graph attention mechanisms.

Graph Neural Networks (GNNs) have achieved remarkable performance in the task of the semi-supervised node classification. However, most existing models learn a deterministic classification function, which lack sufficient flexibility to explore better choices in the presence of kinds of imperfect observed data such as the scarce labeled nodes and noisy graph structure. To improve the rigidness and inflexibility of deterministic classification functions, this paper proposes a novel framework named Graph Stochastic Neural Networks (GSNN), which aims to model the uncertainty of the classification function by simultaneously learning a family of functions, i.e., a stochastic function. Specifically, we introduce a learnable graph neural network coupled with a high-dimensional latent variable to model the distribution of the classification function, and further adopt the amortised variational inference to approximate the intractable joint posterior for missing labels and the latent variable. By maximizing the lower-bound of the likelihood for observed node labels, the instantiated models can be trained in an end-to-end manner effectively. Extensive experiments on three real-world datasets show that GSNN achieves substantial performance gain in different scenarios compared with stat-of-the-art baselines.

Graph Convolutional Networks (GCNs) have shown to be effective in handling unordered data like point clouds and meshes. In this work we propose novel approaches for graph convolution, pooling and unpooling, inspired from finite differences and algebraic multigrid frameworks. We form a parameterized convolution kernel based on discretized differential operators, leveraging the graph mass, gradient and Laplacian. This way, the parameterization does not depend on the graph structure, only on the meaning of the network convolutions as differential operators. To allow hierarchical representations of the input, we propose pooling and unpooling operations that are based on algebraic multigrid methods, which are mainly used to solve partial differential equations on unstructured grids. To motivate and explain our method, we compare it to standard convolutional neural networks, and show their similarities and relations in the case of a regular grid. Our proposed method is demonstrated in various experiments like classification and part-segmentation, achieving on par or better than state of the art results. We also analyze the computational cost of our method compared to other GCNs.

While the self-attention mechanism has been widely used in a wide variety of tasks, it has the unfortunate property of a quadratic cost with respect to the input length, which makes it difficult to deal with long inputs. In this paper, we present a method for accelerating and structuring self-attentions: Sparse Adaptive Connection (SAC). In SAC, we regard the input sequence as a graph and attention operations are performed between linked nodes. In contrast with previous self-attention models with pre-defined structures (edges), the model learns to construct attention edges to improve task-specific performances. In this way, the model is able to select the most salient nodes and reduce the quadratic complexity regardless of the sequence length. Based on SAC, we show that previous variants of self-attention models are its special cases. Through extensive experiments on neural machine translation, language modeling, graph representation learning and image classification, we demonstrate SAC is competitive with state-of-the-art models while significantly reducing memory cost.

Recent research on graph embedding has achieved success in various applications. Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space. We argue an important but neglected problem about this proximity-preserving strategy: Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models. To address the problem, we propose curvature regularization, to enforce flatness for embedding manifolds, thereby preventing the distortion. We present a novel angle-based sectional curvature, termed ABS curvature, and accordingly three kinds of curvature regularization to induce flat embedding manifolds during graph embedding. We integrate curvature regularization into five popular proximity-preserving embedding methods, and empirical results in two applications show significant improvements on a wide range of open graph datasets.

There has been an increased interest in discovering heuristics for combinatorial problems on graphs through machine learning. While existing techniques have primarily focused on obtaining high-quality solutions, scalability to billion-sized graphs has not been adequately addressed. In addition, the impact of a budget-constraint, which is necessary for many practical scenarios, remains to be studied. In this paper, we propose a framework called GCOMB to bridge these gaps. GCOMB trains a Graph Convolutional Network (GCN) using a novel probabilistic greedy mechanism to predict the quality of a node. To further facilitate the combinatorial nature of the problem, GCOMB utilizes a Q-learning framework, which is made efficient through importance sampling. We perform extensive experiments on real graphs to benchmark the efficiency and efficacy of GCOMB. Our results establish that GCOMB is 100 times faster and marginally better in quality than state-of-the-art algorithms for learning combinatorial algorithms. Additionally, a case-study on the practical combinatorial problem of Influence Maximization (IM) shows GCOMB is 150 times faster than the specialized IM algorithm IMM with similar quality.

Deep learning methods for graphs achieve remarkable performance on many node-level and graph-level prediction tasks. However, despite the proliferation of the methods and their success, prevailing Graph Neural Networks (GNNs) neglect subgraphs, rendering subgraph prediction tasks challenging to tackle in many impactful applications. Further, subgraph prediction tasks present several unique challenges: subgraphs can have non-trivial internal topology, but also carry a notion of position and external connectivity information relative to the underlying graph in which they exist. Here, we introduce SUBGNN, a subgraph neural network to learn disentangled subgraph representations. We propose a novel subgraph routing mechanism that propagates neural messages between the subgraph’s components and randomly sampled anchor patches from the underlying graph, yielding highly accurate subgraph representations. SUBGNN specifies three channels, each designed to capture a distinct aspect of subgraph topology, and we provide empirical evidence that the channels encode their intended properties. We design a series of new synthetic and real-world subgraph datasets. Empirical results for subgraph classification on eight datasets show that SUBGNN achieves considerable performance gains, outperforming strong baseline methods, including node-level and graph-level methods, by 19.8% over the strongest baseline. SUBGNN performs exceptionally well on challenging biomedical datasets where subgraphs have complex topology and …

In Graph Neural Networks (GNNs), the graph structure is incorporated into the learning of node representations. This complex structure makes explaining GNNs' predictions become much more challenging. In this paper, we propose PGM-Explainer, a Probabilistic Graphical Model (PGM) model-agnostic explainer for GNNs. Given a prediction to be explained, PGM-Explainer identifies crucial graph components and generates an explanation in form of a PGM approximating that prediction. Different from existing explainers for GNNs where the explanations are drawn from a set of linear functions of explained features, PGM-Explainer is able to demonstrate the dependencies of explained features in form of conditional probabilities. Our theoretical analysis shows that the PGM generated by PGM-Explainer includes the Markov-blanket of the target prediction, i.e. including all its statistical information. We also show that the explanation returned by PGM-Explainer contains the same set of independence statements in the perfect map. Our experiments on both synthetic and real-world datasets show that PGM-Explainer achieves better performance than existing explainers in many benchmark tasks.

Most of the successful deep neural network architectures are structured, often consisting of elements like convolutional neural networks and gated recurrent neural networks. Recently, graph neural networks (GNNs) have been successfully applied to graph-structured data such as point cloud and molecular data. These networks often only consider pairwise dependencies, as they operate on a graph structure. We generalize the GNN into a factor graph neural network (FGNN) providing a simple way to incorporate dependencies among multiple variables. We show that FGNN is able to represent Max-Product belief propagation, an approximate inference method on probabilistic graphical models, providing a theoretical understanding on the capabilities of FGNN and related GNNs. Experiments on synthetic and real datasets demonstrate the potential of the proposed architecture.

Nearest neighbor is a popular nonparametric method for classification and regression with many appealing properties. In the big data era, the sheer volume and spatial/temporal disparity of big data may prohibit centrally processing and storing the data. This has imposed considerable hurdle for nearest neighbor predictions since the entire training data must be memorized. One effective way to overcome this issue is the distributed learning framework. Through majority voting, the distributed nearest neighbor classifier achieves the same rate of convergence as its oracle version in terms of the regret, up to a multiplicative constant that depends solely on the data dimension. The multiplicative difference can be eliminated by replacing majority voting with the weighted voting scheme. In addition, we provide sharp theoretical upper bounds of the number of subsamples in order for the distributed nearest neighbor classifier to reach the optimal convergence rate. It is interesting to note that the weighted voting scheme allows a larger number of subsamples than the majority voting one. Our findings are supported by numerical studies.

Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams are very sensitive to perturbations in the input space. In this work, we develop a framework for constructing robust persistence diagrams from superlevel filtrations of robust density estimators constructed using reproducing kernels. Using an analogue of the influence function on the space of persistence diagrams, we establish the proposed framework to be less sensitive to outliers. The robust persistence diagrams are shown to be consistent estimators in the bottleneck distance, with the convergence rate controlled by the smoothness of the kernel — this, in turn, allows us to construct uniform confidence bands in the space of persistence diagrams. Finally, we demonstrate the superiority of the proposed approach on benchmark datasets.

We investigate the problem of regression where one is allowed to abstain from predicting. We refer to this framework as regression with reject option as an extension of classification with reject option. In this context, we focus on the case where the rejection rate is fixed and derive the optimal rule which relies on thresholding the conditional variance function. We provide a semi-supervised estimation procedure of the optimal rule involving two datasets: a first labeled dataset is used to estimate both regression function and conditional variance function while a second unlabeled dataset is exploited to calibrate the desired rejection rate. The resulting predictor with reject option is shown to be almost as good as the optimal predictor with reject option both in terms of risk and rejection rate. We additionally apply our methodology with kNN algorithm and establish rates of convergence for the resulting kNN predictor under mild conditions. Finally, a numerical study is performed to illustrate the benefit of using the proposed procedure.

In this paper, we present a statistical mechanics framework to understand the effect of sampling properties of training data on the generalization gap of machine learning (ML) algorithms. We connect the generalization gap to the spatial properties of a sample design characterized by the pair correlation function (PCF). In particular, we express generalization gap in terms of the power spectra of the sample design and that of the function to be learned. Using this framework, we show that space-filling sample designs, such as blue noise and Poisson disk sampling, which optimize spectral properties, outperform random designs in terms of the generalization gap and characterize this gain in a closed-form. Our analysis also sheds light on design principles for constructing optimal task-agnostic sample designs that minimize the generalization gap. We corroborate our findings using regression experiments with neural networks on: a) synthetic functions, and b) a complex scientific simulator for inertial confinement fusion (ICF).

Multivariate Hawkes processes are commonly used to model streaming networked event data in a wide variety of applications. However, it remains a challenge to extract reliable inference from complex datasets with uncertainty quantification. Aiming towards this, we develop a statistical inference framework to learn causal relationships between nodes from networked data, where the underlying directed graph implies Granger causality. We provide uncertainty quantification for the maximum likelihood estimate of the network multivariate Hawkes process by providing a non-asymptotic confidence set. The main technique is based on the concentration inequalities of continuous-time martingales. We compare our method to the previously-derived asymptotic Hawkes process confidence interval, and demonstrate the strengths of our method in an application to neuronal connectivity reconstruction.

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP has impractically slow convergence to high frequency signal components. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

We introduce the SE(3)-Transformer, a variant of the self-attention module for 3D point-clouds, which is equivariant under continuous 3D roto-translations. Equivariance is important to ensure stable and predictable performance in the presence of nuisance transformations of the data input. A positive corollary of equivariance is increased weight-tying within the model. The SE(3)-Transformer leverages the benefits of self-attention to operate on large point clouds with varying number of points, while guaranteeing SE(3)-equivariance for robustness. We evaluate our model on a toy N-body particle simulation dataset, showcasing the robustness of the predictions under rotations of the input. We further achieve competitive performance on two real-world datasets, ScanObjectNN and QM9. In all cases, our model outperforms a strong, non-equivariant attention baseline and an equivariant model without attention.

We present a transductive learning algorithm that takes as input training examples from a distribution P and arbitrary (unlabeled) test examples, possibly chosen by an adversary. This is unlike prior work that assumes that test examples are small perturbations of P. Our algorithm outputs a selective classifier, which abstains from predicting on some examples. By considering selective transductive learning, we give the first nontrivial guarantees for learning classes of bounded VC dimension with arbitrary train and test distributions—no prior guarantees were known even for simple classes of functions such as intervals on the line. In particular, for any function in a class C of bounded VC dimension, we guarantee a low test error rate and a low rejection rate with respect to P. Our algorithm is efficient given an Empirical Risk Minimizer (ERM) for C. Our guarantees hold even for test examples chosen by an unbounded white-box adversary. We also give guarantees for generalization, agnostic, and unsupervised settings.

We consider the problem of purchasing data for machine learning or statistical estimation. The data analyst has a budget to purchase datasets from multiple data providers. She does not have any test data that can be used to evaluate the collected data and can assign payments to data providers solely based on the collected datasets. We consider the problem in the standard Bayesian paradigm and in two settings: (1) data are only collected once; (2) data are collected repeatedly and each day's data are drawn independently from the same distribution. For both settings, our mechanisms guarantee that truthfully reporting one's dataset is always an equilibrium by adopting techniques from peer prediction: pay each provider the mutual information between his reported data and other providers' reported data. Depending on the data distribution, the mechanisms can also discourage misreports that would lead to inaccurate predictions. Our mechanisms also guarantee individual rationality and budget feasibility for certain underlying distributions in the first setting and for all distributions in the second setting.

To be well-behaved, systems that process preference data must satisfy certain conditions identified by economic decision theory and by social choice theory. In ML, preferences and rankings are commonly learned by fitting a probabilistic model to noisy preference data. The behavior of this learning process from the view of economic theory has previously been studied for the case where the data consists of rankings. In practice, it is more common to have only pairwise comparison data, and the formal properties of the associated learning problem are more challenging to analyze. We show that a large class of random utility models (including the Thurstone–Mosteller Model), when estimated using the MLE, satisfy a Pareto efficiency condition. These models also satisfy a strong monotonicity property, which implies that the learning process is responsive to input data. On the other hand, we show that these models fail certain other consistency conditions from social choice theory, and in particular do not always follow the majority opinion. Our results inform existing and future applications of random utility models for societal decision making.

Unsupervised and self-supervised learning approaches have become a crucial tool to learn representations for downstream prediction tasks. While these approaches are widely used in practice and achieve impressive empirical gains, their theoretical understanding largely lags behind. Towards bridging this gap, we present a unifying perspective where several such approaches can be viewed as imposing a regularization on the representation via a learnable function using unlabeled data. We propose a discriminative theoretical framework for analyzing the sample complexity of these approaches, which generalizes the framework of (Balcan and Blum, 2010) to allow learnable regularization functions. Our sample complexity bounds show that, with carefully chosen hypothesis classes to exploit the structure in the data, these learnable regularization functions can prune the hypothesis space, and help reduce the amount of labeled data needed. We then provide two concrete examples of functional regularization, one using auto-encoders and the other using masked self-supervision, and apply our framework to quantify the reduction in the sample complexity bound of labeled data. We also provide complementary empirical results to support our analysis.

Stochastic linear bandits with high-dimensional sparse features are a practical model for a variety of domains, such as personalized medicine and online advertising. We derive a novel O(n^{2/3}) dimension-free minimax regret lower bound for sparse linear bandits in the data-poor regime where the horizon is larger than the ambient dimension and where the feature vectors admit a well-conditioned exploration distribution. This is complemented by a nearly matching upper bound for an explore-then-commit algorithm showing that that O(n^{2/3}) is the optimal rate in the data-poor regime. The results complement existing bounds for the data-rich regime and also provide another example where carefully balancing the trade-off between information and regret is necessary. Finally, we prove a dimension-free O(\sqrt{n}) regret upper bound under an additional assumption on the magnitude of the signal for relevant features.

This paper studies adversarial bandits with corruptions. In the basic adversarial bandit setting, the reward of arms is predetermined by an adversary who is oblivious to the learner’s policy. In this paper, we consider an extended setting in which an attacker sits in-between the environment and the learner, and is endowed with a limited budget to corrupt the reward of the selected arm. We have two main results. First, we derive a lower bound on the regret of any bandit algorithm that is aware of the budget of the attacker. Also, for budget-agnostic algorithms, we characterize an impossibility result demonstrating that even when the attacker has a sublinear budget, i.e., a budget growing sublinearly with time horizon T, they fail to achieve a sublinear regret. Second, we propose ExpRb, a bandit algorithm that incorporates a biased estimator and a robustness parameter to deal with corruption. We characterize the regret of ExpRb as a function of the corruption budget and show that for the case of a known corruption budget, the regret of ExpRb is tight.

We study a regret minimization problem with the existence of multiple best/near-optimal arms in the multi-armed bandit setting. We consider the case when the number of arms/actions is comparable or much larger than the time horizon, and make \emph{no} assumptions about the structure of the bandit instance. Our goal is to design algorithms that can automatically adapt to the \emph{unknown} hardness of the problem, i.e., the number of best arms. Our setting captures many modern applications of bandit algorithms where the action space is enormous and the information about the underlying instance/structure is unavailable. We first propose an adaptive algorithm that is agnostic to the hardness level and theoretically derive its regret bound. We then prove a lower bound for our problem setting, which indicates: (1) no algorithm can be minimax optimal simultaneously over all hardness levels; and (2) our algorithm achieves a rate function that is Pareto optimal. With additional knowledge of the expected reward of the best arm, we propose another adaptive algorithm that is minimax optimal, up to polylog factors, over \emph{all} hardness levels. Experimental results confirm our theoretical guarantees and show advantages of our algorithms over the previous state-of-the-art.

In the contextual linear bandit setting, algorithms built on the optimism principle fail to exploit the structure of the problem and have been shown to be asymptotically suboptimal. In this paper, we follow recent approaches of deriving asymptotically optimal algorithms from problem-dependent regret lower bounds and we introduce a novel algorithm improving over the state-of-the-art along multiple dimensions. We build on a reformulation of the lower bound, where context distribution and exploration policy are decoupled, and we obtain an algorithm robust to unbalanced context distributions. Then, using an incremental primal-dual approach to solve the Lagrangian relaxation of the lower bound, we obtain a scalable and computationally efficient algorithm. Finally, we remove forced exploration and build on confidence intervals of the optimization problem to encourage a minimum level of exploration that is better adapted to the problem structure. We demonstrate the asymptotic optimality of our algorithm, while providing both problem-dependent and worst-case finite-time regret guarantees. Our bounds scale with the logarithm of the number of arms, thus avoiding the linear dependence common in all related prior works. Notably, we establish minimax optimality for any learning horizon in the special case of non-contextual linear bandits. Finally, we verify that our algorithm obtains …

We consider the problem of controlling a known linear dynamical system under stochastic noise, adversarially chosen costs, and bandit feedback. Unlike the full feedback setting where the entire cost function is revealed after each decision, here only the cost incurred by the learner is observed. We present a new and efficient algorithm that, for strongly convex and smooth costs, obtains regret that grows with the square root of the time horizon T. We also give extensions of this result to general convex, possibly non-smooth costs, and to non-stochastic system noise. A key component of our algorithm is a new technique for addressing bandit optimization of loss functions with memory.

We consider the problem of robust and adaptive model predictive control (MPC) of a linear system, with unknown parameters that are learned along the way (adaptive), in a critical setting where failures must be prevented (robust). This problem has been studied from different perspectives by different communities. However, the existing theory deals only with the case of quadratic costs (the LQ problem), which limits applications to stabilisation and tracking tasks only. In order to handle more general (non-convex) costs that naturally arise in many practical problems, we carefully select and bring together several tools from different communities, namely non-asymptotic linear regression, recent results in interval prediction, and tree-based planning. Combining and adapting the theoretical guarantees at each layer is non trivial, and we provide the first end-to-end suboptimality analysis for this setting. Interestingly, our analysis naturally adapts to handle many models and combines with a data-driven robust model selection strategy, which enables to relax the modelling assumptions. Last, we strive to preserve tractability at any stage of the method, that we illustrate on two challenging simulated environments.

Entropy regularization, smoothing of Q-values and neural network function approximator are key components of the state-of-the-art reinforcement learning (RL) algorithms, such as Soft Actor-Critic~\cite{haarnoja2018soft}. Despite the widespread use, the impact of these core techniques on the convergence of RL algorithms is not yet fully understood. In this work, we analyse these techniques from error propagation perspective using the approximate dynamic programming framework. In particular, our analysis shows that (1) value smoothing results in increased stability of the algorithm in exchange for slower convergence, (2) entropy regularization reduces overestimation errors at the cost of modifying the original problem, (3) we study a combination of these techniques that describes the Soft Actor-Critic algorithm.

This paper develops a Pontryagin Differentiable Programming (PDP) methodology, which establishes a unified framework to solve a broad class of learning and control tasks. The PDP  distinguishes from existing methods by two novel techniques: first, we  differentiate through  Pontryagin's Maximum Principle, and  this allows  to obtain the analytical derivative of a  trajectory with respect to tunable parameters within an optimal control system,  enabling end-to-end learning of   dynamics, policies, or/and control objective functions; and second, we propose an auxiliary control system in the backward pass of the PDP framework, and  the output of this auxiliary control system is the  analytical derivative of the original system's trajectory with respect to the  parameters, which can be iteratively solved using standard control tools. We investigate three learning modes of the PDP: inverse reinforcement learning,  system identification, and  control/planning. We demonstrate the capability of the PDP in each learning mode on different high-dimensional systems, including multi-link robot arm,  6-DoF maneuvering quadrotor, and 6-DoF rocket powered landing.

Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far. We study KL regularization within an approximate value iteration scheme and show that it implicitly averages q-values. Leveraging this insight, we provide a very strong performance bound, the very first to combine two desirable aspects: a linear dependency to the horizon (instead of quadratic) and an error propagation term involving an averaging effect of the estimation errors (instead of an accumulation effect). We also study the more general case of an additional entropy regularizer. The resulting abstract scheme encompasses many existing RL algorithms. Some of our assumptions do not hold with neural networks, so we complement this theoretical analysis with an extensive empirical study.

