COLT 2022 Papers

Partial observability is ubiquitous in applications of Reinforcement Learning (RL), in which agents learn to make a sequence of decisions despite lacking complete information about the latent states of the controlled system. Partially observable RL is notoriously difficult in theory—well-known complexity-theoretic results show that learning partially observable Markov decision processes (POMDPs) requires an exponential number of samples in the worst case. Yet, this does not rule out the possible existence of interesting subclasses of POMDPs, which include a large set of partial observable applications in practice while being tractable. In this paper we identify a rich family of tractable POMDPs, which we call weakly revealing POMDPs. This family rules out the pathological instances of POMDPs with non-informative observations. We prove that for weakly revealing POMDPs, a simple algorithm combining optimism and Maximum Likelihood Estimation (MLE) is sufficient to guarantee a polynomial sample complexity. To the best of our knowledge, this is the first provably sample-efficient result for learning in overcomplete POMDPs—where the number of latent states can be larger than the number of observations—in settings where exploration is necessary.

Wasserstein GANs with Gradient Penalty (WGAN-GP) are a very popular method for training generative models to produce high quality synthetic data. While WGAN-GP were initially developed to calculate the Wasserstein 1 distance between generated and real data, recent works (e.g. [23]) have provided empirical evidence that this does not occur, and have argued that WGAN-GP perform well not in spite of this issue, but because of it. In this paper we show for the first time that WGAN-GP compute the minimum of a different optimal transport problem, the so-called congested transport [7]. Congested transport determines the cost of moving one distribution to another under a transport model that penalizes congestion. For WGAN-GP, we find that the congestion penalty has a spatially varying component determined by the sampling strategy used in [12] which acts like a local speed limit, making congestion cost less in some regions than others. This aspect of the congested transport problem is new, in that the congestion penalty turns out to be unbounded and depends on the distributions to be transported, and so we provide the necessary mathematical proofs for this setting. One facet of our discovery is a formula connecting the gradient of solutions to the optimization problem in WGAN-GP to the time averaged momentum of the optimal mass flow. This is in contrast to the gradient of Kantorovich potentials for the Wasserstein 1 distance, which is just the normalized direction of flow. Based on this and other considerations, we speculate on how our results explain the observed performance of WGAN-GP. Beyond applications to GANs, our theorems also point to the possibility of approximately solving large scale congested transport problems using neural network techniques.

Consider supervised learning from i.i.d. samples {(y_i, x_i )}_{i≤n} where x_i ∈ R_p are feature vectors and y_i ∈ R are labels. We study empirical risk minimization over a class of functions that are parameterized by k = O(1) vectors θ_1 , . . . , θ_k ∈ R_p, and prove universality results both for the training and test error. Namely, under the proportional asymptotics n, p → ∞ , with n/p = Θ(1), we prove that the training error depends on the random features distribution only through its covariance structure. Further, we prove that the minimum test error over near-empirical risk minimizers enjoys similar universality properties. In particular, the asymptotics of these quantities can be computed —to leading order— under a simpler model in which the feature vectors x_i are replaced by Gaussian vectors g_i with the same covariance. Earlier universality results were limited to strongly convex learning procedures, or to feature vectors x_i with independent entries. Our results do not make any of these assumptions. Our assumptions are general enough to include feature vectors x_i that are produced by randomized featurization maps. In particular we explicitly check the assumptions for certain random features models (computing the output of a one-layer neural network with random weights) and neural tangent models (first-order Taylor approximation of two-layer networks).

We study the subject of universal online learning with non-i.i.d. processes for bounded losses. The notion of universally consistent learning was defined by Hanneke in an effort to study learning theory under minimal assumptions, where the objective is to obtain low long-run average loss for any target function. We are interested in characterizing processes for which learning is possible and whether there exist learning rules guaranteed to be universally consistent given the only assumption that such learning is possible. The case of unbounded losses is very restrictive since the learnable processes almost surely have to visit a finite number of points and as a result, simple memorization is optimistically universal. We focus on the bounded setting and give a complete characterization of the processes admitting strong and weak universal learning. We further show that the k-nearest neighbor algorithm (kNN) is not optimistically universal and present a novel variant of 1NN which is optimistically universal for general input and value spaces in both strong and weak settings. This closes all the COLT 2021 open problems posed on universal online learning.

