COLT 2021 Papers

We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent’s best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under self-play), agnostic (no need to know the actions played by the opponent), symmetric (players taking symmetric roles in the algorithm), and enjoying a finite-time last-iterate convergence guarantee, all of which are desirable properties of decentralized algorithms.

We introduce kernel thinning, a new procedure for compressing a distribution $\mathbb{P}$ more effectively than i.i.d. sampling or standard thinning. Given a suitable reproducing kernel $\mathbf{k}$ and $\mathcal{O}(n^2)$ time, kernel thinning compresses an $n$-point approximation to $\mathbb{P}$ into a $\sqrt{n}$-point approximation with comparable worst-case integration error across the associated reproducing kernel Hilbert space. With high probability, the maximum discrepancy in integration error is $\mathcal{O}_d(n^{-\frac{1}{2}}\sqrt{\log n})$ for compactly supported $\mathbb{P}$ and $\mathcal{O}_d(n^{-\frac{1}{2}} \sqrt{(\log n)^{d+1}\log\log n})$ for sub-exponential $\mathbb{P}$ on $\mathbb{R}^d$. In contrast, an equal-sized i.i.d. sample from $\mathbb{P}$ suffers $\Omega(n^{-\frac14})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\mathbb{P}$ on $[0,1]^d$ but apply to general distributions on $\mathbb{R}^d$ and a wide range of common kernels. We use our results to derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Matérn, and B-spline kernels and present two vignettes illustrating the practical benefits of kernel thinning over i.i.d. sampling and standard Markov chain Monte Carlo thinning.

The seminal result of Johnson and Lindenstrauss on random embeddings has been intensively studied in applied and theoretical computer science. Despite that vast body of literature, we still lack of complete understanding of statistical properties of random projections; a particularly intriguing question is: why are the theoretical bounds that far behind the empirically observed performance? Motivated by this question, this work develops Johnson-Lindenstrauss distributions with optimal, data-oblivious, statistical confidence bounds. These bounds are numerically best possible, for any given data dimension, embedding dimension, and distortion tolerance. They improve upon prior works in terms of statistical accuracy, as well as exactly determine the no-go regimes for data-oblivious approaches. Furthermore, the projection matrices are efficiently samplable. The construction relies on orthogonal matrices, and the proof uses certain elegant properties of the unit sphere. In particular, the following techniques introduced in this work are of independent interest: a) a compact expression for the projection distortion in terms of singular eigenvalues of the projection matrix, b) a parametrization linking the unit sphere and the Dirichlet distribution and c) anti-concentration bounds for the Dirichlet distribution. Besides the technical contribution, the paper presents applications and numerical evaluation along with working implementation in Python (shared as a GitHub repository).

We prove a phase transition known as the “all-or-nothing” phenomenon for noiseless discrete channels. This class of models includes the Bernoulli group testing model and the planted Gaussian perceptron model. Previously, the existence of the all-or-nothing phenomenon for such models was only known in a limited range of parameters. Our work extends the results to all signals with sublinear sparsity. Our main technique is to show that for such models, the “all” half of all-or-nothing implies the “nothing” half, so that a proof of “all” can be turned into a proof of “nothing.” Since the “all” half can often be proven by straightforward means, our equivalence gives a powerful and general approach towards establishing the existence of this phenomenon in other contexts.

Episodic reinforcement learning and contextual bandits are two widely studied sequential decision-making problems. Episodic reinforcement learning generalizes contextual bandits and is often perceived to be more difficult due to long planning horizon and unknown state-dependent transitions. The current paper shows that the long planning horizon and the unknown state-dependent transitions (at most) pose little additional difficulty on sample complexity. We consider the episodic reinforcement learning with S states, A actions, planning horizon H, total reward bounded by 1, and the agent plays for K episodes. We propose a new algorithm, Monotonic Value Propagation (MVP), which relies on a new Bernstein-type bonus. The new bonus only requires tweaking the constants to ensure optimism and thus is significantly simpler than existing bonus constructions. We show MVP enjoys an $O\left(\left(\sqrt{SAK} + S^2A\right) \poly\log \left(SAHK\right)\right)$ regret, approaching the $\Omega\left(\sqrt{SAK}\right)$ lower bound of contextual bandits. Notably, this result 1) exponentially improves the state-of-the-art polynomial-time algorithms by Dann et al. [2019], Zanette et al. [2019], and Zhang et al. [2020] in terms of the dependency on H, and 2) exponentially improves the running time in [Wang et al. 2020] and significantly improves the dependency on S, A and K in sample complexity.