To improve the sample efficiency of policy-gradient based reinforcement learning algorithms, we propose implicit distributional actor-critic (IDAC) that consists of a distributional critic, built on two deep generator networks (DGNs), and a semi-implicit actor (SIA), powered by a flexible policy distribution. We adopt a distributional perspective on the discounted cumulative return and model it with a state-action-dependent implicit distribution, which is approximated by the DGNs that take state-action pairs and random noises as their input. Moreover, we use the SIA to provide a semi-implicit policy distribution, which mixes the policy parameters with a reparameterizable distribution that is not constrained by an analytic density function. In this way, the policy's marginal distribution is implicit, providing the potential to model complex properties such as covariance structure and skewness, but its parameter and entropy can still be estimated. We incorporate these features with an off-policy algorithm framework to solve problems with continuous action space and compare IDAC with state-of-the-art algorithms on representative OpenAI Gym environments. We observe that IDAC outperforms these baselines in most tasks. Python code is provided.

In many game settings, the game is not explicitly given but is only accessible by playing it. While there have been impressive demonstrations in such settings, prior techniques have not offered safety guarantees, that is, guarantees on the game-theoretic exploitability of the computed strategies. In this paper we introduce an approach that shows that it is possible to provide exploitability guarantees in such settings without ever exploring the entire game. We introduce a notion of a certificatae of an extensive-form approximate Nash equilibrium. For verifying a certificate, we give an algorithm that runs in time linear in the size of the certificate rather than the size of the whole game. In zero-sum games, we further show that an optimal certificate---given the exploration so far---can be computed with any standard game-solving algorithm (e.g., using a linear program or counterfactual regret minimization). However, unlike in the cases of normal form or perfect information, we show that certain families of extensive-form games do not have small approximate certificates, even after making extremely nice assumptions on the structure of the game. Despite this difficulty, we find experimentally that very small certificates, even exact ones, often exist in large and even in infinite games. Overall, …

We formulate the novel class of contextual games, a type of repeated games driven by contextual information at each round. By means of kernel-based regularity assumptions, we model the correlation between different contexts and game outcomes and propose a novel online (meta) algorithm that exploits such correlations to minimize the contextual regret of individual players. We define game-theoretic notions of contextual Coarse Correlated Equilibria (c-CCE) and optimal contextual welfare for this new class of games and show that c-CCEs and optimal welfare can be approached whenever players' contextual regrets vanish. Finally, we empirically validate our results in a traffic routing experiment, where our algorithm leads to better performance and higher welfare compared to baselines that do not exploit the available contextual information or the correlations present in the game.

Inference networks of traditional Variational Autoencoders (VAEs) are typically amortized, resulting in relatively inaccurate posterior approximation compared to instance-wise variational optimization. Recent semi-amortized approaches were proposed to address this drawback; however, their iterative gradient update procedures can be computationally demanding. In this paper, we consider a different approach of building a mixture inference model. We propose a novel recursive mixture estimation algorithm for VAEs that iteratively augments the current mixture with new components so as to maximally reduce the divergence between the variational and the true posteriors. Using the functional gradient approach, we devise an intuitive learning criteria for selecting a new mixture component: the new component has to improve the data likelihood (lower bound) and, at the same time, be as divergent from the current mixture distribution as possible, thus increasing representational diversity. Although there have been similar approaches recently, termed boosted variational inference (BVI), our methods differ from BVI in several aspects, most notably that ours deal with recursive inference in VAEs in the form of amortized inference, while BVI is developed within the standard VI framework, leading to a non-amortized single optimization instance, inappropriate for VAEs. A crucial benefit of our approach is that the inference at …

Standard Bayesian inference algorithms are prohibitively expensive in the regime of modern large-scale data. Recent work has found that a small, weighted subset of data (a coreset) may be used in place of the full dataset during inference, taking advantage of data redundancy to reduce computational cost. However, this approach has limitations in the increasingly common setting of sensitive, high-dimensional data. Indeed, we prove that there are situations in which the Kullback-Leibler divergence between the optimal coreset and the true posterior grows with data dimension; and as coresets include a subset of the original data, they cannot be constructed in a manner that preserves individual privacy. We address both of these issues with a single unified solution, Bayesian pseudocoresets --- a small weighted collection of synthetic "pseudodata"---along with a variational optimization method to select both pseudodata and weights. The use of pseudodata (as opposed to the original datapoints) enables both the summarization of high-dimensional data and the differentially private summarization of sensitive data. Real and synthetic experiments on high-dimensional data demonstrate that Bayesian pseudocoresets achieve significant improvements in posterior approximation error compared to traditional coresets, and that pseudocoresets provide privacy without a significant loss in approximation quality.

Variational Bayesian Monte Carlo (VBMC) is a recently introduced framework that uses Gaussian process surrogates to perform approximate Bayesian inference in models with black-box, non-cheap likelihoods. In this work, we extend VBMC to deal with noisy log-likelihood evaluations, such as those arising from simulation-based models. We introduce new 'global' acquisition functions, such as expected information gain (EIG) and variational interquantile range (VIQR), which are robust to noise and can be efficiently evaluated within the VBMC setting. In a novel, challenging, noisy-inference benchmark comprising of a variety of models with real datasets from computational and cognitive neuroscience, VBMC+VIQR achieves state-of-the-art performance in recovering the ground-truth posteriors and model evidence. In particular, our method vastly outperforms 'local' acquisition functions and other surrogate-based inference methods while keeping a small algorithmic cost. Our benchmark corroborates VBMC as a general-purpose technique for sample-efficient black-box Bayesian inference also with noisy models.

Bayesian quadrature (BQ) is a method for solving numerical integration problems in a Bayesian manner, which allows users to quantify their uncertainty about the solution. The standard approach to BQ is based on a Gaussian process (GP) approximation of the integrand. As a result, BQ is inherently limited to cases where GP approximations can be done in an efficient manner, thus often prohibiting very high-dimensional or non-smooth target functions. This paper proposes to tackle this issue with a new Bayesian numerical integration algorithm based on Bayesian Additive Regression Trees (BART) priors, which we call BART-Int. BART priors are easy to tune and well-suited for discontinuous functions. We demonstrate that they also lend themselves naturally to a sequential design setting and that explicit convergence rates can be obtained in a variety of settings. The advantages and disadvantages of this new methodology are highlighted on a set of benchmark tests including the Genz functions, on a rare-event simulation problem and on a Bayesian survey design problem.

Deep Gaussian Processes learn probabilistic data representations for supervised learning by cascading multiple Gaussian Processes. While this model family promises flexible predictive distributions, exact inference is not tractable. Approximate inference techniques trade off the ability to closely resemble the posterior distribution against speed of convergence and computational efficiency. We propose a novel Gaussian variational family that allows for retaining covariances between latent processes while achieving fast convergence by marginalising out all global latent variables. After providing a proof of how this marginalisation can be done for general covariances, we restrict them to the ones we empirically found to be most important in order to also achieve computational efficiency. We provide an efficient implementation of our new approach and apply it to several benchmark datasets. It yields excellent results and strikes a better balance between accuracy and calibrated uncertainty estimates than its state-of-the-art alternatives.

Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians N(0,K) or “whitening” a vector b against covariance matrix K. While existing methods typically require O(N^3) computation, we introduce a highly-efficient quadratic-time algorithm for computing K^{1/2}b, K^{-1/2}b, and their derivatives through matrix-vector multiplication (MVMs). Our method combines Krylov subspace methods with a rational approximation and typically achieves 4 decimal places of accuracy with fewer than 100 MVMs. Moreover, the backward pass requires little additional computation. We demonstrate our method's applicability on matrices as large as 50,000 by 50,000 - well beyond traditional methods - with little approximation error. Applying this increased scalability to variational Gaussian processes, Bayesian optimization, and Gibbs sampling results in more powerful models with higher accuracy. In particular, we perform variational GP inference with up to 10,000 inducing points and perform Gibbs sampling on a 25,000-dimensional problem.

Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo sampling. We exploit the Continuous Piecewise Affine property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first two moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm for gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our new framework will guide the design of new VAE AVI that better approximates the true posterior and open new avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation.

Motivated by decentralized approaches to machine learning, we propose a collaborative Bayesian learning algorithm taking the form of decentralized Langevin dynamics in a non-convex setting. Our analysis show that the initial KL-divergence between the Markov Chain and the target posterior distribution is exponentially decreasing while the error contributions to the overall KL-divergence from the additive noise is decreasing in polynomial time. We further show that the polynomial-term experiences speed-up with number of agents and provide sufficient conditions on the time-varying step-sizes to guarantee convergence to the desired distribution. The performance of the proposed algorithm is evaluated on a wide variety of machine learning tasks. The empirical results show that the performance of individual agents with locally available data is on par with the centralized setting with considerable improvement in the convergence rate.

Gaussian latent variable models are a key class of Bayesian hierarchical models with applications in many fields. Performing Bayesian inference on such models can be challenging as Markov chain Monte Carlo algorithms struggle with the geometry of the resulting posterior distribution and can be prohibitively slow. An alternative is to use a Laplace approximation to marginalize out the latent Gaussian variables and then integrate out the remaining hyperparameters using dynamic Hamiltonian Monte Carlo, a gradient-based Markov chain Monte Carlo sampler. To implement this scheme efficiently, we derive a novel adjoint method that propagates the minimal information needed to construct the gradient of the approximate marginal likelihood. This strategy yields a scalable differentiation method that is orders of magnitude faster than state of the art differentiation techniques when the hyperparameters are high dimensional. We prototype the method in the probabilistic programming framework Stan and test the utility of the embedded Laplace approximation on several models, including one where the dimension of the hyperparameter is ∼6,000. Depending on the cases, the benefits can include an alleviation of the geometric pathologies that frustrate Hamiltonian Monte Carlo and a dramatic speed-up.

To make decisions based on a model fit with auto-encoding variational Bayes (AEVB), practitioners often let the variational distribution serve as a surrogate for the posterior distribution. This approach yields biased estimates of the expected risk, and therefore leads to poor decisions for two reasons. First, the model fit with AEVB may not equal the underlying data distribution. Second, the variational distribution may not equal the posterior distribution under the fitted model. We explore how fitting the variational distribution based on several objective functions other than the ELBO, while continuing to fit the generative model based on the ELBO, affects the quality of downstream decisions. For the probabilistic principal component analysis model, we investigate how importance sampling error, as well as the bias of the model parameter estimates, varies across several approximate posteriors when used as proposal distributions. Our theoretical results suggest that a posterior approximation distinct from the variational distribution should be used for making decisions. Motivated by these theoretical results, we propose learning several approximate proposals for the best model and combining them using multiple importance sampling for decision-making. In addition to toy examples, we present a full-fledged case study of single-cell RNA sequencing. In this challenging instance …

We examine the accuracy of black box variational posterior approximations for parametric models in a probabilistic programming context. The performance of these approximations depends on (1) how well the variational family approximates the true posterior distribution, (2) the choice of divergence, and (3) the optimization of the variational objective. We show that even when the true variational family is used, high-dimensional posteriors can be very poorly approximated using common stochastic gradient descent (SGD) optimizers. Motivated by recent theory, we propose a simple and parallel way to improve SGD estimates for variational inference. The approach is theoretically motivated and comes with a diagnostic for convergence and a novel stopping rule, which is robust to noisy objective functions evaluations. We show empirically, the new workflow works well on a diverse set of models and datasets, or warns if the stochastic optimization fails or if the used variational distribution is not good.

Recent research has seen several advances relevant to black-box VI, but the current state of automatic posterior inference is unclear. One such advance is the use of normalizing flows to define flexible posterior densities for deep latent variable models. Another direction is the integration of Monte-Carlo methods to serve two purposes; first, to obtain tighter variational objectives for optimization, and second, to define enriched variational families through sampling. However, both flows and variational Monte-Carlo methods remain relatively unexplored for black-box VI. Moreover, on a pragmatic front, there are several optimization considerations like step-size scheme, parameter initialization, and choice of gradient estimators, for which there are no clear guidance in the existing literature. In this paper, we postulate that black-box VI is best addressed through a careful combination of numerous algorithmic components. We evaluate components relating to optimization, flows, and Monte-Carlo methods on a benchmark of 30 models from the Stan model library. The combination of these algorithmic components significantly advances the state-of-the-art "out of the box" variational inference.

Bayesian neural networks are enjoying a renaissance driven in part by recent advances in variational inference (VI). The most common form of VI employs a fully factorized or mean-field distribution, but this is known to suffer from several pathologies, especially as we expect posterior distributions with highly correlated parameters. Current algorithms that capture these correlations with a Gaussian approximating family are difficult to scale to large models due to computational costs and high variance of gradient updates. By using a new form of the reparametrization trick, we derive a computationally efficient algorithm for performing VI with a Gaussian family with a low-rank plus diagonal covariance structure. We scale to deep feed-forward and convolutional architectures. We find that adding low-rank terms to parametrized diagonal covariance does not improve predictive performance except on small networks, but low-rank terms added to a constant diagonal covariance improves performance on small and large-scale network architectures.

We challenge the longstanding assumption that the mean-field approximation for variational inference in Bayesian neural networks is severely restrictive, and show this is not the case in deep networks. We prove several results indicating that deep mean-field variational weight posteriors can induce similar distributions in function-space to those induced by shallower networks with complex weight posteriors. We validate our theoretical contributions empirically, both through examination of the weight posterior using Hamiltonian Monte Carlo in small models and by comparing diagonal- to structured-covariance in large settings. Since complex variational posteriors are often expensive and cumbersome to implement, our results suggest that using mean-field variational inference in a deeper model is both a practical and theoretically justified alternative to structured approximations.

Modern variational inference (VI) uses stochastic gradients to avoid intractable expectations, enabling large-scale probabilistic inference in complex models. VI posits a family of approximating distributions q and then finds the member of that family that is closest to the exact posterior p. Traditionally, VI algorithms minimize the “exclusive Kullback-Leibler (KL)” KL(q||p), often for computational convenience. Recent research, however, has also focused on the “inclusive KL” KL(p||q), which has good statistical properties that makes it more appropriate for certain inference problems. This paper develops a simple algorithm for reliably minimizing the inclusive KL using stochastic gradients with vanishing bias. This method, which we call Markovian score climbing (MSC), converges to a local optimum of the inclusive KL. It does not suffer from the systematic errors inherent in existing methods, such as Reweighted Wake-Sleep and Neural Adaptive Sequential Monte Carlo, which lead to bias in their final estimates. We illustrate convergence on a toy model and demonstrate the utility of MSC on Bayesian probit regression for classification as well as a stochastic volatility model for financial data.

The curse of dimensionality is a longstanding challenge in Bayesian inference in high dimensions. In this work, we propose a {projected Stein variational gradient descent} (pSVGD) method to overcome this challenge by exploiting the fundamental property of intrinsic low dimensionality of the data informed subspace stemming from ill-posedness of such problems. We adaptively construct the subspace using a gradient information matrix of the log-likelihood, and apply pSVGD to the much lower-dimensional coefficients of the parameter projection. The method is demonstrated to be more accurate and efficient than SVGD. It is also shown to be more scalable with respect to the number of parameters, samples, data points, and processor cores via experiments with parameters dimensions ranging from the hundreds to the tens of thousands.

Modern gradient boosting software frameworks, such as XGBoost and LightGBM, implement Newton descent in a functional space. At each boosting iteration, their goal is to find the base hypothesis, selected from some base hypothesis class, that is closest to the Newton descent direction in a Euclidean sense. Typically, the base hypothesis class is fixed to be all binary decision trees up to a given depth. In this work, we study a Heterogeneous Newton Boosting Machine (HNBM) in which the base hypothesis class may vary across boosting iterations. Specifically, at each boosting iteration, the base hypothesis class is chosen, from a fixed set of subclasses, by sampling from a probability distribution. We derive a global linear convergence rate for the HNBM under certain assumptions, and show that it agrees with existing rates for Newton's method when the Newton direction can be perfectly fitted by the base hypothesis at each boosting iteration. We then describe a particular realization of a HNBM, SnapBoost, that, at each boosting iteration, randomly selects between either a decision tree of variable depth or a linear regressor with random Fourier features. We describe how SnapBoost is implemented, with a focus on the training complexity. Finally, we present experimental …

Wasserstein gradient flows are continuous time dynamics that define curves of steepest descent to minimize an objective function over the space of probability measures (i.e., the Wasserstein space). This objective is typically a divergence w.r.t. a fixed target distribution. In recent years, these continuous time dynamics have been used to study the convergence of machine learning algorithms aiming at approximating a probability distribution. However, the discrete-time behavior of these algorithms might differ from the continuous time dynamics. Besides, although discretized gradient flows have been proposed in the literature, little is known about their minimization power. In this work, we propose a Forward Backward (FB) discretization scheme that can tackle the case where the objective function is the sum of a smooth and a nonsmooth geodesically convex terms. Using techniques from convex optimization and optimal transport, we analyze the FB scheme as a minimization algorithm on the Wasserstein space. More precisely, we show under mild assumptions that the FB scheme has convergence guarantees similar to the proximal gradient algorithm in Euclidean spaces (resp. similar to the associated Wasserstein gradient flow).

We introduce Unbalanced Sobolev Descent (USD), a particle descent algorithm for transporting a high dimensional source distribution to a target distribution that does not necessarily have the same mass. We define the Sobolev-Fisher discrepancy between distributions and show that it relates to advection-reaction transport equations and the Wasserstein-Fisher-Rao metric between distributions. USD transports particles along gradient flows of the witness function of the Sobolev-Fisher discrepancy (advection step) and reweighs the mass of particles with respect to this witness function (reaction step). The reaction step can be thought of as a birth-death process of the particles with rate of growth proportional to the witness function. When the Sobolev-Fisher witness function is estimated in a Reproducing Kernel Hilbert Space (RKHS), under mild assumptions we show that USD converges asymptotically (in the limit of infinite particles) to the target distribution in the Maximum Mean Discrepancy (MMD) sense. We then give two methods to estimate the Sobolev-Fisher witness with neural networks, resulting in two Neural USD algorithms. The first one implements the reaction step with mirror descent on the weights, while the second implements it through a birth-death process of particles. We show on synthetic examples that USD transports distributions with or without conservation …

Nonlinear dimensionality reduction of high-dimensional data is challenging as the low-dimensional embedding will necessarily contain distortions, and it can be hard to determine which distortions are the most important to avoid. When annotation of data into known relevant classes is available, it can be used to guide the embedding to avoid distortions that worsen class separation. The supervised mapping method introduced in the present paper, called ClassNeRV, proposes an original stress function that takes class annotation into account and evaluates embedding quality both in terms of false neighbors and missed neighbors. ClassNeRV shares the theoretical framework of a family of methods descended from Stochastic Neighbor Embedding (SNE). Our approach has a key advantage over previous ones: in the literature supervised methods often emphasize class separation at the price of distorting the data neighbors' structure; conversely, unsupervised methods provide better preservation of structure at the price of often mixing classes. Experiments show that ClassNeRV can preserve both neighbor structure and class separation, outperforming nine state of the art alternatives.

In the last decade, there has been increasing interest in topological data analysis, a new methodology for using geometric structures in data for inference and learning. A central theme in the area is the idea of persistence, which in its most basic form studies how measures of shape change as a scale parameter varies. There are now a number of frameworks that support statistics and machine learning in this context. However, in many applications there are several different parameters one might wish to vary: for example, scale and density. In contrast to the one-parameter setting, techniques for applying statistics and machine learning in the setting of multiparameter persistence are not well understood due to the lack of a concise representation of the results.

We introduce a new descriptor for multiparameter persistence, which we call the Multiparameter Persistence Image, that is suitable for machine learning and statistical frameworks, is robust to perturbations in the data, has finer resolution than existing descriptors based on slicing, and can be efficiently computed on data sets of realistic size. Moreover, we demonstrate its efficacy by comparing its performance to other multiparameter descriptors on several classification tasks.

Machine learning pipelines often rely on optimizers procedures to make discrete decisions (e.g., sorting, picking closest neighbors, or shortest paths). Although these discrete decisions are easily computed in a forward manner, they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform optimizers into operations that are differentiable and never locally constant. Our approach relies on stochastically perturbed optimizers, and can be used readily within existing solvers. Their derivatives can be evaluated efficiently, and smoothness tuned via the chosen noise amplitude. We also show how this framework can be connected to a family of losses developed in structured prediction, and give theoretical guarantees for their use in learning tasks. We demonstrate experimentally the performance of our approach on various tasks.

Kernel methods augmented with random features give scalable algorithms for learning from big data. But it has been computationally hard to sample random features according to a probability distribution that is optimized for the data, so as to minimize the required number of features for achieving the learning to a desired accuracy. Here, we develop a quantum algorithm for sampling from this optimized distribution over features, in runtime O(D) that is linear in the dimension D of the input data. Our algorithm achieves an exponential speedup in D compared to any known classical algorithm for this sampling task. In contrast to existing quantum machine learning algorithms, our algorithm circumvents sparsity and low-rank assumptions and thus has wide applicability. We also show that the sampled features can be combined with regression by stochastic gradient descent to achieve the learning without canceling out our exponential speedup. Our algorithm based on sampling optimized random features leads to an accelerated framework for machine learning that takes advantage of quantum computers.

Machine learning has proved its ability to produce accurate models -- but the deployment of these models outside the machine learning community has been hindered by the difficulties of interpreting these models. This paper proposes an algorithm that produces a continuous global interpretation of any given continuous black-box function. Our algorithm employs a variation of projection pursuit in which the ridge functions are chosen to be Meijer G-functions, rather than the usual polynomial splines. Because Meijer G-functions are differentiable in their parameters, we can "tune" the parameters of the representation by gradient descent; as a consequence, our algorithm is efficient. Using five familiar data sets from the UCI repository and two familiar machine learning algorithms, we demonstrate that our algorithm produces global interpretations that are both faithful (highly accurate) and parsimonious (involve a small number of terms). Our interpretations permit easy understanding of the relative importance of features and feature interactions. Our interpretation algorithm represents a leap forward from the previous state of the art.