We study universal consistency of non-i.i.d. processes in the context of online learning. A stochastic process is said to admit universal consistency if there exists a learner that achieves vanishing average loss for any measurable response function on this process. When the loss function is unbounded, [1] showed that the only processes admitting strong universal consistency are those taking a finite number of values almost surely. However, when the loss function is bounded, the class of processes admitting strong universal consistency is much richer and its characterization could be dependent on the response setting [2]. In this paper, we show that this class of processes is independent from the response setting thereby closing an open question of [3] (Open Problem 3). Specifically, we show that the class of processes that admit universal online learning is the same for binary classification as for multiclass classification with countable number of classes. Consequently, any output setting with bounded loss can be reduced to binary classification. Our reduction is constructive and practical. Indeed, we show that the nearest neighbor algorithm is transported by our construction. For binary classification on a process admitting strong universal learning, we prove that nearest neighbor successfully learns at least all finite unions of intervals.

We consider the problem of designing uniformly stable first-order optimization algorithms for empirical risk minimization. Uniform stability is often used to obtain generalization error bounds for optimization algorithms, and we are interested in a general approach to achieve it. For Euclidean geometry, we suggest a black-box conversion which given a smooth optimization algorithm, produces a uniformly stable version of the algorithm while maintaining its convergence rate up to logarithmic factors. Using this reduction we obtain a (nearly) optimal algorithm for smooth optimization with convergence rate $\tilde{O}(1/T^2)$ and uniform stability $O(T^2/n)$, resolving an open problem of Chen et al. (2018); Attia and Koren (2021). For more general geometries, we develop a variant of Mirror Descent for smooth optimization with convergence rate $\tilde{O}(1/T)$ and uniform stability $O(T/n)$, leaving open the question of devising a general conversion method as in the Euclidean case.

We contribute to advancing the understanding of Riemannian accelerated gradient methods. In particular, we revisit “\emph{Accelerated Hybrid Proximal Extragradient}” (A-HPE), a powerful framework for obtaining Euclidean accelerated methods \citep{monteiro2013accelerated}. Building on A-HPE, we then propose and analyze Riemannian A-HPE. The core of our analysis consists of two key components: (i) a set of new insights into Euclidean A-HPE itself; and (ii) a careful control of metric distortion caused by Riemannian geometry. We illustrate our framework by obtaining a few existing and new Riemannian accelerated gradient methods as special cases, while characterizing their acceleration as corollaries of our main results.

We study the following problem: Given a variable of interest, we would like to find a best linear predictor for it by choosing a subset of k relevant variables obeying a matroid constraint. This problem is a natural generalization of subset selection problems where it is necessary to spread observations amongst multiple different classes. We derive new, strengthened guarantees for this problem by improving the analysis of the residual random greedy algorithm and by developing a novel distorted local-search algorithm. To quantify our approximation guarantees, we refine the definition of weak submodularity by Das and Kempe (2011) and introduce the notion of an upper submodularity ratio, which we connect to the minimum k-sparse eigenvalue of the covariance matrix. More generally, we look at the problem of maximizing a set function f with lower and upper submodularity ratio $\gamma$ and $\beta$ under a matroid constraint. For this problem, our algorithms have asymptotic approximation guarantee 1/2 and (1 - 1/e) as the function is closer to being submodular. As a second application, we show that the Bayesian A-optimal design objective falls into our framework, leading to new guarantees for this problem as well.

In \emph{bandit with distribution shifts}, one aims to automatically adapt to unknown changes in reward distribution, and \emph{restart} exploration when necessary. While this problem has been studied for many years, a recent breakthrough of Auer et al. (2018, 2019) provides the first adaptive procedure to guarantee an optimal (dynamic) regret $\sqrt{LT}$, for $T$ rounds, and an unknown number $L$ of changes. However, while this rate is tight in the worst case, it remained open whether faster rates are possible, without prior knowledge, if few changes in distribution are actually \emph{severe}. To resolve this question, we propose a new notion of \emph{significant shift}, which only counts very severe changes that clearly necessitate a restart: roughly, these are changes involving not only best arm switches, but also involving large aggregate differences in reward overtime. Thus, our resulting procedure adaptively achieves rates always faster (sometimes significantly) than $O(\sqrt{ST})$, where $S\ll L$ only counts best arm switches, while at the same time, always faster than the optimal $O(V^{\frac{1}{3}}T^{\frac{2}{3}})$ when expressed in terms of \emph{total variation} $V$ (which aggregates differences overtime). Our results are expressed in enough generality to also capture non-stochastic adversarial settings.

