arXiv subjects

Alexander Rakhlin

Publications and source records attributed to Alexander Rakhlin.

At least 55 records · Page 3Linked to original sources

arXiv2023

Model-Free Reinforcement Learning with the Decision-Estimation Coefficient

We consider the problem of interactive decision making, encompassing structured bandits and reinforcement learning with general function approximation. Recently, Foster et al. (2021) introduced the Decision-Estimation Coefficient, a measure of statistical complexity that lower bounds the optimal regret for interactive decision making, as well as a meta-algorithm, Estimation-to-Decisions, which achieves upper bounds in terms of the same quantity. Estimation-to-Decisions is a reduction, which lifts algorithms for (supervised) online estimation into algorithms for decision making. In this paper, we show that by combining Estimation-to-Decisions with a specialized form of optimistic estimation introduced by Zhang (2022), it is possible to obtain guarantees that improve upon those of Foster et al. (2021) by accommodating more lenient notions of estimation error. We use this approach to derive regret bounds for model-free reinforcement learning with value function approximation, and give structural results showing when it can and cannot help more generally.

cs.LG↗

arXiv2023

Tight Bounds for $γ$-Regret via the Decision-Estimation Coefficient

In this work, we give a statistical characterization of the $γ$-regret for arbitrary structured bandit problems, the regret which arises when comparing against a benchmark that is $γ$ times the optimal solution. The $γ$-regret emerges in structured bandit problems over a function class $\mathcal{F}$ where finding an exact optimum of $f \in \mathcal{F}$ is intractable. Our characterization is given in terms of the $γ$-DEC, a statistical complexity parameter for the class $\mathcal{F}$, which is a modification of the constrained Decision-Estimation Coefficient (DEC) of Foster et al., 2023 (and closely related to the original offset DEC of Foster et al., 2021). Our lower bound shows that the $γ$-DEC is a fundamental limit for any model class $\mathcal{F}$: for any algorithm, there exists some $f \in \mathcal{F}$ for which the $γ$-regret of that algorithm scales (nearly) with the $γ$-DEC of $\mathcal{F}$. We provide an upper bound showing that there exists an algorithm attaining a nearly matching $γ$-regret. Due to significant challenges in applying the prior results on the DEC to the $γ$-regret case, both our lower and upper bounds require novel techniques and a new algorithm.

cs.LG↗

arXiv2023

The Statistical Complexity of Interactive Decision Making

A fundamental challenge in interactive learning and decision making, ranging from bandit problems to reinforcement learning, is to provide sample-efficient, adaptive learning algorithms that achieve near-optimal regret. This question is analogous to the classical problem of optimal (supervised) statistical learning, where there are well-known complexity measures (e.g., VC dimension and Rademacher complexity) that govern the statistical complexity of learning. However, characterizing the statistical complexity of interactive learning is substantially more challenging due to the adaptive nature of the problem. The main result of this work provides a complexity measure, the Decision-Estimation Coefficient, that is proven to be both necessary and sufficient for sample-efficient interactive learning. In particular, we provide: 1. a lower bound on the optimal regret for any interactive decision making problem, establishing the Decision-Estimation Coefficient as a fundamental limit. 2. a unified algorithm design principle, Estimation-to-Decisions (E2D), which transforms any algorithm for supervised estimation into an online algorithm for decision making. E2D attains a regret bound that matches our lower bound up to dependence on a notion of estimation performance, thereby achieving optimal sample-efficient learning as characterized by the Decision-Estimation Coefficient. Taken together, these results constitute a theory of learnability for interactive decision making. When applied to reinforcement learning settings, the Decision-Estimation Coefficient recovers essentially all existing hardness results and lower bounds. More broadly, the approach can be viewed as a decision-theoretic analogue of the classical Le Cam theory of statistical estimation; it also unifies a number of existing approaches -- both Bayesian and frequentist.

cs.LG↗

arXiv2023

On the Complexity of Multi-Agent Decision Making: From Learning in Games to Partial Monitoring