Many learning algorithms, such as stochastic gradient descent, are affected by the order in which training examples are used. It is often observed that sampling the training examples without-replacement, also known as random reshuffling, causes learning algorithms to converge faster. We give a counterexample to the Operator Inequality of Noncommutative Arithmetic and Geometric Means, a longstanding conjecture that relates to the performance of random reshuffling in learning algorithms (Recht and Ré, "Toward a noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences," COLT 2012). We use this to give an example of a learning task and algorithm for which with-replacement random sampling actually outperforms random reshuffling.

Optimal Transport (OT) distances are now routinely used as loss functions in ML tasks. Yet, computing OT distances between arbitrary (i.e. not necessarily discrete) probability distributions remains an open problem. This paper introduces a new online estimator of entropy-regularized OT distances between two such arbitrary distributions. It uses streams of samples from both distributions to iteratively enrich a non-parametric representation of the transportation plan. Compared to the classic Sinkhorn algorithm, our method leverages new samples at each iteration, which enables a consistent estimation of the true regularized OT distance. We provide a theoretical analysis of the convergence of the online Sinkhorn algorithm, showing a nearly-1/n asymptotic sample complexity for the iterate sequence. We validate our method on synthetic 1-d to 10-d data and on real 3-d shape data.

Stein Variational Gradient Descent (SVGD), a popular sampling algorithm, is often described as the kernelized gradient flow for the Kullback-Leibler divergence in the geometry of optimal transport. We introduce a new perspective on SVGD that instead views SVGD as the kernelized gradient flow of the chi-squared divergence. Motivated by this perspective, we provide a convergence analysis of the chi-squared gradient flow. We also show that our new perspective provides better guidelines for choosing effective kernels for SVGD.

Policy gradient methods have shown success in learning control policies for high-dimensional dynamical systems. Their biggest downside is the amount of exploration they require before yielding high-performing policies. In a lifelong learning setting, in which an agent is faced with multiple consecutive tasks over its lifetime, reusing information from previously seen tasks can substantially accelerate the learning of new tasks. We provide a novel method for lifelong policy gradient learning that trains lifelong function approximators directly via policy gradients, allowing the agent to benefit from accumulated knowledge throughout the entire training process. We show empirically that our algorithm learns faster and converges to better policies than single-task and lifelong learning baselines, and completely avoids catastrophic forgetting on a variety of challenging domains.

Progress in Reinforcement Learning (RL) algorithms goes hand-in-hand with the development of challenging environments that test the limits of current methods. While existing RL environments are either sufficiently complex or based on fast simulation, they are rarely both. Here, we present the NetHack Learning Environment (NLE), a scalable, procedurally generated, stochastic, rich, and challenging environment for RL research based on the popular single-player terminal-based roguelike game, NetHack. We argue that NetHack is sufficiently complex to drive long-term research on problems such as exploration, planning, skill acquisition, and language-conditioned RL, while dramatically reducing the computational resources required to gather a large amount of experience. We compare NLE and its task suite to existing alternatives, and discuss why it is an ideal medium for testing the robustness and systematic generalization of RL agents. We demonstrate empirical success for early stages of the game using a distributed Deep RL baseline and Random Network Distillation exploration, alongside qualitative analysis of various agents trained in the environment. NLE is open source and available at https://github.com/facebookresearch/nle.

Reinforcement learning (RL) algorithms update an agent’s parameters according to one of several possible rules, discovered manually through years of research. Automating the discovery of update rules from data could lead to more efficient algorithms, or algorithms that are better adapted to specific environments. Although there have been prior attempts at addressing this significant scientific challenge, it remains an open question whether it is feasible to discover alternatives to fundamental concepts of RL such as value functions and temporal-difference learning. This paper introduces a new meta-learning approach that discovers an entire update rule which includes both what to predict' (e.g. value functions) andhow to learn from it' (e.g. bootstrapping) by interacting with a set of environments. The output of this method is an RL algorithm that we call Learned Policy Gradient (LPG). Empirical results show that our method discovers its own alternative to the concept of value functions. Furthermore it discovers a bootstrapping mechanism to maintain and use its predictions. Surprisingly, when trained solely on toy environments, LPG generalises effectively to complex Atari games and achieves non-trivial performance. This shows the potential to discover general RL algorithms from data.

In this work we consider partially observable environments with sparse rewards. We present a self-supervised representation learning method for image-based observations, which arranges embeddings respecting temporal distance of observations. This representation is empirically robust to stochasticity and suitable for novelty detection from the error of a predictive forward model. We consider episodic and life-long uncertainties to guide the exploration. We propose to estimate the missing information about the environment with the world model, which operates in the learned latent space. As a motivation of the method, we analyse the exploration problem in a tabular Partially Observable Labyrinth. We demonstrate the method on image-based hard exploration environments from the Atari benchmark and report significant improvement with respect to prior work. The source code of the method and all the experiments is available at https://github.com/htdt/lwm.

In multi-agent reinforcement learning, discovering successful collective behaviors is challenging as it requires exploring a joint action space that grows exponentially with the number of agents. While the tractability of independent agent-wise exploration is appealing, this approach fails on tasks that require elaborate group strategies. We argue that coordinating the agents' policies can guide their exploration and we investigate techniques to promote such an inductive bias. We propose two policy regularization methods: TeamReg, which is based on inter-agent action predictability and CoachReg that relies on synchronized behavior selection. We evaluate each approach on four challenging continuous control tasks with sparse rewards that require varying levels of coordination as well as on the discrete action Google Research Football environment. Our experiments show improved performance across many cooperative multi-agent problems. Finally, we analyze the effects of our proposed methods on the policies that our agents learn and show that our methods successfully enforce the qualities that we propose as proxies for coordinated behaviors.

Exploration in multi-agent reinforcement learning is a challenging problem, especially in environments with sparse rewards. We propose a general method for efficient exploration by sharing experience amongst agents. Our proposed algorithm, called shared Experience Actor-Critic(SEAC), applies experience sharing in an actor-critic framework by combining the gradients of different agents. We evaluate SEAC in a collection of sparse-reward multi-agent environments and find that it consistently outperforms several baselines and state-of-the-art algorithms by learning in fewer steps and converging to higher returns. In some harder environments, experience sharing makes the difference between learning to solve the task and not learning at all.

The challenge of developing powerful and general Reinforcement Learning (RL) agents has received increasing attention in recent years. Much of this effort has focused on the single-agent setting, in which an agent maximizes a predefined extrinsic reward function. However, a long-term question inevitably arises: how will such independent agents cooperate when they are continually learning and acting in a shared multi-agent environment? Observing that humans often provide incentives to influence others' behavior, we propose to equip each RL agent in a multi-agent environment with the ability to give rewards directly to other agents, using a learned incentive function. Each agent learns its own incentive function by explicitly accounting for its impact on the learning of recipients and, through them, the impact on its own extrinsic objective. We demonstrate in experiments that such agents significantly outperform standard RL and opponent-shaping agents in challenging general-sum Markov games, often by finding a near-optimal division of labor. Our work points toward more opportunities and challenges along the path to ensure the common good in a multi-agent future.

Training multi-agent systems (MAS) to achieve realistic equilibria gives us a useful tool to understand and model real-world systems. We consider a general sum partially observable Markov game where agents of different types share a single policy network, conditioned on agent-specific information. This paper aims at i) formally understanding equilibria reached by such agents, and ii) matching emergent phenomena of such equilibria to real-world targets. Parameter sharing with decentralized execution has been introduced as an efficient way to train multiple agents using a single policy network. However, the nature of resulting equilibria reached by such agents has not been yet studied: we introduce the novel concept of Shared equilibrium as a symmetric pure Nash equilibrium of a certain Functional Form Game (FFG) and prove convergence to the latter for a certain class of games using self-play. In addition, it is important that such equilibria satisfy certain constraints so that MAS are calibrated to real world data for practical use: we solve this problem by introducing a novel dual-Reinforcement Learning based approach that fits emergent behaviors of agents in a Shared equilibrium to externally-specified targets, and apply our methods to a n-player market example. We do so by calibrating parameters governing …

Multi-agent reinforcement learning has recently shown great promise as an approach to networked system control. Arguably, one of the most difficult and important tasks for which large scale networked system control is applicable is common-pool resource management. Crucial common-pool resources include arable land, fresh water, wetlands, wildlife, fish stock, forests and the atmosphere, of which proper management is related to some of society's greatest challenges such as food security, inequality and climate change. Here we take inspiration from a recent research program investigating the game-theoretic incentives of humans in social dilemma situations such as the well-known \textit{tragedy of the commons}. However, instead of focusing on biologically evolved human-like agents, our concern is rather to better understand the learning and operating behaviour of engineered networked systems comprising general-purpose reinforcement learning agents, subject only to nonbiological constraints such as memory, computation and communication bandwidth. Harnessing tools from empirical game-theoretic analysis, we analyse the differences in resulting solution concepts that stem from employing different information structures in the design of networked multi-agent systems. These information structures pertain to the type of information shared between agents as well as the employed communication protocol and network topology. Our analysis contributes new insights into the consequences …

Recent advances in deep reinforcement learning (RL) have led to considerable progress in many 2-player zero-sum games, such as Go, Poker and Starcraft. The purely adversarial nature of such games allows for conceptually simple and principled application of RL methods. However real-world settings are many-agent, and agent interactions are complex mixtures of common-interest and competitive aspects. We consider Diplomacy, a 7-player board game designed to accentuate dilemmas resulting from many-agent interactions. It also features a large combinatorial action space and simultaneous moves, which are challenging for RL algorithms. We propose a simple yet effective approximate best response operator, designed to handle large combinatorial action spaces and simultaneous moves. We also introduce a family of policy iteration methods that approximate fictitious play. With these methods, we successfully apply RL to Diplomacy: we show that our agents convincingly outperform the previous state-of-the-art, and game theoretic equilibrium analysis shows that the new process yields consistent improvements.

The ability to compose learned skills to solve new tasks is an important property for lifelong-learning agents. In this work we formalise the logical composition of tasks as a Boolean algebra. This allows us to formulate new tasks in terms of the negation, disjunction and conjunction of a set of base tasks. We then show that by learning goal-oriented value functions and restricting the transition dynamics of the tasks, an agent can solve these new tasks with no further learning. We prove that by composing these value functions in specific ways, we immediately recover the optimal policies for all tasks expressible under the Boolean algebra. We verify our approach in two domains---including a high-dimensional video game environment requiring function approximation---where an agent first learns a set of base skills, and then composes them to solve a super-exponential number of new tasks.

While Deep Reinforcement Learning (DRL) has emerged as a promising approach to many complex tasks, it remains challenging to train a single DRL agent that is capable of undertaking multiple different continuous control tasks. In this paper, we present a Knowledge Transfer based Multi-task Deep Reinforcement Learning framework (KTM-DRL) for continuous control, which enables a single DRL agent to achieve expert-level performance in multiple different tasks by learning from task-specific teachers. In KTM-DRL, the multi-task agent first leverages an offline knowledge transfer algorithm designed particularly for the actor-critic architecture to quickly learn a control policy from the experience of task-specific teachers, and then it employs an online learning algorithm to further improve itself by learning from new online transition samples under the guidance of those teachers. We perform a comprehensive empirical study with two commonly-used benchmarks in the MuJoCo continuous control task suite. The experimental results well justify the effectiveness of KTM-DRL and its knowledge transfer and online learning algorithms, as well as its superiority over the state-of-the-art by a large margin.

Bootstrapping is a core mechanism in Reinforcement Learning (RL). Most algorithms, based on temporal differences, replace the true value of a transiting state by their current estimate of this value. Yet, another estimate could be leveraged to bootstrap RL: the current policy. Our core contribution stands in a very simple idea: adding the scaled log-policy to the immediate reward. We show that, by slightly modifying Deep Q-Network (DQN) in that way provides an agent that is competitive with the state-of-the-art Rainbow on Atari games, without making use of distributional RL, n-step returns or prioritized replay. To demonstrate the versatility of this idea, we also use it together with an Implicit Quantile Network (IQN). The resulting agent outperforms Rainbow on Atari, installing a new State of the Art with very little modifications to the original algorithm. To add to this empirical study, we provide strong theoretical insights on what happens under the hood -- implicit Kullback-Leibler regularization and increase of the action-gap.

In Meta-Reinforcement Learning (meta-RL) an agent is trained on a set of tasks to prepare for and learn faster in new, unseen, but related tasks. The training tasks are usually hand-crafted to be representative of the expected distribution of target tasks and hence all used in training. We show that given a set of training tasks, learning can be both faster and more effective (leading to better performance in the target tasks), if the training tasks are appropriately selected. We propose a task selection algorithm based on information theory, which optimizes the set of tasks used for training in meta-RL, irrespectively of how they are generated. The algorithm establishes which training tasks are both sufficiently relevant for the target tasks, and different enough from one another. We reproduce different meta-RL experiments from the literature and show that our task selection algorithm improves the final performance in all of them.

Continually solving new, unsolved tasks is the key to learning diverse behaviors. Through reinforcement learning (RL), we have made massive strides towards solving tasks that have a single goal. However, in the multi-task domain, where an agent needs to reach multiple goals, the choice of training goals can largely affect sample efficiency. When biological agents learn, there is often an organized and meaningful order to which learning happens. Inspired by this, we propose setting up an automatic curriculum for goals that the agent needs to solve. Our key insight is that if we can sample goals at the frontier of the set of goals that an agent is able to reach, it will provide a significantly stronger learning signal compared to randomly sampled goals. To operationalize this idea, we introduce a goal proposal module that prioritizes goals that maximize the epistemic uncertainty of the Q-function of the policy. This simple technique samples goals that are neither too hard nor too easy for the agent to solve, hence enabling continual improvement. We evaluate our method across 13 multi-goal robotic tasks and 5 navigation tasks, and demonstrate performance gains over current state-of-the-art methods.

The combination of deep reinforcement learning and search at both training and test time is a powerful paradigm that has led to a number of successes in single-agent settings and perfect-information games, best exemplified by AlphaZero. However, prior algorithms of this form cannot cope with imperfect-information games. This paper presents ReBeL, a general framework for self-play reinforcement learning and search that provably converges to a Nash equilibrium in any two-player zero-sum game. In the simpler setting of perfect-information games, ReBeL reduces to an algorithm similar to AlphaZero. Results in two different imperfect-information games show ReBeL converges to an approximate Nash equilibrium. We also show ReBeL achieves superhuman performance in heads-up no-limit Texas hold'em poker, while using far less domain knowledge than any prior poker AI.

Modern large scale datasets are often plagued with missing entries. For tabular data with missing values, a flurry of imputation algorithms solve for a complete matrix which minimizes some penalized reconstruction error. However, almost none of them can estimate the uncertainty of its imputations. This paper pro- poses a probabilistic and scalable framework for missing value imputation with quantified uncertainty. Our model, the Low Rank Gaussian Copula, augments a standard probabilistic model, Probabilistic Principal Component Analysis, with marginal transformations for each column that allow the model to better match the distribution of the data. It naturally handles Boolean, ordinal, and real-valued observations and quantifies the uncertainty in each imputation. The time required to fit the model scales linearly with the number of rows and the number of columns in the dataset. Empirical results show the method yields state-of-the-art imputation accuracy across a wide range of data types, including those with high rank. Our uncertainty measure predicts imputation error well: entries with lower uncertainty do have lower imputation error (on average). Moreover, for real-valued data, the resulting confidence intervals are well-calibrated.

We consider the task of automated theorem proving, a key AI task. Deep learning has shown promise for training theorem provers, but there are limited human-written theorems and proofs available for supervised learning. To address this limitation, we propose to learn a neural generator that automatically synthesizes theorems and proofs for the purpose of training a theorem prover. Experiments on real-world tasks demonstrate that synthetic data from our approach improves the theorem prover and advances the state of the art of automated theorem proving in Metamath.

In many application settings involving networks, such as messages between users of an on-line social network or transactions between traders in financial markets, the observed data consist of timestamped relational events, which form a continuous-time network. We propose the Community Hawkes Independent Pairs (CHIP) generative model for such networks. We show that applying spectral clustering to an aggregated adjacency matrix constructed from the CHIP model provides consistent community detection for a growing number of nodes and time duration. We also develop consistent and computationally efficient estimators for the model parameters. We demonstrate that our proposed CHIP model and estimation procedure scales to large networks with tens of thousands of nodes and provides superior fits than existing continuous-time network models on several real networks.

Weighted model integration (WMI) is a framework to perform advanced probabilistic inference on hybrid domains, i.e., on distributions over mixed continuous-discrete random variables and in presence of complex logical and arithmetic constraints. In this work, we advance the WMI framework on both the theoretical and algorithmic side. First, we exactly trace the boundaries of tractability for WMI inference by proving that to be amenable to exact and efficient inference a WMI problem has to posses a tree-shaped structure with logarithmic diameter. While this result deepens our theoretical understanding of WMI it hinders the practical applicability of exact WMI solvers to real-world problems. To overcome this, we propose the first approximate WMI solver that does not resort to sampling, but performs exact inference on one approximate models. Our solution performs message passing in a relaxed problem structure iteratively to recover certain lost dependencies and, as our experiments suggest, is competitive with other SOTA WMI solvers.

In this paper, we propose polynomial forms to represent distributions of state variables over time for discrete-time stochastic dynamical systems. This problem arises in a variety of applications in areas ranging from biology to robotics. Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving the state variables. First, we recall ideas from interval arithmetic, and use them to rigorously represent the state variables at time t as a function of the initial state variables and noise symbols that model the random exogenous inputs encountered before time t. Next, we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables. We demonstrate interesting applications that demonstrate how our approach can be useful in some situations to establish mathematically guaranteed bounds that are of a different nature from those obtained through simulations with pseudo-random numbers.

Discrete integration is a fundamental problem in computer science that concerns the computation of discrete sums over exponentially large sets. Despite intense interest from researchers for over three decades, the design of scalable techniques for computing estimates with rigorous guarantees for discrete integration remains the holy grail. The key contribution of this work addresses this scalability challenge via an efficient reduction of discrete integration to model counting. The proposed reduction is achieved via a significant increase in the dimensionality that, contrary to conventional wisdom, leads to solving an instance of the relatively simpler problem of model counting.

Building on the promising approach proposed by Chakraborty et al, our work overcomes the key weakness of their approach: a restriction to dyadic weights. We augment our proposed reduction, called DeWeight, with a state of the art efficient approximate model counter and perform detailed empirical analysis over benchmarks arising from neural network verification domains, an emerging application area of critical importance. DeWeight, to the best of our knowledge, is the first technique to compute estimates with provable guarantees for this class of benchmarks.

Learned neural solvers have successfully been used to solve combinatorial optimization and decision problems. More general counting variants of these problems, however, are still largely solved with hand-crafted solvers. To bridge this gap, we introduce belief propagation neural networks (BPNNs), a class of parameterized operators that operate on factor graphs and generalize Belief Propagation (BP). In its strictest form, a BPNN layer (BPNN-D) is a learned iterative operator that provably maintains many of the desirable properties of BP for any choice of the parameters. Empirically, we show that by training BPNN-D learns to perform the task better than the original BP: it converges 1.7x faster on Ising models while providing tighter bounds. On challenging model counting problems, BPNNs compute estimates 100's of times faster than state-of-the-art handcrafted methods, while returning an estimate of comparable quality.

The ability to leverage large-scale hardware parallelism has been one of the key enablers of the accelerated recent progress in machine learning. Consequently, there has been considerable effort invested into developing efficient parallel variants of classic machine learning algorithms. However, despite the wealth of knowledge on parallelization, some classic machine learning algorithms often prove hard to parallelize efficiently while maintaining convergence.

In this paper, we focus on efficient parallel algorithms for the key machine learning task of inference on graphical models, in particular on the fundamental belief propagation algorithm. We address the challenge of efficiently parallelizing this classic paradigm by showing how to leverage scalable relaxed schedulers, which reduce parallelization overheads, in this context. We investigate the overheads of relaxation analytically, and present an extensive empirical study, showing that our approach outperforms previous parallel belief propagation implementations both in terms of scalability and in terms of wall-clock convergence time, on a range of practical applications.

Inference in discrete graphical models with variational methods is difficult because of the inability to re-parameterize gradients of the Evidence Lower Bound (ELBO). Many sampling-based methods have been proposed for estimating these gradients, but they suffer from high bias or variance. In this paper, we propose a new approach that leverages the tractability of probabilistic circuit models, such as Sum Product Networks (SPN), to compute ELBO gradients exactly (without sampling) for a certain class of densities. In particular, we show that selective-SPNs are suitable as an expressive variational distribution, and prove that when the log-density of the target model is a polynomial the corresponding ELBO can be computed analytically. To scale to graphical models with thousands of variables, we develop an efficient and effective construction of selective-SPNs with size (O(kn)), where (n) is the number of variables and (k) is an adjustable hyperparameter. We demonstrate our approach on three types of graphical models -- Ising models, Latent Dirichlet Allocation, and factor graphs from the UAI Inference Competition. Selective-SPNs give a better lower bound than mean-field and structured mean-field, and is competitive with approximations that do not provide a lower bound, such as Loopy Belief Propagation and Tree-Reweighted Belief Propagation. Our …

We give methods for Bayesian inference of directed acyclic graphs, DAGs, and the induced causal effects from passively observed complete data. Our methods build on a recent Markov chain Monte Carlo scheme for learning Bayesian networks, which enables efficient approximate sampling from the graph posterior, provided that each node is assigned a small number K of candidate parents. We present algorithmic techniques to significantly reduce the space and time requirements, which make the use of substantially larger values of K feasible. Furthermore, we investigate the problem of selecting the candidate parents per node so as to maximize the covered posterior mass. Finally, we combine our sampling method with a novel Bayesian approach for estimating causal effects in linear Gaussian DAG models. Numerical experiments demonstrate the performance of our methods in detecting ancestor–descendant relations, and in causal effect estimation our Bayesian method is shown to outperform previous approaches.