Multi-task learning leverages structural similarities between multiple tasks to learn despite very few samples. Motivated by the recent success of neural networks applied to data-scarce tasks, we consider a linear low-dimensional shared representation model. Despite an extensive literature, existing theoretical results either guarantee weak estimation rates or require a large number of samples per task. This work provides the first estimation error bound for the trace norm regularized estimator when the number of samples per task is small. The advantages of trace norm regularization for learning data-scarce tasks extend to meta-learning and are confirmed empirically on synthetic datasets.

For the task of sampling from a density $\pi \propto \exp(-V)$ on $\R^d$, where $V$ is possibly non-convex but $L$-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with $\varepsilon$-relative Fisher information after $O(L^2 d^2/\varepsilon^2)$ iterations. This is the sampling analogue of complexity bounds for finding an $\varepsilon$-approximate first-order stationary points in non-convex optimization and therefore constitutes a first step towards the general theory of non-log-concave sampling. We discuss numerous extensions and applications of our result; in particular, it yields a new state-of-the-art guarantee for sampling from distributions which satisfy a Poincaré inequality.

We propose a novel algorithm for multi-player multi-armed bandits without collision sensing information. Our algorithm circumvents two problems shared by all state-of-the-art algorithms: it does not need as an input a lower bound on the minimal expected reward of an arm, and its performance does not scale inversely proportionally to the minimal expected reward. We prove a theoretical regret upper bound to justify these claims. We complement our theoretical results with numerical experiments, showing that the proposed algorithm outperforms state-of-the-art in practice.

We revisit the basic problem of quantum state certification: given copies of unknown mixed state ρ∈ℂ^{d×d} and the description of a mixed state σ, decide whether σ=ρ or ‖σ−ρ‖_𝗍𝗋 ≥ ϵ. When σ is maximally mixed, this is mixedness testing, and it is known that Ω(d^{Θ(1)}/ϵ^2) copies are necessary, where the exact exponent depends on the type of measurements the learner can make [OW15, BCL20], and in many of these settings there is a matching upper bound [OW15, BOW19, BCL20]. Can one avoid this d^{Θ(1)} dependence for certain kinds of mixed states σ, e.g. ones which are approximately low rank? More ambitiously, does there exist a simple functional f : ℂ^{d×d} → ℝ_{≥0} for which one can show that Θ(f(σ)/ϵ^2) copies are necessary and sufficient for state certification with respect to any σ? Such instance-optimal bounds are known in the context of classical distribution testing, e.g. [VV17]. Here we give the first bounds of this nature for the quantum setting, showing (up to log factors) that the copy complexity for state certification using nonadaptive incoherent measurements is essentially given by the copy complexity for mixedness testing times the fidelity between σ and the maximally mixed state. Surprisingly, our bound differs substantially from instance optimal bounds for the classical problem, demonstrating a qualitative difference between the two settings.

Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not. We prove that for $n\ge k\ge 2$, learning the components of an $n$-vertex hidden graph with $k$ components requires at least $(k-1)n-\binom k2$ membership queries. Our result improves on the best known information-theoretic bound of $\Omega(n\log k)$ queries, and exactly matches the query complexity of the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem. Additionally, we introduce an oracle that can learn the number of components of $G$ in asymptotically fewer queries than learning the full partition, thus answering another question posed by the same authors. Lastly, we introduce a more applicable version of this oracle, and prove asymptotically tight bounds of $\widetilde\Theta(m)$ queries for both learning and verifying an $m$-edge hidden graph $G$ using it.