A central problem in the theory of multi-agent reinforcement learning (MARL) is to understand what structural conditions and algorithmic principles lead to sample-efficient learning guarantees, and how these considerations change as we move from few to many agents. We study this question in a general framework for interactive decision making with multiple agents, encompassing Markov games with function approximation and normal-form games with bandit feedback. We focus on equilibrium computation, in which a centralized learning algorithm aims to compute an equilibrium by controlling multiple agents that interact with an unknown environment. Our main contributions are: - We provide upper and lower bounds on the optimal sample complexity for multi-agent decision making based on a multi-agent generalization of the Decision-Estimation Coefficient, a complexity measure introduced by Foster et al. (2021) in the single-agent counterpart to our setting. Compared to the best results for the single-agent setting, our bounds have additional gaps. We show that no "reasonable" complexity measure can close these gaps, highlighting a striking separation between single and multiple agents. - We show that characterizing the statistical complexity for multi-agent decision making is equivalent to characterizing the statistical complexity of single-agent decision making, but with hidden (unobserved) rewards, a framework that subsumes variants of the partial monitoring problem. As a consequence, we characterize the statistical complexity for hidden-reward interactive decision making to the best extent possible. Building on this development, we provide several new structural results, including 1) conditions under which the statistical complexity of multi-agent decision making can be reduced to that of single-agent, and 2) conditions under which the so-called curse of multiple agents can be avoided.

cs.LG↗

arXiv2023

Representation Learning with Multi-Step Inverse Kinematics: An Efficient and Optimal Approach to Rich-Observation RL

We study the design of sample-efficient algorithms for reinforcement learning in the presence of rich, high-dimensional observations, formalized via the Block MDP problem. Existing algorithms suffer from either 1) computational intractability, 2) strong statistical assumptions that are not necessarily satisfied in practice, or 3) suboptimal sample complexity. We address these issues by providing the first computationally efficient algorithm that attains rate-optimal sample complexity with respect to the desired accuracy level, with minimal statistical assumptions. Our algorithm, MusIK, combines systematic exploration with representation learning based on multi-step inverse kinematics, a learning objective in which the aim is to predict the learner's own action from the current observation and observations in the (potentially distant) future. MusIK is simple and flexible, and can efficiently take advantage of general-purpose function approximation. Our analysis leverages several new techniques tailored to non-optimistic exploration algorithms, which we anticipate will find broader use.

cs.LG↗

arXiv2022

Generative Modeling with Denoising Auto-Encoders and Langevin Sampling

We study convergence of a generative modeling method that first estimates the score function of the distribution using Denoising Auto-Encoders (DAE) or Denoising Score Matching (DSM) and then employs Langevin diffusion for sampling. We show that both DAE and DSM provide estimates of the score of the Gaussian smoothed population density, allowing us to apply the machinery of Empirical Processes. We overcome the challenge of relying only on $L^2$ bounds on the score estimation error and provide finite-sample bounds in the Wasserstein distance between the law of the population distribution and the law of this sampling scheme. We then apply our results to the homotopy method of arXiv:1907.05600 and provide theoretical justification for its empirical success.

stat.ML↗

arXiv2022

On the Complexity of Adversarial Decision Making

A central problem in online learning and decision making -- from bandits to reinforcement learning -- is to understand what modeling assumptions lead to sample-efficient learning guarantees. We consider a general adversarial decision making framework that encompasses (structured) bandit problems with adversarial rewards and reinforcement learning problems with adversarial dynamics. Our main result is to show -- via new upper and lower bounds -- that the Decision-Estimation Coefficient, a complexity measure introduced by Foster et al. in the stochastic counterpart to our setting, is necessary and sufficient to obtain low regret for adversarial decision making. However, compared to the stochastic setting, one must apply the Decision-Estimation Coefficient to the convex hull of the class of models (or, hypotheses) under consideration. This establishes that the price of accommodating adversarial rewards or dynamics is governed by the behavior of the model class under convexification, and recovers a number of existing results -- both positive and negative. En route to obtaining these guarantees, we provide new structural results that connect the Decision-Estimation Coefficient to variants of other well-known complexity measures, including the Information Ratio of Russo and Van Roy and the Exploration-by-Optimization objective of Lattimore and György.

cs.LG↗

arXiv2022

Intrinsic Dimension Estimation Using Wasserstein Distances

It has long been thought that high-dimensional data encountered in many practical machine learning tasks have low-dimensional structure, i.e., the manifold hypothesis holds. A natural question, thus, is to estimate the intrinsic dimension of a given population distribution from a finite sample. We introduce a new estimator of the intrinsic dimension and provide finite sample, non-asymptotic guarantees. We then apply our techniques to get new sample complexity bounds for Generative Adversarial Networks (GANs) depending only on the intrinsic dimension of the data.