This paper re-examines a continuous optimization framework dubbed NOTEARS for learning Bayesian networks. We first generalize existing algebraic characterizations of acyclicity to a class of matrix polynomials. Next, focusing on a one-parameter-per-edge setting, it is shown that the Karush-Kuhn-Tucker (KKT) optimality conditions for the NOTEARS formulation cannot be satisfied except in a trivial case, which explains a behavior of the associated algorithm. We then derive the KKT conditions for an equivalent reformulation, show that they are indeed necessary, and relate them to explicit constraints that certain edges be absent from the graph. If the score function is convex, these KKT conditions are also sufficient for local minimality despite the non-convexity of the constraint. Informed by the KKT conditions, a local search post-processing algorithm is proposed and shown to substantially and universally improve the structural Hamming distance of all tested algorithms, typically by a factor of 2 or more. Some combinations with local search are both more accurate and more efficient than the original NOTEARS.

We propose the use of hyperedge replacement graph grammars for factor graphs, or actor graph grammars (FGGs) for short. FGGs generate sets of factor graphs and can describe a more general class of models than plate notation, dynamic graphical models, case-factor diagrams, and sum-product networks can. Moreover, inference can be done on FGGs without enumerating all the generated factor graphs. For finite variable domains (but possibly infinite sets of graphs), a generalization of variable elimination to FGGs allows exact and tractable inference in many situations. For finite sets of graphs (but possibly infinite variable domains), a FGG can be converted to a single factor graph amenable to standard inference techniques.

Graphical models are useful tools for describing structured high-dimensional probability distributions. Development of efficient algorithms for learning graphical models with least amount of data remains an active research topic. Reconstruction of graphical models that describe the statistics of discrete variables is a particularly challenging problem, for which the maximum likelihood approach is intractable. In this work, we provide the first sample-efficient method based on the Interaction Screening framework that allows one to provably learn fully general discrete factor models with node-specific discrete alphabets and multi-body interactions, specified in an arbitrary basis. We identify a single condition related to model parametrization that leads to rigorous guarantees on the recovery of model structure and parameters in any error norm, and is readily verifiable for a large class of models. Importantly, our bounds make explicit distinction between parameters that are proper to the model and priors used as an input to the algorithm. Finally, we show that the Interaction Screening framework includes all models previously considered in the literature as special cases, and for which our analysis shows a systematic improvement in sample complexity.

People routinely infer the goals of others by observing their actions over time. Remarkably, we can do so even when those actions lead to failure, enabling us to assist others when we detect that they might not achieve their goals. How might we endow machines with similar capabilities? Here we present an architecture capable of inferring an agent’s goals online from both optimal and non-optimal sequences of actions. Our architecture models agents as boundedly-rational planners that interleave search with execution by replanning, thereby accounting for sub-optimal behavior. These models are specified as probabilistic programs, allowing us to represent and perform efficient Bayesian inference over an agent's goals and internal planning processes. To perform such inference, we develop Sequential Inverse Plan Search (SIPS), a sequential Monte Carlo algorithm that exploits the online replanning assumption of these models, limiting computation by incrementally extending inferred plans as new actions are observed. We present experiments showing that this modeling and inference architecture outperforms Bayesian inverse reinforcement learning baselines, accurately inferring goals from both optimal and non-optimal trajectories involving failure and back-tracking, while generalizing across domains with compositional structure and sparse rewards.

We propose a framework for solving high-dimensional Bayesian inference problems using \emph{structure-exploiting} low-dimensional transport maps or flows. These maps are confined to a low-dimensional subspace (hence, lazy), and the subspace is identified by minimizing an upper bound on the Kullback--Leibler divergence (hence, structured). Our framework provides a principled way of identifying and exploiting low-dimensional structure in an inference problem. It focuses the expressiveness of a transport map along the directions of most significant discrepancy from the posterior, and can be used to build deep compositions of lazy maps, where low-dimensional projections of the parameters are iteratively transformed to match the posterior. We prove weak convergence of the generated sequence of distributions to the posterior, and we demonstrate the benefits of the framework on challenging inference problems in machine learning and differential equations, using inverse autoregressive flows and polynomial maps as examples of the underlying density estimators.

A convolutional neural network strongly robust to adversarial perturbations at reasonable computational and performance cost has not yet been demonstrated. The primate visual ventral stream seems to be robust to small perturbations in visual stimuli but the underlying mechanisms that give rise to this robust perception are not understood. In this work, we investigate the role of two biologically plausible mechanisms in adversarial robustness. We demonstrate that the non-uniform sampling performed by the primate retina and the presence of multiple receptive fields with a range of receptive field sizes at each eccentricity improve the robustness of neural networks to small adversarial perturbations. We verify that these two mechanisms do not suffer from gradient obfuscation and study their contribution to adversarial robustness through ablation studies.

A fundamental question in multiclass classification concerns understanding the consistency properties of surrogate risk minimization algorithms, which minimize a (often convex) surrogate to the multiclass 0-1 loss. In particular, the framework of calibrated surrogates has played an important role in analyzing the Bayes consistency properties of such algorithms, i.e. in studying convergence to a Bayes optimal classifier (Zhang, 2004; Tewari and Bartlett, 2007). However, follow-up work has suggested this framework can be of limited value when studying H-consistency; in particular, concerns have been raised that even when the data comes from an underlying linear model, minimizing certain convex calibrated surrogates over linear scoring functions fails to recover the true model (Long and Servedio, 2013). In this paper, we investigate this apparent conundrum. We find that while some calibrated surrogates can indeed fail to provide H-consistency when minimized over a natural-looking but naively chosen scoring function class F, the situation can potentially be remedied by minimizing them over a more carefully chosen class of scoring functions F. In particular, for the popular one-vs-all hinge and logistic surrogates, both of which are calibrated (and therefore provide Bayes consistency) under realizable models, but were previously shown to pose problems for realizable H-consistency, we …

The Information Bottleneck (IB) objective uses information theory to formulate a task-performance versus robustness trade-off. It has been successfully applied in the standard discriminative classification setting. We pose the question whether the IB can also be used to train generative likelihood models such as normalizing flows. Since normalizing flows use invertible network architectures (INNs), they are information-preserving by construction. This seems contradictory to the idea of a bottleneck. In this work, firstly, we develop the theory and methodology of IB-INNs, a class of conditional normalizing flows where INNs are trained using the IB objective: Introducing a small amount of controlled information loss allows for an asymptotically exact formulation of the IB, while keeping the INN's generative capabilities intact. Secondly, we investigate the properties of these models experimentally, specifically used as generative classifiers. This model class offers advantages such as improved uncertainty quantification and out-of-distribution detection, but traditional generative classifier solutions suffer considerably in classification accuracy. We find the trade-off parameter in the IB controls a mix of generative capabilities and accuracy close to standard classifiers. Empirically, our uncertainty estimates in this mixed regime compare favourably to conventional generative and discriminative classifiers. Code is provided in the supplement.

Stationary stochastic processes (SPs) are a key component of many probabilistic models, such as those for off-the-grid spatio-temporal data. They enable the statistical symmetry of underlying physical phenomena to be leveraged, thereby aiding generalization. Prediction in such models can be viewed as a translation equivariant map from observed data sets to predictive SPs, emphasizing the intimate relationship between stationarity and equivariance. Building on this, we propose the Convolutional Neural Process (ConvNP), which endows Neural Processes (NPs) with translation equivariance and extends convolutional conditional NPs to allow for dependencies in the predictive distribution. The latter enables ConvNPs to be deployed in settings which require coherent samples, such as Thompson sampling or conditional image completion. Moreover, we propose a new maximum-likelihood objective to replace the standard ELBO objective in NPs, which conceptually simplifies the framework and empirically improves performance. We demonstrate the strong performance and generalization capabilities of ConvNPs on 1D regression, image completion, and various tasks with real-world spatio-temporal data.

Miscalibration -- a mismatch between a model's confidence and its correctness -- of Deep Neural Networks (DNNs) makes their predictions hard to rely on. Ideally, we want networks to be accurate, calibrated and confident. We show that, as opposed to the standard cross-entropy loss, focal loss (Lin et al., 2017) allows us to learn models that are already very well calibrated. When combined with temperature scaling, whilst preserving accuracy, it yields state-of-the-art calibrated models. We provide a thorough analysis of the factors causing miscalibration, and use the insights we glean from this to justify the empirically excellent performance of focal loss. To facilitate the use of focal loss in practice, we also provide a principled approach to automatically select the hyperparameter involved in the loss function. We perform extensive experiments on a variety of computer vision and NLP datasets, and with a wide variety of network architectures, and show that our approach achieves state-of-the-art calibration without compromising on accuracy in almost all cases. Code is available at https://github.com/torrvision/focal_calibration.

We study three notions of uncertainty quantification---calibration, confidence intervals and prediction sets---for binary classification in the distribution-free setting, that is without making any distributional assumptions on the data. With a focus towards calibration, we establish a 'tripod' of theorems that connect these three notions for score-based classifiers. A direct implication is that distribution-free calibration is only possible, even asymptotically, using a scoring function whose level sets partition the feature space into at most countably many sets. Parametric calibration schemes such as variants of Platt scaling do not satisfy this requirement, while nonparametric schemes based on binning do. To close the loop, we derive distribution-free confidence intervals for binned probabilities for both fixed-width and uniform-mass binning. As a consequence of our 'tripod' theorems, these confidence intervals for binned probabilities lead to distribution-free calibration. We also derive extensions to settings with streaming data and covariate shift.

Distribution alignment has many applications in deep learning, including domain adaptation and unsupervised image-to-image translation. Most prior work on unsupervised distribution alignment relies either on minimizing simple non-parametric statistical distances such as maximum mean discrepancy or on adversarial alignment. However, the former fails to capture the structure of complex real-world distributions, while the latter is difficult to train and does not provide any universal convergence guarantees or automatic quantitative validation procedures. In this paper, we propose a new distribution alignment method based on a log-likelihood ratio statistic and normalizing flows. We show that, under certain assumptions, this combination yields a deep neural likelihood-based minimization objective that attains a known lower bound upon convergence. We experimentally verify that minimizing the resulting objective results in domain alignment that preserves the local structure of input domains.

Reasoning over an instance composed of a set of vectors, like a point cloud, requires that one accounts for intra-set dependent features among elements. However, since such instances are unordered, the elements' features should remain unchanged when the input's order is permuted. This property, permutation equivariance, is a challenging constraint for most neural architectures. While recent work has proposed global pooling and attention-based solutions, these may be limited in the way that intradependencies are captured in practice. In this work we propose a more general formulation to achieve permutation equivariance through ordinary differential equations (ODE). Our proposed module, Exchangeable Neural ODE (ExNODE), can be seamlessly applied for both discriminative and generative tasks. We also extend set modeling in the temporal dimension and propose a VAE based model for temporal set modeling. Extensive experiments demonstrate the efficacy of our method over strong baselines.

Deep generative networks trained via maximum likelihood on a natural image dataset like CIFAR10 often assign high likelihoods to images from datasets with different objects (e.g., SVHN). We refine previous investigations of this failure at anomaly detection for invertible generative networks and provide a clear explanation of it as a combination of model bias and domain prior: Convolutional networks learn similar low-level feature distributions when trained on any natural image dataset and these low-level features dominate the likelihood. Hence, when the discriminative features between inliers and outliers are on a high-level, e.g., object shapes, anomaly detection becomes particularly challenging. To remove the negative impact of model bias and domain prior on detecting high-level differences, we propose two methods, first, using the log likelihood ratios of two identical models, one trained on the in-distribution data (e.g., CIFAR10) and the other one on a more general distribution of images (e.g., 80 Million Tiny Images). We also derive a novel outlier loss for the in-distribution network on samples from the more general distribution to further improve the performance. Secondly, using a multi-scale model like Glow, we show that low-level features are mainly captured at early scales. Therefore, using only the likelihood contribution of …

The recent, counter-intuitive discovery that deep generative models (DGMs) can frequently assign a higher likelihood to outliers has implications for both outlier detection applications as well as our overall understanding of generative modeling. In this work, we present a possible explanation for this phenomenon, starting from the observation that a model's typical set and high-density region may not conincide. From this vantage point we propose a novel outlier test, the empirical success of which suggests that the failure of existing likelihood-based outlier tests does not necessarily imply that the corresponding generative model is uncalibrated. We also conduct additional experiments to help disentangle the impact of low-level texture versus high-level semantics in differentiating outliers. In aggregate, these results suggest that modifications to the standard evaluation practices and benchmarks commonly applied in the literature are needed.

There is a growing interest in societal concerns in machine learning systems, especially in fairness. Multicalibration gives a comprehensive methodology to address group fairness. In this work, we address the multicalibration error and decouple it from the prediction error. The importance of decoupling the fairness metric (multicalibration) and the accuracy (prediction error) is due to the inherent trade-off between the two, and the societal decision regarding the ``right tradeoff'' (as imposed many times by regulators). Our work gives sample complexity bounds for uniform convergence guarantees of multicalibration error, which implies that regardless of the accuracy, we can guarantee that the empirical and (true) multicalibration errors are close. We emphasize that our results: (1) are more general than previous bounds, as they apply to both agnostic and realizable settings, and do not rely on a specific type of algorithm (such as differentially private), (2) improve over previous multicalibration sample complexity bounds and (3) implies uniform convergence guarantees for the classical calibration error.

Group fairness is measured via parity of quantitative metrics across different protected demographic groups. In this paper, we investigate the problem of reliably assessing group fairness metrics when labeled examples are few but unlabeled examples are plentiful. We propose a general Bayesian framework that can augment labeled data with unlabeled data to produce more accurate and lower-variance estimates compared to methods based on labeled data alone. Our approach estimates calibrated scores (for unlabeled examples) of each group using a hierarchical latent variable model conditioned on labeled examples. This in turn allows for inference of posterior distributions for an array of group fairness metrics with a notion of uncertainty. We demonstrate that our approach leads to significant and consistent reductions in estimation error across multiple well-known fairness datasets, sensitive attributes, and predictive models. The results clearly show the benefits of using both unlabeled data and Bayesian inference in assessing whether a prediction model is fair or not.

Among existing uncertainty estimation approaches, Dirichlet Prior Network (DPN) distinctly models different predictive uncertainty types. However, for in-domain examples with high data uncertainties among multiple classes, even a DPN model often produces indistinguishable representations from the out-of-distribution (OOD) examples, compromising their OOD detection performance. We address this shortcoming by proposing a novel loss function for DPN to maximize the representation gap between in-domain and OOD examples. Experimental results demonstrate that our proposed approach consistently improves OOD detection performance.

We study the problem of reducing adversarially robust learning to standard PAC learning, i.e. the complexity of learning adversarially robust predictors using access to only a black-box non-robust learner. We give a reduction that can robustly learn any hypothesis class C using any non-robust learner A for C. The number of calls to A depends logarithmically on the number of allowed adversarial perturbations per example, and we give a lower bound showing this is unavoidable.

Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees, based on solving large semidefinite programs which are impractical to run, or make strong assumptions such as requiring the factors to be nearly orthogonal. In this paper we introduce a new variant of alternating minimization, which in turn is inspired by understanding how the progress measures that guide convergence of alternating minimization in the matrix setting need to be adapted to the tensor setting. We show strong provable guarantees, including showing that our algorithm converges linearly to the true tensors even when the factors are highly correlated and can be implemented in nearly linear time. Moreover our algorithm is also highly practical and we show that we can complete third order tensors with a thousand dimensions from observing a tiny fraction of its entries. In contrast, and somewhat surprisingly, we show that the standard version of alternating minimization, without our new twist, can converge at a drastically slower rate in practice.

We give an online algorithm and prove novel mistake and regret bounds for online binary matrix completion with side information. The mistake bounds we prove are of the form \tilde{O}(D/\gamma^2). The term 1/\gamma^2 is analogous to the usual margin term in SVM (perceptron) bounds. More specifically, if we assume that there is some factorization of the underlying m x n matrix into PQ^T, where the rows of P are interpreted as "classifiers" in R^d and the rows of Q as "instances" in R^d, then gamma is the maximum (normalized) margin over all factorizations PQ^T consistent with the observed matrix. The quasi-dimension term D measures the quality of side information. In the presence of vacuous side information, D = m+n. However, if the side information is predictive of the underlying factorization of the matrix, then in an ideal case, D \in O(k + l) where k is the number of distinct row factors and l is the number of distinct column factors. We additionally provide a generalization of our algorithm to the inductive setting. In this setting, we provide an example where the side information is not directly specified in advance. For this example, the quasi-dimension D is now bounded by …

As in standard linear regression, in truncated linear regression, we are given access to observations (Ai, yi)i whose dependent variable equals yi= Ai^{\rm T} \cdot x^* + \etai, where x^* is some fixed unknown vector of interest and \etai is independent noise; except we are only given an observation if its dependent variable yi lies in some "truncation set" S \subset \mathbb{R}. The goal is to recover x^* under some favorable conditions on the Ai's and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering k-sparse n-dimensional vectors x^* from m truncated samples, which attains an optimal \ell2 reconstruction error of O(\sqrt{(k \log n)/m}). As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, such as that which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaption of the LASSO optimization problem that accommodates truncation. This generalizes the works of both: (1) [Daskalakis et al. 2018], where no regularization is needed due to the low dimensionality of …

When analyzing communities of microorganisms from their sequenced DNA, an important task is taxonomic profiling: enumerating the presence and relative abundance of all organisms, or merely of all taxa, contained in the sample. This task can be tackled via compressive-sensing-based approaches, which favor communities featuring the fewest organisms among those consistent with the observed DNA data. Despite their successes, these parsimonious approaches sometimes conflict with biological realism by overlooking organism similarities. Here, we leverage a recently developed notion of biological diversity that simultaneously accounts for organism similarities and retains the optimization strategy underlying compressive-sensing-based approaches. We demonstrate that minimizing biological diversity still produces sparse taxonomic profiles and we experimentally validate superiority to existing compressive-sensing-based approaches. Despite showing that the objective function is almost never convex and often concave, generally yielding NP-hard problems, we exhibit ways of representing organism similarities for which minimizing diversity can be performed via a sequence of linear programs guaranteed to decrease diversity. Better yet, when biological similarity is quantified by k-mer co-occurrence (a popular notion in bioinformatics), minimizing diversity actually reduces to one linear program that can utilize multiple k-mer sizes to enhance performance. In proof-of-concept experiments, we verify that the latter procedure can lead to …

Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks). It is an open question whether norms can explain the implicit regularization in matrix factorization. The current paper resolves this open question in the negative, by proving that there exist natural matrix factorization problems on which the implicit regularization drives all norms (and quasi-norms) towards infinity. Our results suggest that, rather than perceiving the implicit regularization via norms, a potentially more useful interpretation is minimization of rank. We demonstrate empirically that this interpretation extends to a certain class of non-linear neural networks, and hypothesize that it may be key to explaining generalization in deep learning.

The robustness of a neural network to adversarial examples can be provably certified by solving a convex relaxation. If the relaxation is loose, however, then the resulting certificate can be too conservative to be practically useful. Recently, a less conservative robustness certificate was proposed, based on a semidefinite programming (SDP) relaxation of the ReLU activation function. In this paper, we describe a geometric technique that determines whether this SDP certificate is exact, meaning whether it provides both a lower-bound on the size of the smallest adversarial perturbation, as well as a globally optimal perturbation that attains the lower-bound. Concretely, we show, for a least-squares restriction of the usual adversarial attack problem, that the SDP relaxation amounts to the nonconvex projection of a point onto a hyperbola. The resulting SDP certificate is exact if and only if the projection of the point lies on the major axis of the hyperbola. Using this geometric technique, we prove that the certificate is exact over a single hidden layer under mild assumptions, and explain why it is usually conservative for several hidden layers. We experimentally confirm our theoretical insights using a general-purpose interior-point method and a custom rank-2 Burer-Monteiro algorithm.

Understanding of GAN training is still very limited. One major challenge is its non-convex-non-concave min-max objective, which may lead to sub-optimal local minima. In this work, we perform a global landscape analysis of the empirical loss of GANs. We prove that a class of separable-GAN, including the original JS-GAN, has exponentially many bad basins which are perceived as mode-collapse. We also study the relativistic pairing GAN (RpGAN) loss which couples the generated samples and the true samples. We prove that RpGAN has no bad basins. Experiments on synthetic data show that the predicted bad basin can indeed appear in training. We also perform experiments to support our theory that RpGAN has a better landscape than separable-GAN. For instance, we empirically show that RpGAN performs better than separable-GAN with relatively narrow neural nets. The code is available at \url{https://github.com/AilsaF/RS-GAN}.

We develop a framework that leverages the smoothed complexity analysis by Spielman and Teng to circumvent paradoxes and impossibility theorems in social choice, motivated by modern applications of social choice powered by AI and ML. For Condrocet’s paradox, we prove that the smoothed likelihood of the paradox either vanishes at an exponential rate as the number of agents increases, or does not vanish at all. For the ANR impossibility on the non-existence of voting rules that simultaneously satisfy anonymity, neutrality, and resolvability, we characterize the rate for the impossibility to vanish, to be either polynomially fast or exponentially fast. We also propose a novel easy-to-compute tie-breaking mechanism that optimally preserves anonymity and neutrality for even number of alternatives in natural settings. Our results illustrate the smoothed possibility of social choice—even though the paradox and the impossibility theorem hold in the worst case, they may not be a big concern in practice.