Thompson Sampling (TS) is an efficient method for decision-making under uncertainty, where an action is sampled from a carefully prescribed distribution which is updated based on the observed data. In this work, we study the problem of adaptive control of stabilizable linear-quadratic regulators (LQRs) using TS, where the system dynamics are unknown. Previous works have established that $\tilde{O}(\sqrt{T})$ frequentist regret is optimal for the adaptive control of LQRs. However, the existing methods either work only in restrictive settings, require a priori known stabilizing controllers or utilize computationally intractable approaches. We propose an efficient TS algorithm for the adaptive control of LQRs, TS-based Adaptive Control, TSAC, that attains $\tilde{O}(\sqrt{T})$ regret, even for multidimensional systems, thereby solving the open problem posed in Abeille and Lazaric (2018). TSAC does not require a priori known stabilizing controller and achieves fast stabilization of the underlying system by effectively exploring the environment in the early stages. Our result hinges on developing a novel lower bound on the probability that the TS provides an optimistic sample. By carefully prescribing an early exploration strategy and a policy update rule, we show that TS achieves order-optimal regret in adaptive control of multidimensional stabilizable LQRs. We empirically demonstrate the performance and the efficiency of the proposed algorithm in several adaptive control tasks.

We establish the first tight lower bound of $\Omega(\log\log\kappa)$ on the query complexity of sampling from the class of strongly log-concave and log-smooth distributions with condition number $\kappa$ in one dimension. Whereas existing guarantees for MCMC-based algorithms scale polynomially in $\kappa$, we introduce a novel algorithm based on rejection sampling that closes this doubly exponential gap.

It is currently known how to characterize functions that neural networks can learn with SGD for two extremal parametrizations: neural networks in the linear regime, and neural networks with no structural constraints. However, for the main parametrization of interest —non-linear but regular networks— no tight characterization has yet been achieved, despite significant developments. We take a step in this direction by considering depth-2 neural networks trained by SGD in the mean-field regime. We consider functions on binary inputs that depend on a latent low-dimensional subspace (i.e., small number of coordinates). This regime is of interest since it is poorly understood how neural networks routinely tackle high-dimensional datasets and adapt to latent low-dimensional structure without suffering from the curse of dimensionality. Accordingly, we study SGD-learnability with $O(d)$ sample complexity in a large ambient dimension $d$. Our main results characterize a hierarchical property —the merged-staircase property— that is both \emph{necessary and nearly sufficient} for learning in this setting. We further show that non-linear training is necessary: for this class of functions, linear methods on any feature map (e.g., the NTK) are not capable of learning efficiently. The key tools are a new “dimension-free” dynamics approximation result that applies to functions defined on a latent space of low-dimension, a proof of global convergence based on polynomial identity testing, and an improvement of lower bounds against linear methods for non-almost orthogonal functions.

The Lovász hinge is a convex surrogate recently proposed for structured binary classification, in which k binary predictions are made simultaneously and the error is judged by a submodular set function. Despite its wide usage in image segmentation and related problems, its consistency has remained open. We resolve this open question, showing that the Lovász hinge is inconsistent for its desired target unless the set function is modular. Leveraging a recent embedding framework, we instead derive the target loss for which the Lovász hinge is consistent. This target, which we call the structured abstain problem, allows one to abstain on any subset of the k predictions. We derive two link functions, each of which are consistent for all submodular set functions simultaneously.

We study high-dimensional sparse estimation under three natural constraints: communication constraints, local privacy constraints, and linear measurements (compressive sensing). Without sparsity assumptions, it has been established that interactivity cannot improve the minimax rates of estimation under these information constraints. The question of whether interactivity helps with natural inference tasks has been a topic of active research. We settle this question in the affirmative for the prototypical problems of high-dimensional sparse mean estimation and compressive sensing, by demonstrating a gap between interactive and noninteractive protocols. We further establish that the gap increases when we have more structured sparsity: for \emph{block sparsity} this gap can be as large as \emph{polynomial} in the dimensionality. Thus, the more structured the sparsity is, the greater is the advantage of interaction. Proving the lower bounds requires a careful breaking of a sum of correlated random variables into independent components using Baranyai’s theorem on decomposition of hypergraphs, which might be of independent interest.