In the classical multi-armed bandit problem, instance-dependent algorithms attain improved performance on "easy" problems with a gap between the best and second-best arm. Are similar guarantees possible for contextual bandits? While positive results are known for certain special cases, there is no general theory characterizing when and how instance-dependent regret bounds for contextual bandits can be achieved for rich, general classes of policies. We introduce a family of complexity measures that are both sufficient and necessary to obtain instance-dependent regret bounds. We then introduce new oracle-efficient algorithms which adapt to the gap whenever possible, while also attaining the minimax rate in the worst case. Finally, we provide structural results that tie together a number of complexity measures previously proposed throughout contextual bandits, reinforcement learning, and active learning and elucidate their role in determining the optimal instance-dependent regret. In a large-scale empirical evaluation, we find that our approach often gives superior results for challenging exploration problems. Turning our focus to reinforcement learning with function approximation, we develop new oracle-efficient algorithms for reinforcement learning with rich observations that obtain optimal gap-dependent sample complexity.

We study the generalization properties of the popular stochastic optimization method known as stochastic gradient descent (SGD) for optimizing general non-convex loss functions. Our main contribution is providing upper bounds on the generalization error that depend on local statistics of the stochastic gradients evaluated along the path of iterates calculated by SGD. The key factors our bounds depend on are the variance of the gradients (with respect to the data distribution) and the local smoothness of the objective function along the SGD path, and the sensitivity of the loss function to perturbations to the final output. Our key technical tool is combining the information-theoretic generalization bounds previously used for analyzing randomized variants of SGD with a perturbation analysis of the iterates.

We present an efficient algorithm for stochastic bandit convex optimisation with no assumptions on smoothness or strong convexity and for which the regret is bounded by O(d^(4.5) sqrt(n) polylog(n)), where n is the number of interactions and d is the dimension.

We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(\Delta,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{\Delta_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $\Delta_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the corruption magnitude, and $\log_+(x) = \max\left(1,\log x\right)$. The regime includes stochastic bandits, stochastically constrained adversarial bandits, and stochastic bandits with adversarial corruptions as special cases. Additionally, we provide a general analysis, which allows to achieve the same kind of improvement for generalizations of Tsallis-INF to other settings beyond multiarmed bandits.

We give a computationally-efficient PAC active learning algorithm for $d$-dimensional homogeneous halfspaces that can tolerate Massart noise (Massart and Nedelec, 2006) and Tsybakov noise (Tsybakov, 2004). Specialized to the $\eta$-Massart noise setting, our algorithm achieves an information-theoretically near-optimal label complexity of $\tilde{O}\left( \frac{d}{(1-2\eta)^2} \mathrm{polylog}(\frac1\epsilon) \right)$ under a wide range of unlabeled data distributions (specifically, the family of ``structured distributions' defined in Diakonikolas et al. (2020)). Under the more challenging Tsybakov noise condition, we identify two subfamilies of noise conditions, under which our efficient algorithm provides label complexity guarantees strictly lower than passive learning algorithms.

We resolve the long-standing "impossible tuning" issue for the classic expert problem and show that, it is in fact possible to achieve regret $O\left(\sqrt{(\ln d)\sum_t \ell_{t,i}^2}\right)$ simultaneously for all expert $i$ in a $T$-round $d$-expert problem where $\ell_{t,i}$ is the loss for expert $i$ in round $t$. Our algorithm is based on the Mirror Descent framework with a correction term and a weighted entropy regularizer. While natural, the algorithm has not been studied before and requires a careful analysis. We also generalize the bound to $O\left(\sqrt{(\ln d)\sum_t (\ell_{t,i}-m_{t,i})^2}\right)$ for any prediction vector $m_t$ that the learner receives, and recover or improve many existing results by choosing different $m_t$. Furthermore, we use the same framework to create a master algorithm that combines a set of base algorithms and learns the best one with little overhead. The new guarantee of our master allows us to derive many new results for both the expert problem and more generally Online Linear Optimization.