Recent results in the ML community have revealed that learning algorithms used to compute the optimal strategy for the leader to commit to in a Stackelberg game, are susceptible to manipulation by the follower. Such a learning algorithm operates by querying the best responses or the payoffs of the follower, who consequently can deceive the algorithm by responding as if their payoffs were much different than what they actually are. For this strategic behavior to be successful, the main challenge faced by the follower is to pinpoint the payoffs that would make the learning algorithm compute a commitment so that best responding to it maximizes the follower's utility, according to the true payoffs. While this problem has been considered before, the related literature only focused on the simplified scenario in which the payoff space is finite, thus leaving the general version of the problem unanswered. In this paper, we fill this gap by showing that it is always possible for the follower to efficiently compute (near-)optimal payoffs for various scenarios of learning interaction between the leader and the follower.

We study the dynamics of optimization and the generalization properties of one-hidden layer neural networks with quadratic activation function in the overparametrized regime where the layer width m is larger than the input dimension d.

We consider a teacher-student scenario where the teacher has the same structure as the student with a hidden layer of smaller width m*<=m.

We describe how the empirical loss landscape is affected by the number n of data samples and the width m* of the teacher network. In particular we determine how the probability that there be no spurious minima on the empirical loss depends on n, d, and m*, thereby establishing conditions under which the neural network can in principle recover the teacher.

We also show that under the same conditions gradient descent dynamics on the empirical loss converges and leads to small generalization error, i.e. it enables recovery in practice.

Finally we characterize the time-convergence rate of gradient descent in the limit of a large number of samples.

These results are confirmed by numerical experiments.

Recently, there have been significant interests in studying the so-called "double-descent" of the generalization error of linear regression models under the overparameterized and overfitting regime, with the hope that such analysis may provide the first step towards understanding why overparameterized deep neural networks (DNN) still generalize well. However, to date most of these studies focused on the min L2-norm solution that overfits the data. In contrast, in this paper we study the overfitting solution that minimizes the L1-norm, which is known as Basis Pursuit (BP) in the compressed sensing literature. Under a sparse true linear regression model with p i.i.d. Gaussian features, we show that for a large range of p up to a limit that grows exponentially with the number of samples n, with high probability the model error of BP is upper bounded by a value that decreases with p. To the best of our knowledge, this is the first analytical result in the literature establishing the double-descent of overfitting BP for finite n and p. Further, our results reveal significant differences between the double-descent of BP and min L2-norm solutions. Specifically, the double-descent upper-bound of BP is independent of the signal strength, and for high SNR and …

Humans easily interpret expressions that describe unfamiliar situations composed from familiar parts ("greet the pink brontosaurus by the ferris wheel"). Modern neural networks, by contrast, struggle to interpret novel compositions. In this paper, we introduce a new benchmark, gSCAN, for evaluating compositional generalization in situated language understanding. Going beyond a related benchmark that focused on syntactic aspects of generalization, gSCAN defines a language grounded in the states of a grid world, facilitating novel evaluations of acquiring linguistically motivated rules. For example, agents must understand how adjectives such as 'small' are interpreted relative to the current world state or how adverbs such as 'cautiously' combine with new verbs. We test a strong multi-modal baseline model and a state-of-the-art compositional method finding that, in most cases, they fail dramatically when generalization requires systematic compositional rules.

Despite being the workhorse of deep learning, the backpropagation algorithm is no panacea. It enforces sequential layer updates, thus preventing efficient parallelization of the training process. Furthermore, its biological plausibility is being challenged. Alternative schemes have been devised; yet, under the constraint of synaptic asymmetry, none have scaled to modern deep learning tasks and architectures. Here, we challenge this perspective, and study the applicability of Direct Feedback Alignment (DFA) to neural view synthesis, recommender systems, geometric learning, and natural language processing. In contrast with previous studies limited to computer vision tasks, our findings show that it successfully trains a large range of state-of-the-art deep learning architectures, with performance close to fine-tuned backpropagation. When a larger gap between DFA and backpropagation exists, like in Transformers, we attribute this to a need to rethink common practices for large and complex architectures. At variance with common beliefs, our work supports that challenging tasks can be tackled in the absence of weight transport.

The state-of-the art machine learning approach to training deep neural networks, backpropagation, is implausible for real neural networks: neurons need to know their outgoing weights; training alternates between a bottom-up forward pass (computation) and a top-down backward pass (learning); and the algorithm often needs precise labels of many data points. Biologically plausible approximations to backpropagation, such as feedback alignment, solve the weight transport problem, but not the other two. Thus, fully biologically plausible learning rules have so far remained elusive. Here we present a family of learning rules that does not suffer from any of these problems. It is motivated by the information bottleneck principle (extended with kernel methods), in which networks learn to compress the input as much as possible without sacrificing prediction of the output. The resulting rules have a 3-factor Hebbian structure: they require pre- and post-synaptic firing rates and an error signal - the third factor - consisting of a global teaching signal and a layer-specific term, both available without a top-down pass. They do not require precise labels; instead, they rely on the similarity between pairs of desired outputs. Moreover, to obtain good performance on hard problems and retain biological plausibility, our rules need divisive …

Lifelong learning and adaptability are two defining aspects of biological agents. Modern reinforcement learning (RL) approaches have shown significant progress in solving complex tasks, however once training is concluded, the found solutions are typically static and incapable of adapting to new information or perturbations. While it is still not completely understood how biological brains learn and adapt so efficiently from experience, it is believed that synaptic plasticity plays a prominent role in this process. Inspired by this biological mechanism, we propose a search method that, instead of optimizing the weight parameters of neural networks directly, only searches for synapse-specific Hebbian learning rules that allow the network to continuously self-organize its weights during the lifetime of the agent. We demonstrate our approach on several reinforcement learning tasks with different sensory modalities and more than 450K trainable plasticity parameters. We find that starting from completely random weights, the discovered Hebbian rules enable an agent to navigate a dynamical 2D-pixel environment; likewise they allow a simulated 3D quadrupedal robot to learn how to walk while adapting to morphological damage not seen during training and in the absence of any explicit reward or error signal in less than 100 timesteps.

Primate vision depends on recurrent processing for reliable perception. A growing body of literature also suggests that recurrent connections improve the learning efficiency and generalization of vision models on classic computer vision challenges. Why then, are current large-scale challenges dominated by feedforward networks? We posit that the effectiveness of recurrent vision models is bottlenecked by the standard algorithm used for training them, "back-propagation through time" (BPTT), which has O(N) memory-complexity for training an N step model. Thus, recurrent vision model design is bounded by memory constraints, forcing a choice between rivaling the enormous capacity of leading feedforward models or trying to compensate for this deficit through granular and complex dynamics. Here, we develop a new learning algorithm, "contractor recurrent back-propagation" (C-RBP), which alleviates these issues by achieving constant O(1) memory-complexity with steps of recurrent processing. We demonstrate that recurrent vision models trained with C-RBP can detect long-range spatial dependencies in a synthetic contour tracing task that BPTT-trained models cannot. We further show that recurrent vision models trained with C-RBP to solve the large-scale Panoptic Segmentation MS-COCO challenge outperform the leading feedforward approach, with fewer free parameters. C-RBP is a general-purpose learning algorithm for any application that can benefit from expansive …

The brain modifies its synaptic strengths during learning in order to better adapt to its environment. However, the underlying plasticity rules that govern learning are unknown. Many proposals have been suggested, including Hebbian mechanisms, explicit error backpropagation, and a variety of alternatives. It is an open question as to what specific experimental measurements would need to be made to determine whether any given learning rule is operative in a real biological system. In this work, we take a "virtual experimental" approach to this problem. Simulating idealized neuroscience experiments with artificial neural networks, we generate a large-scale dataset of learning trajectories of aggregate statistics measured in a variety of neural network architectures, loss functions, learning rule hyperparameters, and parameter initializations. We then take a discriminative approach, training linear and simple non-linear classifiers to identify learning rules from features based on these observables. We show that different classes of learning rules can be separated solely on the basis of aggregate statistics of the weights, activations, or instantaneous layer-wise activity changes, and that these results generalize to limited access to the trajectory and held-out architectures and learning curricula. We identify the statistics of each observable that are most relevant for rule identification, finding …

Active inference is a Bayesian framework for understanding biological intelligence. The underlying theory brings together perception and action under one single imperative: minimizing free energy. However, despite its theoretical utility in explaining intelligence, computational implementations have been restricted to low-dimensional and idealized situations. In this paper, we present a neural architecture for building deep active inference agents operating in complex, continuous state-spaces using multiple forms of Monte-Carlo (MC) sampling. For this, we introduce a number of techniques, novel to active inference. These include: i) selecting free-energy-optimal policies via MC tree search, ii) approximating this optimal policy distribution via a feed-forward `habitual' network, iii) predicting future parameter belief updates using MC dropouts and, finally, iv) optimizing state transition precision (a high-end form of attention). Our approach enables agents to learn environmental dynamics efficiently, while maintaining task performance, in relation to reward-based counterparts. We illustrate this in a new toy environment, based on the dSprites data-set, and demonstrate that active inference agents automatically create disentangled representations that are apt for modeling state transitions. In a more complex Animal-AI environment, our agents (using the same neural architecture) are able to simulate future state transitions and actions (i.e., plan), to evince reward-directed navigation - …

Many aspects of human reasoning, including language, require learning rules from very little data. Humans can do this, often learning systematic rules from very few examples, and combining these rules to form compositional rule-based systems. Current neural architectures, on the other hand, often fail to generalize in a compositional manner, especially when evaluated in ways that vary systematically from training. In this work, we present a neuro-symbolic model which learns entire rule systems from a small set of examples. Instead of directly predicting outputs from inputs, we train our model to induce the explicit system of rules governing a set of previously seen examples, drawing upon techniques from the neural program synthesis literature. Our rule-synthesis approach outperforms neural meta-learning techniques in three domains: an artificial instruction-learning domain used to evaluate human learning, the SCAN challenge datasets, and learning rule-based translations of number words into integers for a wide range of human languages.

Recent theoretical and experimental results suggest that the dopamine system implements distributional temporal difference backups, allowing learning of the entire distributions of the long-run values of states rather than just their expected values. However, the distributional codes explored so far rely on a complex imputation step which crucially relies on spatial non-locality: in order to compute reward prediction errors, units must know not only their own state but also the states of the other units. It is far from clear how these steps could be implemented in realistic neural circuits. Here, we introduce the Laplace code: a local temporal difference code for distributional reinforcement learning that is representationally powerful and computationally straightforward. The code decomposes value distributions and prediction errors across three separated dimensions: reward magnitude (related to distributional quantiles), temporal discounting (related to the Laplace transform of future rewards) and time horizon (related to eligibility traces). Besides lending itself to a local learning rule, the decomposition recovers the temporal evolution of the immediate reward distribution, indicating all possible rewards at all future times. This increases representational capacity and allows for temporally-flexible computations that immediately adjust to changing horizons or discount factors.

How do animals learn? This remains an elusive question in neuroscience. Whereas reinforcement learning often focuses on the design of algorithms that enable artificial agents to efficiently learn new tasks, here we develop a modeling framework to directly infer the empirical learning rules that animals use to acquire new behaviors. Our method efficiently infers the trial-to-trial changes in an animal’s policy, and decomposes those changes into a learning component and a noise component. Specifically, this allows us to: (i) compare different learning rules and objective functions that an animal may be using to update its policy; (ii) estimate distinct learning rates for different parameters of an animal’s policy; (iii) identify variations in learning across cohorts of animals; and (iv) uncover trial-to-trial changes that are not captured by normative learning rules. After validating our framework on simulated choice data, we applied our model to data from rats and mice learning perceptual decision-making tasks. We found that certain learning rules were far more capable of explaining trial-to-trial changes in an animal's policy. Whereas the average contribution of the conventional REINFORCE learning rule to the policy update for mice learning the International Brain Laboratory's task was just 30%, we found that adding baseline …

Learning latent features from time series data is an important problem in both machine learning and brain function. One approach, called Slow Feature Analysis (SFA), leverages the slowness of many salient features relative to the rapidly varying input signals. Furthermore, when trained on naturalistic stimuli, SFA reproduces interesting properties of cells in the primary visual cortex and hippocampus, suggesting that the brain uses temporal slowness as a computational principle for learning latent features. However, despite the potential relevance of SFA for modeling brain function, there is currently no SFA algorithm with a biologically plausible neural network implementation, by which we mean an algorithm operates in the online setting and can be mapped onto a neural network with local synaptic updates. In this work, starting from an SFA objective, we derive an SFA algorithm, called Bio-SFA, with a biologically plausible neural network implementation. We validate Bio-SFA on naturalistic stimuli.

In real-world settings, we repeatedly decide whether to pursue better conditions or to keep things unchanged. Examples include time investment, employment, entertainment preferences etc. How do we make such decisions? To address this question, the field of behavioral ecology has developed foraging paradigms – the model settings in which human and non-human subjects decided when to leave depleting food resources. Foraging theory, represented by the marginal value theorem (MVT), provided accurate average-case stay-or-leave rules consistent with behaviors of subjects towards depleting resources. Yet, the algorithms underlying individual choices and ways to learn such algorithms remained unclear. In this work, we build interpretable deep actor-critic models to show that R-learning – a reinforcement learning (RL) approach balancing short-term and long-term rewards – is consistent with the way real-life agents may learn making stay-or-leave decisions. Specifically we show that deep R-learning predicts choice patterns consistent with behavior of mice in foraging tasks; its TD error, the training signal in our model, correlates with dopamine activity of ventral tegmental area (VTA) neurons in the brain. Our theoretical and experimental results show that deep R-learning agents leave depleting reward resources when reward intake rates fall below their exponential averages over past trials. This individual-case …

Noninvasive behavioral tracking of animals is crucial for many scientific investigations. Recent transfer learning approaches for behavioral tracking have considerably advanced the state of the art. Typically these methods treat each video frame and each object to be tracked independently. In this work, we improve on these methods (particularly in the regime of few training labels) by leveraging the rich spatiotemporal structures pervasive in behavioral video --- specifically, the spatial statistics imposed by physical constraints (e.g., paw to elbow distance), and the temporal statistics imposed by smoothness from frame to frame. We propose a probabilistic graphical model built on top of deep neural networks, Deep Graph Pose (DGP), to leverage these useful spatial and temporal constraints, and develop an efficient structured variational approach to perform inference in this model. The resulting semi-supervised model exploits both labeled and unlabeled frames to achieve significantly more accurate and robust tracking while requiring users to label fewer training frames. In turn, these tracking improvements enhance performance on downstream applications, including robust unsupervised segmentation of behavioral syllables,'' and estimation of interpretabledisentangled'' low-dimensional representations of the full behavioral video. Open source code is available at \href{\CodeLink}{https://github.com/paninski-lab/deepgraphpose}.

How are sensory representations learned via experience? Deep learning offers a theoretical toolkit for studying how neural codes emerge under different learning rules. Studies suggesting that representations in deep networks resemble those in biological brains have mostly relied on one specific learning rule: gradient descent, the workhorse behind modern deep learning. However, it remains unclear how robust these emergent representations in deep networks are to this specific choice of learning algorithm. Here we present a continuous two-dimensional space of candidate learning rules, parameterized by levels of top-down feedback and Hebbian learning. We show that this space contains five important candidate learning algorithms as specific points--Gradient Descent, Contrastive Hebbian, quasi-Predictive Coding, Hebbian & Anti-Hebbian. Next, we exhaustively characterize the properties of each rule during learning about hierarchically structured data, and identify zones within this space where deep networks exhibit qualitative signatures of biological learning. We find that while a large set of algorithms achieve zero training error at convergence, only a subset show hallmarks of human semantic development like progressive differentiation and illusory correlations. Further, only a subset adjust intermediate neural representations toward task-relevant representations, indicative of backpropagation-like behavior. Finally, we show that algorithms can dramatically differ in their learned neural …

To guide behavior, the brain extracts relevant features from high-dimensional data streamed by sensory organs. Neuroscience experiments demonstrate that the processing of sensory inputs by cortical neurons is modulated by instructive signals which provide context and task-relevant information. Here, adopting a normative approach, we model these instructive signals as supervisory inputs guiding the projection of the feedforward data. Mathematically, we start with a family of Reduced-Rank Regression (RRR) objective functions which include Reduced Rank (minimum) Mean Square Error (RRMSE) and Canonical Correlation Analysis (CCA), and derive novel offline and online optimization algorithms, which we call Bio-RRR. The online algorithms can be implemented by neural networks whose synaptic learning rules resemble calcium plateau potential dependent plasticity observed in the cortex. We detail how, in our model, the calcium plateau potential can be interpreted as a backpropagating error signal. We demonstrate that, despite relying exclusively on biologically plausible local learning rules, our algorithms perform competitively with existing implementations of RRMSE and CCA.

The search for biologically faithful synaptic plasticity rules has resulted in a large body of models. They are usually inspired by -- and fitted to -- experimental data, but they rarely produce neural dynamics that serve complex functions. These failures suggest that current plasticity models are still under-constrained by existing data. Here, we present an alternative approach that uses meta-learning to discover plausible synaptic plasticity rules. Instead of experimental data, the rules are constrained by the functions they implement and the structure they are meant to produce. Briefly, we parameterize synaptic plasticity rules by a Volterra expansion and then use supervised learning methods (gradient descent or evolutionary strategies) to minimize a problem-dependent loss function that quantifies how effectively a candidate plasticity rule transforms an initially random network into one with the desired function. We first validate our approach by re-discovering previously described plasticity rules, starting at the single-neuron level and ``Oja’s rule'', a simple Hebbian plasticity rule that captures the direction of most variability of inputs to a neuron (i.e., the first principal component). We expand the problem to the network level and ask the framework to find Oja’s rule together with an anti-Hebbian rule such that an initially random …

Randomized smoothing has been shown to provide good certified-robustness guarantees for high-dimensional classification problems. It uses the probabilities of predicting the top two most-likely classes around an input point under a smoothing distribution to generate a certified radius for a classifier's prediction. However, most smoothing methods do not give us any information about the \emph{confidence} with which the underlying classifier (e.g., deep neural network) makes a prediction. In this work, we propose a method to generate certified radii for the prediction confidence of the smoothed classifier. We consider two notions for quantifying confidence: average prediction score of a class and the margin by which the average prediction score of one class exceeds that of another. We modify the Neyman-Pearson lemma (a key theorem in randomized smoothing) to design a procedure for computing the certified radius where the confidence is guaranteed to stay above a certain threshold. Our experimental results on CIFAR-10 and ImageNet datasets show that using information about the distribution of the confidence scores allows us to achieve a significantly better certified radius than ignoring it. Thus, we demonstrate that extra information about the base classifier at the input point can help improve certified guarantees for the smoothed classifier. …

Perturbations targeting the graph structure have proven to be extremely effective in reducing the performance of Graph Neural Networks (GNNs), and traditional defenses such as adversarial training do not seem to be able to improve robustness. This work is motivated by the observation that adversarially injected edges effectively can be viewed as additional samples to a node's neighborhood aggregation function, which results in distorted aggregations accumulating over the layers. Conventional GNN aggregation functions, such as a sum or mean, can be distorted arbitrarily by a single outlier. We propose a robust aggregation function motivated by the field of robust statistics. Our approach exhibits the largest possible breakdown point of 0.5, which means that the bias of the aggregation is bounded as long as the fraction of adversarial edges of a node is less than 50%. Our novel aggregation function, Soft Medoid, is a fully differentiable generalization of the Medoid and therefore lends itself well for end-to-end deep learning. Equipping a GNN with our aggregation improves the robustness with respect to structure perturbations on Cora ML by a factor of 3 (and 5.5 on Citeseer) and by a factor of 8 for low-degree nodes.

We propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, we show differentiability with respect to layer inputs, for a general persistent homology with arbitrary filtration. Thus, our proposed layer can be placed anywhere in the network and feed critical information on the topological features of input data into subsequent layers to improve the learnability of the networks toward a given task. A task-optimal structure of PLLay is learned during training via backpropagation, without requiring any input featurization or data preprocessing. We provide a novel adaptation for the DTM function-based filtration, and show that the proposed layer is robust against noise and outliers through a stability analysis. We demonstrate the effectiveness of our approach by classification experiments on various datasets.

We propose a novel learning framework based on neural mean-field dynamics for inference and estimation problems of diffusion on networks. Our new framework is derived from the Mori-Zwanzig formalism to obtain an exact evolution of the node infection probabilities, which renders a delay differential equation with memory integral approximated by learnable time convolution operators, resulting in a highly structured and interpretable RNN. Directly using cascade data, our framework can jointly learn the structure of the diffusion network and the evolution of infection probabilities, which are cornerstone to important downstream applications such as influence maximization. Connections between parameter learning and optimal control are also established. Empirical study shows that our approach is versatile and robust to variations of the underlying diffusion network models, and significantly outperform existing approaches in accuracy and efficiency on both synthetic and real-world data.

We propose a general physics-based deep learning architecture for wave-based imaging problems. A key difficulty in imaging problems with a varying background wave speed is that the medium ``bends'' the waves differently depending on their position and direction. This space-bending geometry makes the equivariance to translations of convolutional networks an undesired inductive bias. We build an interpretable neural architecture inspired by Fourier integral operators (FIOs) which approximate the wave physics. FIOs model a wide range of imaging modalities, from seismology and radar to Doppler and ultrasound. We focus on learning the geometry of wave propagation captured by FIOs, which is implicit in the data, via a loss based on optimal transport. The proposed FIONet performs significantly better than the usual baselines on a number of imaging inverse problems, especially in out-of-distribution tests.