stat.ML↗

arXiv2022

Smoothed Online Learning is as Easy as Statistical Learning

Much of modern learning theory has been split between two regimes: the classical offline setting, where data arrive independently, and the online setting, where data arrive adversarially. While the former model is often both computationally and statistically tractable, the latter requires no distributional assumptions. In an attempt to achieve the best of both worlds, previous work proposed the smooth online setting where each sample is drawn from an adversarially chosen distribution, which is smooth, i.e., it has a bounded density with respect to a fixed dominating measure. We provide tight bounds on the minimax regret of learning a nonparametric function class, with nearly optimal dependence on both the horizon and smoothness parameters. Furthermore, we provide the first oracle-efficient, no-regret algorithms in this setting. In particular, we propose an oracle-efficient improper algorithm whose regret achieves optimal dependence on the horizon and a proper algorithm requiring only a single oracle call per round whose regret has the optimal horizon dependence in the classification setting and is sublinear in general. Both algorithms have exponentially worse dependence on the smoothness parameter of the adversary than the minimax rate. We then prove a lower bound on the oracle complexity of any proper learning algorithm, which matches the oracle-efficient upper bounds up to a polynomial factor, thus demonstrating the existence of a statistical-computational gap in smooth online learning. Finally, we apply our results to the contextual bandit setting to show that if a function class is learnable in the classical setting, then there is an oracle-efficient, no-regret algorithm for contextual bandits in the case that contexts arrive in a smooth manner.

stat.ML↗

arXiv2022

Damped Online Newton Step for Portfolio Selection

We revisit the classic online portfolio selection problem, where at each round a learner selects a distribution over a set of portfolios to allocate its wealth. It is known that for this problem a logarithmic regret with respect to Cover's loss is achievable using the Universal Portfolio Selection algorithm, for example. However, all existing algorithms that achieve a logarithmic regret for this problem have per-round time and space complexities that scale polynomially with the total number of rounds, making them impractical. In this paper, we build on the recent work by Haipeng et al. 2018 and present the first practical online portfolio selection algorithm with a logarithmic regret and whose per-round time and space complexities depend only logarithmically on the horizon. Behind our approach are two key technical novelties of independent interest. We first show that the Damped Online Newton steps can approximate mirror descent iterates well, even when dealing with time-varying regularizers. Second, we present a new meta-algorithm that achieves an adaptive logarithmic regret (i.e. a logarithmic regret on any sub-interval) for mixable losses.

cs.LG↗

arXiv2021

On Submodular Contextual Bandits

We consider the problem of contextual bandits where actions are subsets of a ground set and mean rewards are modeled by an unknown monotone submodular function that belongs to a class $\mathcal{F}$. We allow time-varying matroid constraints to be placed on the feasible sets. Assuming access to an online regression oracle with regret $\mathsf{Reg}(\mathcal{F})$, our algorithm efficiently randomizes around local optima of estimated functions according to the Inverse Gap Weighting strategy. We show that cumulative regret of this procedure with time horizon $n$ scales as $O(\sqrt{n \mathsf{Reg}(\mathcal{F})})$ against a benchmark with a multiplicative factor $1/2$. On the other hand, using the techniques of (Filmus and Ward 2014), we show that an $ε$-Greedy procedure with local randomization attains regret of $O(n^{2/3} \mathsf{Reg}(\mathcal{F})^{1/3})$ against a stronger $(1-e^{-1})$ benchmark.

cs.LG↗

arXiv2021

Deep learning: a statistical viewpoint

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

math.ST↗

arXiv2021

On the Minimal Error of Empirical Risk Minimization

We study the minimal error of the Empirical Risk Minimization (ERM) procedure in the task of regression, both in the random and the fixed design settings. Our sharp lower bounds shed light on the possibility (or impossibility) of adapting to simplicity of the model generating the data. In the fixed design setting, we show that the error is governed by the global complexity of the entire class. In contrast, in random design, ERM may only adapt to simpler models if the local neighborhoods around the regression function are nearly as complex as the class itself, a somewhat counter-intuitive conclusion. We provide sharp lower bounds for performance of ERM for both Donsker and non-Donsker classes. We also discuss our results through the lens of recent studies on interpolation in overparameterized models.