Random graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For the correlated Erdös-Rényi model, we prove the first impossibility result for partial recovery in the sparse regime (with constant average degree). Our bound is tight in the noiseless case (the graph isomorphism problem) and we conjecture that it is still tight with noise. Our proof technique relies on a careful application of the probabilistic method to build automorphisms between tree components of a subcritical Erdös-Rényi graph.

Understanding the implicit regularization (or implicit bias) of gradient descent has recently been a very active research area. However, the implicit regularization in nonlinear neural networks is still poorly understood, especially for regression losses such as the square loss. Perhaps surprisingly, we prove that even for a single ReLU neuron, it is impossible to characterize the implicit regularization with the square loss by any explicit function of the model parameters (although on the positive side, we show it can be characterized approximately). For one hidden-layer networks, we prove a similar result, where in general it is impossible to characterize implicit regularization properties in this manner, except for the “balancedness” property identified in Du et al. (2018). Our results suggest that a more general framework than the one considered so far may be needed to understand implicit regularization for nonlinear predictors, and provides some clues on what this framework should be.

Analysis of low-degree polynomial algorithms is a powerful, newly-popular method for predicting computational thresholds in hypothesis testing problems. One limitation of current techniques for this analysis is their restriction to Bernoulli and Gaussian distributions. We expand this range of possibilities by performing the low-degree analysis of hypothesis testing for the Morris class of natural exponential families with quadratic variance function, giving a unified treatment of Gaussian, Poisson, gamma (including exponential and chi-squared), binomial (including Bernoulli), negative binomial (including geometric), and generalized hyperbolic secant distributions. We then give several algorithmic applications. 1. In models where a random signal is observed through coordinatewise-independent noise applied in an exponential family, the success or failure of low-degree polynomials is governed by the z-score overlap, the inner product of z-score vectors with respect to the null distribution of two independent copies of the signal. 2. In the same models, testing with low-degree polynomials exhibits channel monotonicity: the above distributions admit a total ordering by computational cost of hypothesis testing, according to a scalar parameter describing how the variance depends on the mean in an exponential family. 3. In a spiked matrix model with a particular non-Gaussian noise distribution, the low-degree prediction is incorrect unless polynomials with arbitrarily large degree in individual matrix entries are permitted. This shows that polynomials summing over self-avoiding walks and variants thereof, as proposed recently by Ding, Hopkins, and Steurer (2020) for spiked matrix models with heavy-tailed noise, are strictly suboptimal for this model. Thus low-degree polynomials appear to offer a tradeoff between robustness and strong performance fine-tuned to specific models. Inspired by this, we suggest that a class of problems requiring "exploration before inference," where an algorithm must first examine the input and then use some intermediate computation to choose a suitable inference subroutine, appears especially difficult for low-degree polynomials.

Group testing is a fundamental problem in statistical inference with many real-world applications, including the need for massive group testing during the ongoing COVID-19 pandemic. In this paper we study the task of approximate recovery, in which we tolerate having a small number of incorrectly classified items. One of the most well-known, optimal, and easy to implement testing procedures is the non-adaptive Bernoulli group testing, where all tests are conducted in parallel, and each item is chosen to be part of any certain test independently with some fixed probability. In this setting, there is an observed gap between the number of tests above which recovery is information theoretically possible, and the number of tests required by the currently best known efficient algorithms to succeed. In this paper we seek to understand whether this computational-statistical gap can be closed. Our main contributions are the following: 1.Often times such gaps are explained by a phase transition in the landscape of the solution space of the problem (an Overlap Gap Property (OGP) phase transition). We provide first moment evidence that, perhaps surprisingly, such a phase transition does not take place throughout the regime for which recovery is information theoretically possible. This fact suggests that the model is in fact amenable to local search algorithms. 2. We prove the complete absence of “bad” local minima for a part of the “hard” regime, a fact which implies an improvement over known theoretical results on the performance of efficient algorithms for approximate recovery without false-negatives. 3. Finally, motivated by the evidence for the abscence for the OGP, we present extensive simulations that strongly suggest that a very simple local algorithm known as Glauber Dynamics does indeed succeed, and can be used to efficiently implement the well-known (theoretically optimal) Smallest Satisfying Set (SSS) estimator. Given that practical algorithms for this task utilize Branch and Bound and Linear Programming relaxation techniques, our finding could potentially be of practical interest.