We develop Neuron Shapley as a new framework to quantify the contribution of individual neurons to the prediction and performance of a deep network. By accounting for interactions across neurons, Neuron Shapley is more effective in identifying important filters compared to common approaches based on activation patterns. Interestingly, removing just 30 filters with the highest Shapley scores effectively destroys the prediction accuracy of Inception-v3 on ImageNet. Visualization of these few critical filters provides insights into how the network functions. Neuron Shapley is a flexible framework and can be applied to identify responsible neurons in many tasks. We illustrate additional applications of identifying filters that are responsible for biased prediction in facial recognition and filters that are vulnerable to adversarial attacks. Removing these filters is a quick way to repair models. Computing exact Shapley values is computationally infeasible and therefore sampling-based approximations are used in practice. We introduce a new multi-armed bandit algorithm that is able to efficiently  detect neurons with the largest Shapley value orders of magnitude faster than existing Shapley value approximation methods.

Graph convolution networks (GCNs) have become effective models for graph classification. Similar to many deep networks, GCNs are vulnerable to adversarial attacks on graph topology and node attributes. Recently, a number of effective attack and defense algorithms have been developed, but certificates of robustness against \emph{topological perturbations} are currently available only for PageRank and label/feature propagation, while none has been designed for GCNs. We propose the first algorithm for certifying the robustness of GCNs to topological attacks in the application of \emph{graph classification}. Our method is based on Lagrange dualization and convex envelope, which result in tight approximation bounds that are efficiently computable by dynamic programming. When used in conjunction with robust training, it allows an increased number of graphs to be certified as robust.

We prove dimension free representation results for neural networks with D ReLU layers under square loss for a class of functions G_D defined in the paper. These results capture the precise benefits of depth in the following sense:

    1. The rates for representing the class of functions G_D via D ReLU layers is sharp up to constants, as shown by matching lower bounds.
    2.G_D is a proper subset of G_{D+1} and as D grows the class of functions G_D grow to contain less smooth functions. 
    3. If D^{\prime} < D, then the approximation rate achieved by depth D^{\prime} networks is strictly worse than that achieved by depth D networks for the class G_D.

This constitutes a fine-grained characterization of the representation power of feedforward networks of arbitrary depth D and number of neurons N, in contrast to existing representation results which either require D growing quickly with N or assume that the function being represented is highly smooth. In the latter case similar rates can be obtained with a single nonlinear layer. Our results confirm the prevailing hypothesis that deeper networks are better at representing less smooth functions, and indeed, the main technical novelty is to fully exploit the fact …

Spiking neural networks (SNNs) are well suited for spatio-temporal learning and implementations on energy-efficient event-driven neuromorphic processors. However, existing SNN error backpropagation (BP) methods lack proper handling of spiking discontinuities and suffer from low performance compared with the BP methods for traditional artificial neural networks. In addition, a large number of time steps are typically required to achieve decent performance, leading to high latency and rendering spike-based computation unscalable to deep architectures. We present a novel Temporal Spike Sequence Learning Backpropagation (TSSL-BP) method for training deep SNNs, which breaks down error backpropagation across two types of inter-neuron and intra-neuron dependencies and leads to improved temporal learning precision. It captures inter-neuron dependencies through presynaptic firing times by considering the all-or-none characteristics of firing activities and captures intra-neuron dependencies by handling the internal evolution of each neuronal state in time. TSSL-BP efficiently trains deep SNNs within a much shortened temporal window of a few steps while improving the accuracy for various image classification datasets including CIFAR10.

Deep probabilistic generative models enable modeling the likelihoods of very high dimensional data. An important application of generative modeling should be the ability to detect out-of-distribution (OOD) samples by setting a threshold on the likelihood. However, a recent study shows that probabilistic generative models can, in some cases, assign higher likelihoods on certain types of OOD samples, making the OOD detection rules based on likelihood threshold problematic. To address this issue, several OOD detection methods have been proposed for deep generative models. In this paper, we make the observation that some of these methods fail when applied to generative models based on Variational Auto-encoders (VAE). As an alternative, we propose Likelihood Regret, an efficient OOD score for VAEs. We benchmark our proposed method over existing approaches, and empirical results suggest that our method obtains the best overall OOD detection performances compared with other OOD method applied on VAE.

Many important problems in science and engineering, such as drug design, involve optimizing an expensive black-box objective function over a complex, high-dimensional, and structured input space. Although machine learning techniques have shown promise in solving such problems, existing approaches substantially lack sample efficiency. We introduce an improved method for efficient black-box optimization, which performs the optimization in the low-dimensional, continuous latent manifold learned by a deep generative model. In contrast to previous approaches, we actively steer the generative model to maintain a latent manifold that is highly useful for efficiently optimizing the objective. We achieve this by periodically retraining the generative model on the data points queried along the optimization trajectory, as well as weighting those data points according to their objective function value. This weighted retraining can be easily implemented on top of existing methods, and is empirically shown to significantly improve their efficiency and performance on synthetic and real-world optimization problems.

We develop a general approach to distill symbolic representations of a learned deep model by introducing strong inductive biases. We focus on Graph Neural Networks (GNNs). The technique works as follows: we first encourage sparse latent representations when we train a GNN in a supervised setting, then we apply symbolic regression to components of the learned model to extract explicit physical relations. We find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example—a detailed dark matter simulation—and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution-data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.

We describe a procedure for explaining neurons in deep representations by identifying compositional logical concepts that closely approximate neuron behavior. Compared to prior work that uses atomic labels as explanations, analyzing neurons compositionally allows us to more precisely and expressively characterize their behavior. We use this procedure to answer several questions on interpretability in models for vision and natural language processing. First, we examine the kinds of abstractions learned by neurons. In image classification, we find that many neurons learn highly abstract but semantically coherent visual concepts, while other polysemantic neurons detect multiple unrelated features; in natural language inference (NLI), neurons learn shallow lexical heuristics from dataset biases. Second, we see whether compositional explanations give us insight into model performance: vision neurons that detect human-interpretable concepts are positively correlated with task performance, while NLI neurons that fire for shallow heuristics are negatively correlated with task performance. Finally, we show how compositional explanations provide an accessible way for end users to produce simple "copy-paste" adversarial examples that change model behavior in predictable ways.

Linear relaxation based perturbation analysis (LiRPA) for neural networks, which computes provable linear bounds of output neurons given a certain amount of input perturbation, has become a core component in robustness verification and certified defense. The majority of LiRPA-based methods focus on simple feed-forward networks and need particular manual derivations and implementations when extended to other architectures. In this paper, we develop an automatic framework to enable perturbation analysis on any neural network structures, by generalizing existing LiRPA algorithms such as CROWN to operate on general computational graphs. The flexibility, differentiability and ease of use of our framework allow us to obtain state-of-the-art results on LiRPA based certified defense for fairly complicated networks like DenseNet, ResNeXt and Transformer that are not supported by prior works. Our framework also enables loss fusion, a technique that significantly reduces the computational complexity of LiRPA for certified defense. For the first time, we demonstrate LiRPA based certified defense on Tiny ImageNet and Downscaled ImageNet where previous approaches cannot scale to due to the relatively large number of classes. Our work also yields an open-source library for the community to apply LiRPA to areas beyond adversarial robustness without much LiRPA expertise, e.g., we create a …

Density-ratio estimation via classification is a cornerstone of unsupervised learning. It has provided the foundation for state-of-the-art methods in representation learning and generative modelling, with the number of use-cases continuing to proliferate. However, it suffers from a critical limitation: it fails to accurately estimate ratios p/q for which the two densities differ significantly. Empirically, we find this occurs whenever the KL divergence between p and q exceeds tens of nats. To resolve this limitation, we introduce a new framework, telescoping density-ratio estimation (TRE), that enables the estimation of ratios between highly dissimilar densities in high-dimensional spaces. Our experiments demonstrate that TRE can yield substantial improvements over existing single-ratio methods for mutual information estimation, representation learning and energy-based modelling.

Given a set of items F and a weight function W: F -> (0,1), the problem of sampling seeks to sample an item proportional to its weight. Sampling is a fundamental problem in machine learning. The daunting computational complexity of sampling with formal guarantees leads designers to propose heuristics-based techniques for which no rigorous theoretical analysis exists to quantify the quality of the generated distributions. This poses a challenge in designing a testing methodology to test whether a sampler under test generates samples according to a given distribution. Only recently, Chakraborty and Meel (2019) designed the first scalable verifier, called Barbarik, for samplers in the special case when the weight function W is constant, that is, when the sampler is supposed to sample uniformly from F. The techniques in Barbarik, however, fail to handle general weight functions. The primary contribution of this paper is an affirmative answer to the above challenge: motivated by Barbarik, but using different techniques and analysis, we design Barbarik2, an algorithm to test whether the distribution generated by a sampler is epsilon-close or eta-far from any target distribution. In contrast to black-box sampling techniques that require a number of samples proportional to |F|, Barbarik2 requires only …

In neural networks with binary activations and or binary weights the training by gradient descent is complicated as the model has piecewise constant response. We consider stochastic binary networks, obtained by adding noises in front of activations. The expected model response becomes a smooth function of parameters, its gradient is well defined but it is challenging to estimate it accurately. We propose a new method for this estimation problem combining sampling and analytic approximation steps. The method has a significantly reduced variance at the price of a small bias which gives a very practical tradeoff in comparison with existing unbiased and biased estimators. We further show that one extra linearization step leads to a deep straight-through estimator previously known only as an ad-hoc heuristic. We experimentally show higher accuracy in gradient estimation and demonstrate a more stable and better performing training in deep convolutional models with both proposed methods.

Predictive modeling often uses black box machine learning methods, such as deep neural networks, to achieve state-of-the-art performance. In scientific domains, the scientist often wishes to discover which features are actually important for making the predictions. These discoveries may lead to costly follow-up experiments and as such it is important that the error rate on discoveries is not too high. Model-X knockoffs enable important features to be discovered with control of the false discovery rate (FDR). However, knockoffs require rich generative models capable of accurately modeling the knockoff features while ensuring they obey the so-called "swap" property. We develop Deep Direct Likelihood Knockoffs (DDLK), which directly minimizes the KL divergence implied by the knockoff swap property. DDLK consists of two stages: it first maximizes the explicit likelihood of the features, then minimizes the KL divergence between the joint distribution of features and knockoffs and any swap between them. To ensure that the generated knockoffs are valid under any possible swap, DDLK uses the Gumbel-Softmax trick to optimize the knockoff generator under the worst-case swap. We find DDLK has higher power than baselines while controlling the false discovery rate on a variety of synthetic and real benchmarks including a task involving …

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift---a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input)---which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real-world data.

In many real-world scenarios, decision makers seek to efficiently optimize multiple competing objectives in a sample-efficient fashion. Multi-objective Bayesian optimization (BO) is a common approach, but many of the best-performing acquisition functions do not have known analytic gradients and suffer from high computational overhead. We leverage recent advances in programming models and hardware acceleration for multi-objective BO using Expected Hypervolume Improvement (EHVI)---an algorithm notorious for its high computational complexity. We derive a novel formulation of q-Expected Hypervolume Improvement (qEHVI), an acquisition function that extends EHVI to the parallel, constrained evaluation setting. qEHVI is an exact computation of the joint EHVI of q new candidate points (up to Monte-Carlo (MC) integration error). Whereas previous EHVI formulations rely on gradient-free acquisition optimization or approximated gradients, we compute exact gradients of the MC estimator via auto-differentiation, thereby enabling efficient and effective optimization using first-order and quasi-second-order methods. Our empirical evaluation demonstrates that qEHVI is computationally tractable in many practical scenarios and outperforms state-of-the-art multi-objective BO algorithms at a fraction of their wall time.

Sampling from a log-concave distribution function is one core problem that has wide applications in Bayesian statistics and machine learning. While most gradient free methods have slow convergence rate, the Langevin Monte Carlo (LMC) that provides fast convergence requires the computation of gradients. In practice one uses finite-differencing approximations as surrogates, and the method is expensive in high-dimensions.

A natural strategy to reduce computational cost in each iteration is to utilize random gradient approximations, such as random coordinate descent (RCD) or simultaneous perturbation stochastic approximation (SPSA).We show by a counterexamplethat blindly applying RCD does not achieve the goal in the most general setting. The high variance induced by the randomness means a larger number of iterations are needed, and this balances out the saving in each iteration.

We then introduce a new variance reduction approach, termed Randomized Coordinates Averaging Descent (RCAD), and incorporate it with both overdamped and underdamped LMC. The methods are termed RCAD-O-LMC and RCAD-U-LMC respectively. The methods still sit in the random gradient approximation framework, and thus the computational cost in each iteration is low. However, by employing RCAD, the variance is reduced, so the methods converge within the same number of iterations as the classical overdamped …

We propose an adaptively weighted stochastic gradient Langevin dynamics algorithm (SGLD), so-called contour stochastic gradient Langevin dynamics (CSGLD), for Bayesian learning in big data statistics. The proposed algorithm is essentially a scalable dynamic importance sampler, which automatically flattens the target distribution such that the simulation for a multi-modal distribution can be greatly facilitated. Theoretically, we prove a stability condition and establish the asymptotic convergence of the self-adapting parameter to a unique fixed-point, regardless of the non-convexity of the original energy function; we also present an error analysis for the weighted averaging estimators. Empirically, the CSGLD algorithm is tested on multiple benchmark datasets including CIFAR10 and CIFAR100. The numerical results indicate its superiority over the existing state-of-the-art algorithms in training deep neural networks.

Mutual information (MI) is a commonly adopted utility function in Bayesian optimal experimental design (BOED). While theoretically appealing, MI evaluation poses a significant computational burden for most real world applications. As a result, many algorithms utilize MI bounds as proxies that lack regret-style guarantees. Here, we utilize two-sided bounds to provide such guarantees. Bounds are successively refined/tightened through additional computation until a desired guarantee is achieved. We consider the problem of adaptively allocating computational resources in BOED. Our approach achieves the same guarantee as existing methods, but with fewer evaluations of the costly MI reward. We adapt knapsack optimization of best arm identification problems, with important differences that impact overall algorithm design and performance. First, observations of MI rewards are biased. Second, evaluating experiments incurs shared costs amongst all experiments (posterior sampling) in addition to per experiment costs that may vary with increasing evaluation. We propose and demonstrate an algorithm that accounts for these variable costs in the refinement decision.

Metropolis-Hastings (MH) is a commonly-used MCMC algorithm, but it can be intractable on large datasets due to requiring computations over the whole dataset. In this paper, we study \emph{minibatch MH} methods, which instead use subsamples to enable scaling. We observe that most existing minibatch MH methods are inexact (i.e. they may change the target distribution), and show that this inexactness can cause arbitrarily large errors in inference. We propose a new exact minibatch MH method, \emph{TunaMH}, which exposes a tunable trade-off between its minibatch size and its theoretically guaranteed convergence rate. We prove a lower bound on the batch size that any minibatch MH method \emph{must} use to retain exactness while guaranteeing fast convergence---the first such bound for minibatch MH---and show TunaMH is asymptotically optimal in terms of the batch size. Empirically, we show TunaMH outperforms other exact minibatch MH methods on robust linear regression, truncated Gaussian mixtures, and logistic regression.

In this paper, we present a new practical method for Bayesian learning that can rapidly draw representative samples from complex posterior distributions with multiple isolated modes in the presence of mini-batch noise. This is achieved by simulating a collection of replicas in parallel with different temperatures and periodically swapping them. When evolving the replicas' states, the Nos\'e-Hoover dynamics is applied, which adaptively neutralizes the mini-batch noise. To perform proper exchanges, a new protocol is developed with a noise-aware test of acceptance, by which the detailed balance is reserved in an asymptotic way. While its efficacy on complex multimodal posteriors has been illustrated by testing over synthetic distributions, experiments with deep Bayesian neural networks on large-scale datasets have shown its significant improvements over strong baselines.

We propose a new Stein self-repulsive dynamics for obtaining diversified samples from intractable un-normalized distributions. Our idea is to introduce Stein variational gradient as a repulsive force to push the samples of Langevin dynamics away from the past trajectories. This simple idea allows us to significantly decrease the auto-correlation in Langevin dynamics and hence increase the effective sample size. Importantly, as we establish in our theoretical analysis, the asymptotic stationary distribution remains correct even with the addition of the repulsive force, thanks to the special properties of the Stein variational gradient. We perform extensive empirical studies of our new algorithm, showing that our method yields much higher sample efficiency and better uncertainty estimation than vanilla Langevin dynamics.

Learning-based methodologies increasingly find applications in safety-critical domains like autonomous driving and medical robotics. Due to the rare nature of dangerous events, real-world testing is prohibitively expensive and unscalable. In this work, we employ a probabilistic approach to safety evaluation in simulation, where we are concerned with computing the probability of dangerous events. We develop a novel rare-event simulation method that combines exploration, exploitation, and optimization techniques to find failure modes and estimate their rate of occurrence. We provide rigorous guarantees for the performance of our method in terms of both statistical and computational efficiency. Finally, we demonstrate the efficacy of our approach on a variety of scenarios, illustrating its usefulness as a tool for rapid sensitivity analysis and model comparison that are essential to developing and testing safety-critical autonomous systems.

We establish a general form of explicit, input-dependent, measure-valued warpings for learning nonstationary kernels. While stationary kernels are uniquitous and simple to use, they struggle to adapt to functions that vary in smoothness with respect to the input. The proposed learning algorithm warps inputs as conditional Gaussian measures that control the smoothness of a standard stationary kernel. This construction allows us to capture non-stationary patterns in the data and provides intuitive inductive bias. The resulting method is based on sparse spectrum Gaussian processes, enabling closed-form solutions, and is extensible to a stacked construction to capture more complex patterns. The method is extensively validated alongside related algorithms on synthetic and real world datasets. We demonstrate a remarkable efficiency in the number of parameters of the warping functions in learning problems with both small and large data regimes.

Unlike in the traditional statistical modeling for which a user typically hand-specify a prior, Neural Processes (NPs) implicitly define a broad class of stochastic processes with neural networks. Given a data stream, NP learns a stochastic process that best describes the data. While this ``data-driven'' way of learning stochastic processes has proven to handle various types of data, NPs still relies on an assumption that uncertainty in stochastic processes is modeled by a single latent variable, which potentially limits the flexibility. To this end, we propose the Bootstrapping Neural Process (BNP), a novel extension of the NP family using the bootstrap. The bootstrap is a classical data-driven technique for estimating uncertainty, which allows BNP to learn the stochasticity in NPs without assuming a particular form. We demonstrate the efficacy of BNP on various types of data and its robustness in the presence of model-data mismatch.

Decision trees with binary splits are popularly constructed using Classification and Regression Trees (CART) methodology. For regression models, this approach recursively divides the data into two near-homogenous daughter nodes according to a split point that maximizes the reduction in sum of squares error (the impurity) along a particular variable. This paper aims to study the statistical properties of regression trees constructed with CART methodology. In doing so, we find that the training error is governed by the Pearson correlation between the optimal decision stump and response data in each node, which we bound by constructing a prior distribution on the split points and solving a nonlinear optimization problem. We leverage this connection between the training error and Pearson correlation to show that CART with cost-complexity pruning achieves an optimal complexity/goodness-of-fit tradeoff when the depth scales with the logarithm of the sample size. Data dependent quantities, which adapt to the dimensionality and latent structure of the regression model, are seen to govern the convergence rates of the prediction error.

We propose here a generalization of regression trees, referred to as Probabilistic Regression (PR) trees, that adapt to the smoothness of the prediction function relating input and output variables while preserving the interpretability of the prediction and being robust to noise. In PR trees, an observation is associated to all regions of a tree through a probability distribution that reflects how far the observation is to a region. We show that such trees are consistent, meaning that their error tends to 0 when the sample size tends to infinity, a property that has not been established for similar, previous proposals as Soft trees and Smooth Transition Regression trees. We further explain how PR trees can be used in different ensemble methods, namely Random Forests and Gradient Boosted Trees. Lastly, we assess their performance through extensive experiments that illustrate their benefits in terms of performance, interpretability and robustness to noise.

Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model’s ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer. On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches.

We explore the link between deep ensembles and Gaussian processes (GPs) through the lens of the Neural Tangent Kernel (NTK): a recent development in understanding the training dynamics of wide neural networks (NNs). Previous work has shown that even in the infinite width limit, when NNs become GPs, there is no GP posterior interpretation to a deep ensemble trained with squared error loss. We introduce a simple modification to standard deep ensembles training, through addition of a computationally-tractable, randomised and untrainable function to each ensemble member, that enables a posterior interpretation in the infinite width limit. When ensembled together, our trained NNs give an approximation to a posterior predictive distribution, and we prove that our Bayesian deep ensembles make more conservative predictions than standard deep ensembles in the infinite width limit. Finally, using finite width NNs we demonstrate that our Bayesian deep ensembles faithfully emulate the analytic posterior predictive when available, and outperform standard deep ensembles in various out-of-distribution settings, for both regression and classification tasks.