math.ST↗

arXiv2021

Top-$k$ eXtreme Contextual Bandits with Arm Hierarchy

Motivated by modern applications, such as online advertisement and recommender systems, we study the top-$k$ extreme contextual bandits problem, where the total number of arms can be enormous, and the learner is allowed to select $k$ arms and observe all or some of the rewards for the chosen arms. We first propose an algorithm for the non-extreme realizable setting, utilizing the Inverse Gap Weighting strategy for selecting multiple arms. We show that our algorithm has a regret guarantee of $O(k\sqrt{(A-k+1)T \log (|\mathcal{F}|T)})$, where $A$ is the total number of arms and $\mathcal{F}$ is the class containing the regression function, while only requiring $\tilde{O}(A)$ computation per time step. In the extreme setting, where the total number of arms can be in the millions, we propose a practically-motivated arm hierarchy model that induces a certain structure in mean rewards to ensure statistical and computational efficiency. The hierarchical structure allows for an exponential reduction in the number of relevant arms for each context, thus resulting in a regret guarantee of $O(k\sqrt{(\log A-k+1)T \log (|\mathcal{F}|T)})$. Finally, we implement our algorithm using a hierarchical linear function class and show superior performance with respect to well-known benchmarks on simulated bandit feedback experiments using extreme multi-label classification datasets. On a dataset with three million arms, our reduction scheme has an average inference time of only 7.9 milliseconds, which is a 100x improvement.

stat.ML↗

arXiv2020

Learning the Linear Quadratic Regulator from Nonlinear Observations

We introduce a new problem setting for continuous control called the LQR with Rich Observations, or RichLQR. In our setting, the environment is summarized by a low-dimensional continuous latent state with linear dynamics and quadratic costs, but the agent operates on high-dimensional, nonlinear observations such as images from a camera. To enable sample-efficient learning, we assume that the learner has access to a class of decoder functions (e.g., neural networks) that is flexible enough to capture the mapping from observations to latent states. We introduce a new algorithm, RichID, which learns a near-optimal policy for the RichLQR with sample complexity scaling only with the dimension of the latent state space and the capacity of the decoder function class. RichID is oracle-efficient and accesses the decoder class only through calls to a least-squares regression oracle. Our results constitute the first provable sample complexity guarantee for continuous control with an unknown nonlinearity in the system model and general function approximation.

cs.LG↗

arXiv2020

Instance-Dependent Complexity of Contextual Bandits and Reinforcement Learning: A Disagreement-Based Perspective

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.

cs.LG↗

arXiv2020

Beyond UCB: Optimal and Efficient Contextual Bandits with Regression Oracles

A fundamental challenge in contextual bandits is to develop flexible, general-purpose algorithms with computational requirements no worse than classical supervised learning tasks such as classification and regression. Algorithms based on regression have shown promising empirical success, but theoretical guarantees have remained elusive except in special cases. We provide the first universal and optimal reduction from contextual bandits to online regression. We show how to transform any oracle for online regression with a given value function class into an algorithm for contextual bandits with the induced policy class, with no overhead in runtime or memory requirements. We characterize the minimax rates for contextual bandits with general, potentially nonparametric function classes, and show that our algorithm is minimax optimal whenever the oracle obtains the optimal rate for regression. Compared to previous results, our algorithm requires no distributional assumptions beyond realizability, and works even when contexts are chosen adversarially.

cs.LG↗

arXiv2020

Fast Mixing of Multi-Scale Langevin Dynamics under the Manifold Hypothesis

Recently, the task of image generation has attracted much attention. In particular, the recent empirical successes of the Markov Chain Monte Carlo (MCMC) technique of Langevin Dynamics have prompted a number of theoretical advances; despite this, several outstanding problems remain. First, the Langevin Dynamics is run in very high dimension on a nonconvex landscape; in the worst case, due to the NP-hardness of nonconvex optimization, it is thought that Langevin Dynamics mixes only in time exponential in the dimension. In this work, we demonstrate how the manifold hypothesis allows for the considerable reduction of mixing time, from exponential in the ambient dimension to depending only on the (much smaller) intrinsic dimension of the data. Second, the high dimension of the sampling space significantly hurts the performance of Langevin Dynamics; we leverage a multi-scale approach to help ameliorate this issue and observe that this multi-resolution algorithm allows for a trade-off between image quality and computational expense in generation.

stat.ML↗