This paper explores a theory of generalization for learning problems on product distributions, complementing the existing learning theories in the sense that it does not rely on any complexity measures of the hypothesis classes. The main contributions are two general sample complexity bounds: (1) $\tilde{O} \big( \frac{nk}{\epsilon^2} \big)$ samples are sufficient and necessary for learning an $\epsilon$-optimal hypothesis in \emph{any problem} on an $n$-dimensional product distribution, whose marginals have finite supports of sizes at most $k$; (2) $\tilde{O} \big( \frac{n}{\epsilon^2} \big)$ samples are sufficient and necessary for any problem on $n$-dimensional product distributions if it satisfies a notion of strong monotonicity from the algorithmic game theory literature. As applications of these theories, we match the optimal sample complexity for single-parameter revenue maximization (Guo et al., STOC 2019), improve the state-of-the-art for multi-parameter revenue maximization (Gonczarowski and Weinberg, FOCS 2018) and prophet inequality (Correa et al., EC 2019; Rubinstein et al., ITCS 2020), and provide the first and tight sample complexity bound for Pandora’s problem.

We initiate a program of average smoothness analysis for efficiently learning real-valued functions on metric spaces. Rather than using the Lipschitz constant as the regularizer, we define a local slope at each point and gauge the function complexity as the average of these values. Since the mean can be dramatically smaller than the maximum, this complexity measure can yield considerably sharper generalization bounds — assuming that these admit a refinement where the Lipschitz constant is replaced by our average of local slopes. In addition to the usual average, we also examine a “weak” average that is more forgiving and yields a much wider function class. Our first major contribution is to obtain just such distribution-sensitive bounds. This required overcoming a number of technical challenges, perhaps the most formidable of which was bounding the {\em empirical} covering numbers, which can be much worse-behaved than the ambient ones. Our combinatorial results are accompanied by efficient algorithms for smoothing the labels of the random sample, as well as guarantees that the extension from the sample to the whole space will continue to be, with high probability, smooth on average. Along the way we discover a surprisingly rich combinatorial and analytic structure in the function class we define.

We prove hardness-of-learning results under a well-studied assumption on the existence of local pseudorandom generators. As we show, this assumption allows us to surpass the current state of the art, and prove hardness of various basic problems, with no hardness results to date. Our results include: hardness of learning shallow ReLU neural networks under the Gaussian distribution and other distributions; hardness of learning intersections of $\omega(1)$ halfspaces, DNF formulas with $\omega(1)$ terms, and ReLU networks with $\omega(1)$ hidden neurons; hardness of weakly learning deterministic finite automata under the uniform distribution; hardness of weakly learning depth-$3$ Boolean circuits under the uniform distribution, as well as distribution-specific hardness results for learning DNF formulas and intersections of halfspaces. We also establish lower bounds on the complexity of learning intersections of a constant number of halfspaces, and ReLU networks with a constant number of hidden neurons. Moreover, our results imply the hardness of virtually all improper PAC-learning problems (both distribution-free and distribution-specific) that were previously shown hard under other assumptions.

We consider variants of the classical Frank-Wolfe algorithm for constrained smooth convex minimization, that instead of access to the standard oracle for minimizing a linear function over the feasible set, have access to an oracle that can find an extreme point of the feasible set that is closest in Euclidean distance to a given vector. We first show that for many feasible sets of interest, such an oracle can be implemented with the same complexity as the standard linear optimization oracle. We then show that with such an oracle we can design new Frank-Wolfe variants which enjoy significantly improved complexity bounds in case the set of optimal solutions lies in the convex hull of a subset of extreme points with small diameter (e.g., a low-dimensional face of a polytope). In particular, for many $0\text{–}1$ polytopes, under quadratic growth and strict complementarity conditions, we obtain the first linearly convergent variant with rate that depends only on the dimension of the optimal face and not on the ambient dimension.