Ensemble learning is widely used in applications to make predictions in complex decision problems---for example, averaging models fitted to a sequence of samples bootstrapped from the available training data. While such methods offer more accurate, stable, and robust predictions and model estimates, much less is known about how to perform valid, assumption-lean inference on the output of these types of procedures. In this paper, we propose the jackknife+-after-bootstrap (J+aB), a procedure for constructing a predictive interval, which uses only the available bootstrapped samples and their corresponding fitted models, and is therefore "free" in terms of the cost of model fitting. The J+aB offers a predictive coverage guarantee that holds with no assumptions on the distribution of the data, the nature of the fitted model, or the way in which the ensemble of models are aggregated---at worst, the failure rate of the predictive interval is inflated by a factor of 2. Our numerical experiments verify the coverage and accuracy of the resulting predictive intervals on real data.

Ensembles over neural network weights trained from different random initialization, known as deep ensembles, achieve state-of-the-art accuracy and calibration. The recently introduced batch ensembles provide a drop-in replacement that is more parameter efficient. In this paper, we design ensembles not only over weights, but over hyperparameters to improve the state of the art in both settings. For best performance independent of budget, we propose hyper-deep ensembles, a simple procedure that involves a random search over different hyperparameters, themselves stratified across multiple random initializations. Its strong performance highlights the benefit of combining models with both weight and hyperparameter diversity. We further propose a parameter efficient version, hyper-batch ensembles, which builds on the layer structure of batch ensembles and self-tuning networks. The computational and memory costs of our method are notably lower than typical ensembles. On image classification tasks, with MLP, LeNet, ResNet 20 and Wide ResNet 28-10 architectures, we improve upon both deep and batch ensembles.

Existing methods for estimating uncertainty in deep learning tend to require multiple forward passes, making them unsuitable for applications where computational resources are limited. To solve this, we perform probabilistic reasoning over the depth of neural networks. Different depths correspond to subnetworks which share weights and whose predictions are combined via marginalisation, yielding model uncertainty. By exploiting the sequential structure of feed-forward networks, we are able to both evaluate our training objective and make predictions with a single forward pass. We validate our approach on real-world regression and image classification tasks. Our approach provides uncertainty calibration, robustness to dataset shift, and accuracies competitive with more computationally expensive baselines.

We take a Bayesian perspective to illustrate a connection between training speed and the marginal likelihood in linear models. This provides two major insights: first, that a measure of a model's training speed can be used to estimate its marginal likelihood. Second, that this measure, under certain conditions, predicts the relative weighting of models in linear model combinations trained to minimize a regression loss. We verify our results in model selection tasks for linear models and for the infinite-width limit of deep neural networks. We further provide encouraging empirical evidence that the intuition developed in these settings also holds for deep neural networks trained with stochastic gradient descent. Our results suggest a promising new direction towards explaining why neural networks trained with stochastic gradient descent are biased towards functions that generalize well.

Virtually any model we use in machine learning to make predictions does not perfectly represent reality. So, most of the learning happens under model misspecification. In this work, we present a novel analysis of the generalization performance of Bayesian model averaging under model misspecification and i.i.d. data using a new family of second-order PAC-Bayes bounds. This analysis shows, in simple and intuitive terms, that Bayesian model averaging provides suboptimal generalization performance when the model is misspecified. In consequence, we provide strong theoretical arguments showing that Bayesian methods are not optimal for learning predictive models, unless the model class is perfectly specified. Using novel second-order PAC-Bayes bounds, we derive a new family of Bayesian-like algorithms, which can be implemented as variational and ensemble methods. The output of these algorithms is a new posterior distribution, different from the Bayesian posterior, which induces a posterior predictive distribution with better generalization performance. Experiments with Bayesian neural networks illustrate these findings.

Algorithms are commonly used to predict outcomes under a particular decision or intervention, such as predicting likelihood of default if a loan is approved. Generally, to learn such counterfactual prediction models from observational data on historical decisions and corresponding outcomes, one must measure all factors that jointly affect the outcome and the decision taken. Motivated by decision support applications, we study the counterfactual prediction task in the setting where all relevant factors are captured in the historical data, but it is infeasible, undesirable, or impermissible to use some such factors in the prediction model. We refer to this setting as runtime confounding. We propose a doubly-robust procedure for learning counterfactual prediction models in this setting. Our theoretical analysis and experimental results suggest that our method often outperforms competing approaches. We also present a validation procedure for evaluating the performance of counterfactual prediction methods.

Gaussian processes are an effective model class for learning unknown functions, particularly in settings where accurately representing predictive uncertainty is of key importance. Motivated by applications in the physical sciences, the widely-used Matérn class of Gaussian processes has recently been generalized to model functions whose domains are Riemannian manifolds, by re-expressing said processes as solutions of stochastic partial differential equations. In this work, we propose techniques for computing the kernels of these processes on compact Riemannian manifolds via spectral theory of the Laplace-Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods. We also extend the generalization from the Matérn to the widely-used squared exponential Gaussian process. By allowing Riemannian Matérn Gaussian processes to be trained using well-understood techniques, our work enables their use in mini-batch, online, and non-conjugate settings, and makes them more accessible to machine learning practitioners.

We introduce a new family of non-linear neural network activation functions that mimic the properties induced by the widely-used Mat\'ern family of kernels in Gaussian process (GP) models. This class spans a range of locally stationary models of various degrees of mean-square differentiability. We show an explicit link to the corresponding GP models in the case that the network consists of one infinitely wide hidden layer. In the limit of infinite smoothness the Mat\'ern family results in the RBF kernel, and in this case we recover RBF activations. Mat\'ern activation functions result in similar appealing properties to their counterparts in GP models, and we demonstrate that the local stationarity property together with limited mean-square differentiability shows both good performance and uncertainty calibration in Bayesian deep learning tasks. In particular, local stationarity helps calibrate out-of-distribution (OOD) uncertainty. We demonstrate these properties on classification and regression benchmarks and a radar emitter classification task.

Gaussian processes (GPs) are flexible priors for modeling functions. However, their success depends on the kernel accurately reflecting the properties of the data. One of the appeals of the GP framework is that the marginal likelihood of the kernel hyperparameters is often available in closed form, enabling optimization and sampling procedures to fit these hyperparameters to data. Unfortunately, point-wise evaluation of the marginal likelihood is expensive due to the need to solve a linear system; searching or sampling the space of hyperparameters thus often dominates the practical cost of using GPs. We introduce an approach to the identification of kernel hyperparameters in GP regression and related problems that sidesteps the need for costly marginal likelihoods. Our strategy is to "amortize" inference over hyperparameters by training a single neural network, which consumes a set of regression data and produces an estimate of the kernel function, useful across different tasks. To accommodate the varying dimension and cardinality of different regression problems, we use a hierarchical self-attention-based neural network that produces estimates of the hyperparameters which are invariant to the order of the input data points and data dimensions. We show that a single neural model trained on synthetic data is able to …

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood. We study the quality of common variational methods in approximating the Bayesian predictive distribution. For single-hidden layer ReLU BNNs, we prove a fundamental limitation in function-space of two of the most commonly used distributions defined in weight-space: mean-field Gaussian and Monte Carlo dropout. We find there are simple cases where neither method can have substantially increased uncertainty in between well-separated regions of low uncertainty. We provide strong empirical evidence that exact inference does not have this pathology, hence it is due to the approximation and not the model. In contrast, for deep networks, we prove a universality result showing that there exist approximate posteriors in the above classes which provide flexible uncertainty estimates. However, we find empirically that pathologies of a similar form as in the single-hidden layer case can persist when performing variational inference in deeper networks. Our results motivate careful consideration of the implications of approximate inference methods in BNNs.

Extracting relevant features from data sets where the number of observations (n) is much smaller then the number of predictors (p) is a major challenge in modern statistics. Sorted L-One Penalized Estimation (SLOPE)—a generalization of the lasso—is a promising method within this setting. Current numerical procedures for SLOPE, however, lack the efficiency that respective tools for the lasso enjoy, particularly in the context of estimating a complete regularization path. A key component in the efficiency of the lasso is predictor screening rules: rules that allow predictors to be discarded before estimating the model. This is the first paper to establish such a rule for SLOPE. We develop a screening rule for SLOPE by examining its subdifferential and show that this rule is a generalization of the strong rule for the lasso. Our rule is heuristic, which means that it may discard predictors erroneously. In our paper, however, we show that such situations are rare and easily safeguarded against by a simple check of the optimality conditions. Our numerical experiments show that the rule performs well in practice, leading to improvements by orders of magnitude for data in the (p >> n) domain, as well as incurring no additional computational overhead …

Social media has created new ways for citizens to stay informed on societal matters and participate in political discourse. However, with its algorithmically-curated and virally-propagating content, social media has contributed further to the polarization of opinions by reinforcing users' existing viewpoints. An emerging line of research seeks to understand how content-recommendation algorithms can be re-designed to mitigate societal polarization amplified by social-media interactions. In this paper, we study the problem of allocating seed users to opposing campaigns: by drawing on the equal-time rule of political campaigning on traditional media, our goal is to allocate seed users to campaigners with the aim to maximize the expected number of users who are co-exposed to both campaigns.

We show that the problem of maximizing co-exposure is NP-hard and its objective function is neither submodular nor supermodular. However, by exploiting a connection to a submodular function that acts as a lower bound to the objective, we are able to devise a greedy algorithm with provable approximation guarantee. We further provide a scalable instantiation of our approximation algorithm by introducing a novel extension to the notion of random reverse-reachable sets for efficiently estimating the expected co-exposure. We experimentally demonstrate the quality of our proposal on …

We study MinMax solution methods for a general class of optimization problems related to (and including) optimal transport. Theoretically, the focus is on fitting a large class of problems into a single MinMax framework and generalizing regularization techniques known from classical optimal transport. We show that regularization techniques justify the utilization of neural networks to solve such problems by proving approximation theorems and illustrating fundamental issues if no regularization is used. We further study the relation to the literature on generative adversarial nets, and analyze which algorithmic techniques used therein are particularly suitable to the class of problems studied in this paper. Several numerical experiments showcase the generality of the setting and highlight which theoretical insights are most beneficial in practice.

The Lipschitz constant of a network plays an important role in many applications of deep learning, such as robustness certification and Wasserstein Generative Adversarial Network. We introduce a semidefinite programming hierarchy to estimate the global and local Lipschitz constant of a multiple layer deep neural network. The novelty is to combine a polynomial lifting for ReLU functions derivatives with a weak generalization of Putinar's positivity certificate. This idea could also apply to other, nearly sparse, polynomial optimization problems in machine learning. We empirically demonstrate that our method provides a trade-off with respect to state of the art linear programming approach, and in some cases we obtain better bounds in less time.

A major challenge in current optimization research for deep learning is to automatically find optimal step sizes for each update step. The optimal step size is closely related to the shape of the loss in the update step direction. However, this shape has not yet been examined in detail. This work shows empirically that the sample loss over lines in negative gradient direction is mostly convex and well suited for one-dimensional parabolic approximations. Exploiting this parabolic property we introduce a simple and robust line search approach, which performs loss-shape dependent update steps. Our approach combines well-known methods such as parabolic approximation, line search and conjugate gradient, to perform efficiently. It successfully competes with common and state-of-the-art optimization methods on a large variety of experiments without the need of hand-designed step size schedules. Thus, it is of interest for objectives where step-size schedules are unknown or do not perform well. Our excessive evaluation includes multiple comprehensive hyperparameter grid searches on several datasets and architectures. We provide proof of convergence for an adapted scenario. Finally, we give a general investigation of exact line searches in the context of sample losses and exact losses, including their relation to our line search approach.

Randomly initialized neural networks are known to become harder to train with increasing depth, unless architectural enhancements like residual connections and batch normalization are used. We here investigate this phenomenon by revisiting the connection between random initialization in deep networks and spectral instabilities in products of random matrices. Given the rich literature on random matrices, it is not surprising to find that the rank of the intermediate representations in unnormalized networks collapses quickly with depth. In this work we highlight the fact that batch normalization is an effective strategy to avoid rank collapse for both linear and ReLU networks. Leveraging tools from Markov chain theory, we derive a meaningful lower rank bound in deep linear networks. Empirically, we also demonstrate that this rank robustness generalizes to ReLU nets. Finally, we conduct an extensive set of experiments on real-world data sets, which confirm that rank stability is indeed a crucial condition for training modern-day deep neural architectures.

Langevin dynamics (LD) has been proven to be a powerful technique for optimizing a non-convex objective as an efficient algorithm to find local minima while eventually visiting a global minimum on longer time-scales. LD is based on the first-order Langevin diffusion which is reversible in time. We study two variants that are based on non-reversible Langevin diffusions: the underdamped Langevin dynamics (ULD) and the Langevin dynamics with a non-symmetric drift (NLD). Adopting the techniques of Tzen et al. (2018) for LD to non-reversible diffusions, we show that for a given local minimum that is within an arbitrary distance from the initialization, with high probability, either the ULD trajectory ends up somewhere outside a small neighborhood of this local minimum within a recurrence time which depends on the smallest eigenvalue of the Hessian at the local minimum or they enter this neighborhood by the recurrence time and stay there for a potentially exponentially long escape time. The ULD algorithm improves upon the recurrence time obtained for LD in Tzen et al. (2018) with respect to the dependency on the smallest eigenvalue of the Hessian at the local minimum. Similar results and improvements are obtained for the NLD algorithm. We also show …

In federated learning, heterogeneity in the clients' local datasets and computation speeds results in large variations in the number of local updates performed by each client in each communication round. Naive weighted aggregation of such models causes objective inconsistency, that is, the global model converges to a stationary point of a mismatched objective function which can be arbitrarily different from the true objective. This paper provides a general framework to analyze the convergence of federated heterogeneous optimization algorithms. It subsumes previously proposed methods such as FedAvg and FedProx and provides the first principled understanding of the solution bias and the convergence slowdown due to objective inconsistency. Using insights from this analysis, we propose FedNova, a normalized averaging method that eliminates objective inconsistency while preserving fast error convergence.

We present an extension of the conditional gradient method to problems whose feasible sets are convex cones. We provide a convergence analysis for the method and for variants with nonconvex objectives, and we extend the analysis to practical cases with effective line search strategies. For the specific case of the positive semidefinite cone, we present a memory-efficient version based on randomized matrix sketches and advocate a heuristic greedy step that greatly improves its practical performance. Numerical results on phase retrieval and matrix completion problems indicate that our method can offer substantial advantages over traditional conditional gradient and Burer-Monteiro approaches.

Finding Nash equilibria in two-player zero-sum continuous games is a central problem in machine learning, e.g. for training both GANs and robust models. The existence of pure Nash equilibria requires strong conditions which are not typically met in practice. Mixed Nash equilibria exist in greater generality and may be found using mirror descent. Yet this approach does not scale to high dimensions. To address this limitation, we parametrize mixed strategies as mixtures of particles, whose positions and weights are updated using gradient descent-ascent. We study this dynamics as an interacting gradient flow over measure spaces endowed with the Wasserstein-Fisher-Rao metric. We establish global convergence to an approximate equilibrium for the related Langevin gradient-ascent dynamic. We prove a law of large numbers that relates particle dynamics to mean-field dynamics. Our method identifies mixed equilibria in high dimensions and is demonstrably effective for training mixtures of GANs.

Federated learning is a distributed paradigm that aims at training models using samples distributed across multiple users in a network while keeping the samples on users’ devices with the aim of efficiency and protecting users privacy. In such settings, the training data is often statistically heterogeneous and manifests various distribution shifts across users, which degrades the performance of the learnt model. The primary goal of this paper is to develop a robust federated learning algorithm that achieves satisfactory performance against distribution shifts in users' samples. To achieve this goal, we first consider a structured affine distribution shift in users' data that captures the device-dependent data heterogeneity in federated settings. This perturbation model is applicable to various federated learning problems such as image classification where the images undergo device-dependent imperfections, e.g. different intensity, contrast, and brightness. To address affine distribution shifts across users, we propose a Federated Learning framework Robust to Affine distribution shifts (FLRA) that is provably robust against affine Wasserstein shifts to the distribution of observed samples. To solve the FLRA's distributed minimax optimization problem, we propose a fast and efficient optimization method and provide convergence and performance guarantees via a gradient Descent Ascent (GDA) method. We further prove …

Compositionality is a basic structural feature of both biological and artificial neural networks. Learning compositional functions via gradient descent incurs well known problems like vanishing and exploding gradients, making careful learning rate tuning essential for real-world applications. This paper proves that multiplicative weight updates satisfy a descent lemma tailored to compositional functions. Based on this lemma, we derive Madam---a multiplicative version of the Adam optimiser---and show that it can train state of the art neural network architectures without learning rate tuning. We further show that Madam is easily adapted to train natively compressed neural networks by representing their weights in a logarithmic number system. We conclude by drawing connections between multiplicative weight updates and recent findings about synapses in biology.

Arguably, the two most popular accelerated or momentum-based optimization methods are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order differential equation with a friction term. Such connections with continuous-time dynamical systems have been instrumental in demystifying acceleration phenomena in optimization. Here we study structure-preserving discretizations for a certain class of dissipative (conformal) Hamiltonian systems, allowing us to analyze the symplectic structure of both Nesterov and heavy ball, besides providing several new insights into these methods. Moreover, we propose a new algorithm based on a dissipative relativistic system that normalizes the momentum and may result in more stable/faster optimization. Importantly, such a method generalizes both Nesterov and heavy ball, each being recovered as distinct limiting cases, and has potential advantages at no additional cost.

Ensembles of deep neural networks are known to achieve state-of-the-art performance in uncertainty estimation and lead to accuracy improvement. In this work, we focus on a classification problem and investigate the behavior of both non-calibrated and calibrated negative log-likelihood (CNLL) of a deep ensemble as a function of the ensemble size and the member network size. We indicate the conditions under which CNLL follows a power law w. r. t. ensemble size or member network size, and analyze the dynamics of the parameters of the discovered power laws. Our important practical finding is that one large network may perform worse than an ensemble of several medium-size networks with the same total number of parameters (we call this ensemble a memory split). Using the detected power law-like dependencies, we can predict (1) the possible gain from the ensembling of networks with given structure, (2) the optimal memory split given a memory budget, based on a relatively small number of trained networks.

The key distinguishing property of a Bayesian approach is marginalization, rather than using a single setting of weights. Bayesian marginalization can particularly improve the accuracy and calibration of modern deep neural networks, which are typically underspecified by the data, and can represent many compelling but different solutions. We show that deep ensembles provide an effective mechanism for approximate Bayesian marginalization, and propose a related approach that further improves the predictive distribution by marginalizing within basins of attraction, without significant overhead. We also investigate the prior over functions implied by a vague distribution over neural network weights, explaining the generalization properties of such models from a probabilistic perspective. From this perspective, we explain results that have been presented as mysterious and distinct to neural network generalization, such as the ability to fit images with random labels, and show that these results can be reproduced with Gaussian processes. We also show that Bayesian model averaging alleviates double descent, resulting in monotonic performance improvements with increased flexibility.

It has been recently demonstrated that multi-generational self-distillation can improve generalization. Despite this intriguing observation, reasons for the enhancement remain poorly understood. In this paper, we first demonstrate experimentally that the improved performance of multi-generational self-distillation is in part associated with the increasing diversity in teacher predictions. With this in mind, we offer a new interpretation for teacher-student training as amortized MAP estimation, such that teacher predictions enable instance-specific regularization. Our framework allows us to theoretically relate self-distillation to label smoothing, a commonly used technique that regularizes predictive uncertainty, and suggests the importance of predictive diversity in addition to predictive uncertainty. We present experimental results using multiple datasets and neural network architectures that, overall, demonstrate the utility of predictive diversity. Finally, we propose a novel instance-specific label smoothing technique that promotes predictive diversity without the need for a separately trained teacher model. We provide an empirical evaluation of the proposed method, which, we find, often outperforms classical label smoothing.

Batch normalization dramatically increases the largest trainable depth of residual networks, and this benefit has been crucial to the empirical success of deep residual networks on a wide range of benchmarks. We show that this key benefit arises because, at initialization, batch normalization downscales the residual branch relative to the skip connection, by a normalizing factor on the order of the square root of the network depth. This ensures that, early in training, the function computed by normalized residual blocks in deep networks is close to the identity function (on average). We use this insight to develop a simple initialization scheme that can train deep residual networks without normalization. We also provide a detailed empirical study of residual networks, which clarifies that, although batch normalized networks can be trained with larger learning rates, this effect is only beneficial in specific compute regimes, and has minimal benefits when the batch size is small.

We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs). Our method is based on reformulating the numerical approximation of a whole family of Kolmogorov PDEs as a single statistical learning problem using the Feynman-Kac formula. Successful numerical experiments are presented, which empirically confirm the functionality and efficiency of our proposed algorithm in the case of heat equations and Black-Scholes option pricing models parametrized by affine-linear coefficient functions. We show that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region. Most notably, our numerical observations and theoretical results also demonstrate that the proposed method does not suffer from the curse of dimensionality, distinguishing it from almost all standard numerical methods for PDEs.

Several works have aimed to explain why overparameterized neural networks generalize well when trained by Stochastic Gradient Descent (SGD). The consensus explanation that has emerged credits the randomized nature of SGD for the bias of the training process towards low-complexity models and, thus, for implicit regularization. We take a careful look at this explanation in the context of image classification with common deep neural network architectures. We find that if we do not regularize \emph{explicitly}, then SGD can be easily made to converge to poorly-generalizing, high-complexity models: all it takes is to first train on a random labeling on the data, before switching to properly training with the correct labels. In contrast, we find that in the presence of explicit regularization, pretraining with random labels has no detrimental effect on SGD. We believe that our results give evidence that explicit regularization plays a far more important role in the success of overparameterized neural networks than what has been understood until now. Specifically, in suppressing complicated models that got lucky with the training data, regularization not only makes simple models that fit the data well the global optima, but it also clears the way to make them discoverable by local methods, …

Modern neural networks are often regarded as complex black-box functions whose behavior is difficult to understand owing to their nonlinear dependence on the data and the nonconvexity in their loss landscapes. In this work, we show that these common perceptions can be completely false in the early phase of learning. In particular, we formally prove that, for a class of well-behaved input distributions, the early-time learning dynamics of a two-layer fully-connected neural network can be mimicked by training a simple linear model on the inputs. We additionally argue that this surprising simplicity can persist in networks with more layers and with convolutional architecture, which we verify empirically. Key to our analysis is to bound the spectral norm of the difference between the Neural Tangent Kernel (NTK) and an affine transform of the data kernel; however, unlike many previous results utilizing the NTK, we do not require the network to have disproportionately large width, and the network is allowed to escape the kernel regime later in training.

Federated Learning (FL) is a machine learning setting where many devices collaboratively train a machine learning model while keeping the training data decentralized. In most of the current training schemes the central model is refined by averaging the parameters of the server model and the updated parameters from the client side. However, directly averaging model parameters is only possible if all models have the same structure and size, which could be a restrictive constraint in many scenarios.

In this work we investigate more powerful and more flexible aggregation schemes for FL. Specifically, we propose ensemble distillation for model fusion, i.e. training the central classifier through unlabeled data on the outputs of the models from the clients. This knowledge distillation technique mitigates privacy risk and cost to the same extent as the baseline FL algorithms, but allows flexible aggregation over heterogeneous client models that can differ e.g. in size, numerical precision or structure. We show in extensive empirical experiments on various CV/NLP datasets (CIFAR-10/100, ImageNet, AG News, SST2) and settings (heterogeneous models/data) that the server model can be trained much faster, requiring fewer communication rounds than any existing FL technique so far.

In many real-world deployments of machine learning systems, data arrive piecemeal. These learning scenarios may be passive, where data arrive incrementally due to structural properties of the problem (e.g., daily financial data) or active, where samples are selected according to a measure of their quality (e.g., experimental design). In both of these cases, we are building a sequence of models that incorporate an increasing amount of data. We would like each of these models in the sequence to be performant and take advantage of all the data that are available to that point. Conventional intuition suggests that when solving a sequence of related optimization problems of this form, it should be possible to initialize using the solution of the previous iterate---to ``warm start'' the optimization rather than initialize from scratch---and see reductions in wall-clock time. However, in practice this warm-starting seems to yield poorer generalization performance than models that have fresh random initializations, even though the final training losses are similar. While it appears that some hyperparameter settings allow a practitioner to close this generalization gap, they seem to only do so in regimes that damage the wall-clock gains of the warm start. Nevertheless, it is highly desirable to be …

We tackle the problem of predicting the number of optimization steps that a pre-trained deep network needs to converge to a given value of the loss function. To do so, we leverage the fact that the training dynamics of a deep network during fine-tuning are well approximated by those of a linearized model. This allows us to approximate the training loss and accuracy at any point during training by solving a low-dimensional Stochastic Differential Equation (SDE) in function space. Using this result, we are able to predict the time it takes for Stochastic Gradient Descent (SGD) to fine-tune a model to a given loss without having to perform any training.
In our experiments, we are able to predict training time of a ResNet within a 20\% error margin on a variety of datasets and hyper-parameters, at a 30 to 45-fold reduction in cost compared to actual training. We also discuss how to further reduce the computational and memory cost of our method, and in particular we show that by exploiting the spectral properties of the gradients' matrix it is possible to predict training time on a large dataset while processing only a subset of the samples.

In this paper, we show that although the minimizers of cross-entropy and related classification losses are off at infinity, network weights learned by gradient flow converge in direction, with an immediate corollary that network predictions, training errors, and the margin distribution also converge. This proof holds for deep homogeneous networks — a broad class of networks allowing for ReLU, max-pooling, linear, and convolutional layers — and we additionally provide empirical support not just close to the theory (e.g., the AlexNet), but also on non-homogeneous networks (e.g., the DenseNet). If the network further has locally Lipschitz gradients, we show that these gradients also converge in direction, and asymptotically align with the gradient flow path, with consequences on margin maximization, convergence of saliency maps, and a few other settings. Our analysis complements and is distinct from the well-known neural tangent and mean-field theories, and in particular makes no requirements on network width and initialization, instead merely requiring perfect classification accuracy. The proof proceeds by developing a theory of unbounded nonsmooth Kurdyka-Łojasiewicz inequalities for functions definable in an o-minimal structure, and is also applicable outside deep learning.

We study the task of replicating the functionality of black-box neural models, for which we only know the output class probabilities provided for a set of input images. We assume back-propagation through the black-box model is not possible and its training images are not available, e.g. the model could be exposed only through an API. In this context, we present a teacher-student framework that can distill the black-box (teacher) model into a student model with minimal accuracy loss. To generate useful data samples for training the student, our framework (i) learns to generate images on a proxy data set (with images and classes different from those used to train the black-box) and (ii) applies an evolutionary strategy to make sure that each generated data sample exhibits a high response for a specific class when given as input to the black box. Our framework is compared with several baseline and state-of-the-art methods on three benchmark data sets. The empirical evidence indicates that our model is superior to the considered baselines. Although our method does not back-propagate through the black-box network, it generally surpasses state-of-the-art methods that regard the teacher as a glass-box model. Our code is available at: https://github.com/antoniobarbalau/black-box-ripper.

One of the most important steps toward interpretability and explainability of neural network models is feature selection, which aims to identify the subset of relevant features. Theoretical results in the field have mostly focused on the prediction aspect of the problem with virtually no work on feature selection consistency for deep neural networks due to the model's severe nonlinearity and unidentifiability. This lack of theoretical foundation casts doubt on the applicability of deep learning to contexts where correct interpretations of the features play a central role.

In this work, we investigate the problem of feature selection for analytic deep networks. We prove that for a wide class of networks, including deep feed-forward neural networks, convolutional neural networks and a major sub-class of residual neural networks, the Adaptive Group Lasso selection procedure with Group Lasso as the base estimator is selection-consistent. The work provides further evidence that Group Lasso might be inefficient for feature selection with neural networks and advocates the use of Adaptive Group Lasso over the popular Group Lasso.

Methods for adversarial robustness certification aim to provide an upper bound on the test error of a classifier under adversarial manipulation of its input. Current certification methods are computationally expensive and limited to attacks that optimize the manipulation with respect to a norm. We overcome these limitations by investigating the robustness properties of Nearest Prototype Classifiers (NPCs) like learning vector quantization and large margin nearest neighbor. For this purpose, we study the hypothesis margin. We prove that if NPCs use a dissimilarity measure induced by a seminorm, the hypothesis margin is a tight lower bound on the size of adversarial attacks and can be calculated in constant time—this provides the first adversarial robustness certificate calculable in reasonable time. Finally, we show that each NPC trained by a triplet loss maximizes the hypothesis margin and is therefore optimized for adversarial robustness. In the presented evaluation, we demonstrate that NPCs optimized for adversarial robustness are competitive with state-of-the-art methods and set a new benchmark with respect to computational complexity for robustness certification.

Determinantal point processes (DPPs) are a useful probabilistic model for selecting a small diverse subset out of a large collection of items, with applications in summarization, recommendation, stochastic optimization, experimental design and more. Given a kernel function and a subset size k, our goal is to sample k out of n items with probability proportional to the determinant of the kernel matrix induced by the subset (a.k.a. k-DPP). Existing k-DPP sampling algorithms require an expensive preprocessing step which involves multiple passes over all n items, making it infeasible for large datasets. A naïve heuristic addressing this problem is to uniformly subsample a fraction of the data and perform k-DPP sampling only on those items, however this method offers no guarantee that the produced sample will even approximately resemble the target distribution over the original dataset. In this paper, we develop alpha-DPP, an algorithm which adaptively builds a sufficiently large uniform sample of data that is then used to efficiently generate a smaller set of k items, while ensuring that this set is drawn exactly from the target distribution defined on all n items. We show empirically that our algorithm produces a k-DPP sample after observing only a small fraction of …

Standard machine learning algorithms typically assume that data is sampled independently from the distribution of interest. In attempts to relax this assumption, fields such as adversarial learning typically assume that data is provided by an adversary, whose sole objective is to fool a learning algorithm. However, in reality, it is often the case that data comes from self-interested agents, with less malicious goals and intentions which lie somewhere between the two settings described above. To tackle this problem, we present a Stackelberg competition model for least squares regression, in which data is provided by agents who wish to achieve specific predictions for their data. Although the resulting optimisation problem is nonconvex, we derive an algorithm which converges globally, outperforming current approaches which only guarantee convergence to local optima. We also provide empirical results on two real-world datasets, the medical personal costs dataset and the red wine dataset, showcasing the performance of our algorithm relative to algorithms which are optimal under adversarial assumptions, outperforming the state of the art.

We propose an estimator and confidence interval for computing the value of a policy from off-policy data in the contextual bandit setting. To this end we apply empirical likelihood techniques to formulate our estimator and confidence interval as simple convex optimization problems. Using the lower bound of our confidence interval, we then propose an off-policy policy optimization algorithm that searches for policies with large reward lower bound. We empirically find that both our estimator and confidence interval improve over previous proposals in finite sample regimes. Finally, the policy optimization algorithm we propose outperforms a strong baseline system for learning from off-policy data.

Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport. We study the asymptotic properties of POTD and show that in the cases when the class labels contain no error, POTD estimates the SDR subspace exclusively. Empirical studies show POTD outperforms most of the state-of-the-art linear dimension reduction methods.

Biased regularization and fine tuning are two recent meta-learning approaches. They have been shown to be effective to tackle distributions of tasks, in which the tasks’ target vectors are all close to a common meta-parameter vector. However, these methods may perform poorly on heterogeneous environments of tasks, where the complexity of the tasks’ distribution cannot be captured by a single meta- parameter vector. We address this limitation by conditional meta-learning, inferring a conditioning function mapping task’s side information into a meta-parameter vector that is appropriate for that task at hand. We characterize properties of the environment under which the conditional approach brings a substantial advantage over standard meta-learning and we highlight examples of environments, such as those with multiple clusters, satisfying these properties. We then propose a convex meta-algorithm providing a comparable advantage also in practice. Numerical experiments confirm our theoretical findings.

Learning graphical structures based on Directed Acyclic Graphs (DAGs) is a challenging problem, partly owing to the large search space of possible graphs. A recent line of work formulates the structure learning problem as a continuous constrained optimization task using the least squares objective and an algebraic characterization of DAGs. However, the formulation requires a hard DAG constraint and may lead to optimization difficulties. In this paper, we study the asymptotic role of the sparsity and DAG constraints for learning DAG models in the linear Gaussian and non-Gaussian cases, and investigate their usefulness in the finite sample regime. Based on the theoretical results, we formulate a likelihood-based score function, and show that one only has to apply soft sparsity and DAG constraints to learn a DAG equivalent to the ground truth DAG. This leads to an unconstrained optimization problem that is much easier to solve. Using gradient-based optimization and GPU acceleration, our procedure can easily handle thousands of nodes while retaining a high accuracy. Extensive experiments validate the effectiveness of our proposed method and show that the DAG-penalized likelihood objective is indeed favorable over the least squares one with the hard DAG constraint.

We introduce a deep learning model for speech denoising, a long-standing challenge in audio analysis arising in numerous applications. Our approach is based on a key observation about human speech: there is often a short pause between each sentence or word. In a recorded speech signal, those pauses introduce a series of time periods during which only noise is present. We leverage these incidental silent intervals to learn a model for automatic speech denoising given only mono-channel audio. Detected silent intervals over time expose not just pure noise but its time-varying features, allowing the model to learn noise dynamics and suppress it from the speech signal. Experiments on multiple datasets confirm the pivotal role of silent interval detection for speech denoising, and our method outperforms several state-of-the-art denoising methods, including those that accept only audio input (like ours) and those that denoise based on audiovisual input (and hence require more information). We also show that our method enjoys excellent generalization properties, such as denoising spoken languages not seen during training.

Haplotype assembly and viral quasispecies reconstruction are challenging tasks concerned with analysis of genomic mixtures using sequencing data. High-throughput sequencing technologies generate enormous amounts of short fragments (reads) which essentially oversample components of a mixture; the representation redundancy enables reconstruction of the components (haplotypes, viral strains). The reconstruction problem, known to be NP-hard, boils down to grouping together reads originating from the same component in a mixture. Existing methods struggle to solve this problem with required level of accuracy and low runtimes; the problem is becoming increasingly more challenging as the number and length of the components increase. This paper proposes a read clustering method based on a convolutional auto-encoder designed to first project sequenced fragments to a low-dimensional space and then estimate the probability of the read origin using learned embedded features. The components are reconstructed by finding consensus sequences that agglomerate reads from the same origin. Mini-batch stochastic gradient descent and dimension reduction of reads allow the proposed method to efficiently deal with massive numbers of long reads. Experiments on simulated, semi-experimental and experimental data demonstrate the ability of the proposed method to accurately reconstruct haplotypes and viral quasispecies, often demonstrating superior performance compared to state-of-the-art methods. Source …

Boolean tensor has been broadly utilized in representing high dimensional logical data collected on spatial, temporal and/or other relational domains. Boolean Tensor Decomposition (BTD) factorizes a binary tensor into the Boolean sum of multiple rank-1 tensors, which is an NP-hard problem. Existing BTD methods have been limited by their high computational cost, in applications to large scale or higher order tensors. In this work, we presented a computationally efficient BTD algorithm, namely Geometric Expansion for all-order Tensor Factorization (GETF), that sequentially identifies the rank-1 basis components for a tensor from a geometric perspective. We conducted rigorous theoretical analysis on the validity as well as algorithemic efficiency of GETF in decomposing all-order tensor. Experiments on both synthetic and real-world data demonstrated that GETF has significantly improved performance in reconstruction accuracy, extraction of latent structures and it is an order of magnitude faster than other state-of-the-art methods.

We consider the parameter estimation problem of a probabilistic generative model prescribed using a natural exponential family of distributions. For this problem, the typical maximum likelihood estimator usually overfits under limited training sample size, is sensitive to noise and may perform poorly on downstream predictive tasks. To mitigate these issues, we propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric Kullback-Leibler ball around a parametric nominal distribution. Leveraging the analytical expression of the Kullback-Leibler divergence between two distributions in the same natural exponential family, we show that the min-max estimation problem is tractable in a broad setting, including the robust training of generalized linear models. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.

A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothness - which shows how sensitive it is to changes of its input. Our goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approximation and smoothness. This leads to novel soft-max functions, each of which is optimal for a different application. The most commonly used soft-max function, called exponential mechanism, has optimal tradeoff between approximation measured in terms of expected additive approximation and smoothness measured with respect to Renyi Divergence. We introduce a soft-max function, called piece-wise linear soft-max, with optimal tradeoff between approximation, measured in terms of worst-case additive approximation and smoothness, measured with respect to l_q-norm. The worst-case approximation guarantee of the piece-wise linear mechanism enforces sparsity in the output of our soft-max function, a property that is known to be important in Machine Learning applications Martins et al. 16, Laha et al. 18 and is not satisfied by the exponential mechanism. Moreover, the l_q-smoothness is suitable for applications in Mechanism Design and Game Theory where the piece-wise linear mechanism outperforms the exponential mechanism. Finally, we investigate another soft-max function, called power mechanism, …

Recommendation Systems like YouTube are vibrant ecosystems with two types of users: Content consumers (those who watch videos) and content providers (those who create videos). While the computational task of recommending relevant content is largely solved, designing a system that guarantees high social welfare for \textit{all} stakeholders is still in its infancy. In this work, we investigate the dynamics of content creation using a game-theoretic lens. Employing a stylized model that was recently suggested by other works, we show that the dynamics will always converge to a pure Nash Equilibrium (PNE), but the convergence rate can be exponential. We complement the analysis by proposing an efficient PNE computation algorithm via a combinatorial optimization problem that is of independent interest.

Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems where they are used, and are simple to evaluate, differentiate and integrate. However, for modeling non-negative functions, which are crucial for unsupervised learning, density estimation, or non-parametric Bayesian methods, linear models are not applicable directly. Moreover, current state-of-the-art models like generalized linear models either lead to non-convex optimization problems, or cannot be easily integrated. In this paper we provide the first model for non-negative functions which benefits from the same good properties of linear models. In particular, we prove that it admits a representer theorem and provide an efficient dual formulation for convex problems. We study its representation power, showing that the resulting space of functions is strictly richer than that of generalized linear models. Finally we extend the model and the theoretical results to functions with outputs in convex cones. The paper is complemented by an experimental evaluation of the model showing its effectiveness in terms of formulation, algorithmic derivation and practical results on the problems of density estimation, regression with heteroscedastic errors, and multiple …

Program synthesis techniques construct or infer programs from user-provided specifications, such as input-output examples. Yet most specifications, especially those given by end-users, leave the synthesis problem radically ill-posed, because many programs may simultaneously satisfy the specification. Prior work resolves this ambiguity by using various inductive biases, such as a preference for simpler programs. This work introduces a new inductive bias derived by modeling the program synthesis task as rational communication, drawing insights from recursive reasoning models of pragmatics. Given a specification, we score a candidate program both on its consistency with the specification, and also whether a rational speaker would chose this particular specification to communicate that program. We develop efficient algorithms for such an approach when learning from input-output examples, and build a pragmatic program synthesizer over a simple grid-like layout domain. A user study finds that end-user participants communicate more effectively with the pragmatic program synthesizer over a non-pragmatic one.

Detecting statistical interactions between input features is a crucial and challenging task. Recent advances demonstrate that it is possible to extract learned interactions from trained neural networks. It has also been observed that, in neural networks, any interacting features must follow a strongly weighted connection to common hidden units. Motivated by the observation, in this paper, we propose to investigate the interaction detection problem from a novel topological perspective by analyzing the connectivity in neural networks. Specially, we propose a new measure for quantifying interaction strength, based upon the well-received theory of persistent homology. Based on this measure, a Persistence Interaction Dection (PID) algorithm is developed to efficiently detect interactions. Our proposed algorithm is evaluated across a number of interaction detection tasks on several synthetic and real-world datasets with different hyperparameters. Experimental results validate that the PID algorithm outperforms the state-of-the-art baselines.

Neural populations encode the sensory world imperfectly: their capacity is limited by the number of neurons, availability of metabolic and other biophysical resources, and intrinsic noise. The brain is presumably shaped by these limitations, improving efficiency by discarding some aspects of incoming sensory streams, while preferentially preserving commonly occurring, behaviorally-relevant information. Here we construct a stochastic recurrent neural circuit model that can learn efficient, task-specific sensory codes using a novel form of reward-modulated Hebbian synaptic plasticity. We illustrate the flexibility of the model by training an initially unstructured neural network to solve two different tasks: stimulus estimation, and stimulus discrimination. The network achieves high performance in both tasks by appropriately allocating resources and using its recurrent circuitry to best compensate for different levels of noise. We also show how the interaction between stimulus priors and task structure dictates the emergent network representations.

Group studies involving large cohorts of subjects are important to draw general conclusions about brain functional organization. However, the aggregation of data coming from multiple subjects is challenging, since it requires accounting for large variability in anatomy, functional topography and stimulus response across individuals. Data modeling is especially hard for ecologically relevant conditions such as movie watching, where the experimental setup does not imply well-defined cognitive operations. We propose a novel MultiView Independent Component Analysis (ICA) model for group studies, where data from each subject are modeled as a linear combination of shared independent sources plus noise. Contrary to most group-ICA procedures, the likelihood of the model is available in closed form. We develop an alternate quasi-Newton method for maximizing the likelihood, which is robust and converges quickly. We demonstrate the usefulness of our approach first on fMRI data, where our model demonstrates improved sensitivity in identifying common sources among subjects. Moreover, the sources recovered by our model exhibit lower between-sessions variability than other methods. On magnetoencephalography (MEG) data, our method yields more accurate source localization on phantom data. Applied on 200 subjects from the Cam-CAN dataset, it reveals a clear sequence of evoked activity in sensor and source space.

Diffusion-weighted magnetic resonance imaging (DWI) is the only non-invasive method for quantifying microstructure and reconstructing white-matter pathways in the living human brain. Fluctuations from multiple sources create significant noise in DWI data which must be suppressed before subsequent microstructure analysis. We introduce a self-supervised learning method for denoising DWI data, Patch2Self, which uses the entire volume to learn a full-rank locally linear denoiser for that volume. By taking advantage of the oversampled q-space of DWI data, Patch2Self can separate structure from noise without requiring an explicit model for either. We demonstrate the effectiveness of Patch2Self via quantitative and qualitative improvements in microstructure modeling, tracking (via fiber bundle coherency) and model estimation relative to other unsupervised methods on real and simulated data.

Functional magnetic resonance imaging (fMRI) is a crucial technology for gaining insights into cognitive processes in humans. Data amassed from fMRI measurements result in volumetric data sets that vary over time. However, analysing such data presents a challenge due to the large degree of noise and person-to-person variation in how information is represented in the brain. To address this challenge, we present a novel topological approach that encodes each time point in an fMRI data set as a persistence diagram of topological features, i.e. high-dimensional voids present in the data. This representation naturally does not rely on voxel-by-voxel correspondence and is robust towards noise. We show that these time-varying persistence diagrams can be clustered to find meaningful groupings between participants, and that they are also useful in studying within-subject brain state trajectories of subjects performing a particular task. Here, we apply both clustering and trajectory analysis techniques to a group of participants watching the movie 'Partly Cloudy'. We observe significant differences in both brain state trajectories and overall topological activity between adults and children watching the same movie.