Diffusion probabilistic models have shown significant progress in video generation; however, their computational efficiency is limited by the large number of sampling steps required. Reducing sampling steps often compromises video quality or generation diversity. In this work, we introduce a distillation method that combines variational score distillation and consistency distillation to achieve few-step video generation, maintaining both high quality and diversity. We also propose a latent reward model fine-tuning approach to further enhance video generation performance according to any specified reward metric. This approach reduces memory usage and does not require the reward to be differentiable. Our method demonstrates state-of-the-art performance in few-step generation for 10-second videos (128 frames at 12 FPS). The distilled student model achieves a score of 82.57 on VBench, surpassing the teacher model as well as baseline models Gen-3, T2V-Turbo, and Kling. One-step distillation accelerates the teacher model's diffusion sampling by up to 278.6 times, enabling near real-time generation. Human evaluations further validate the superior performance of our 4-step student models compared to teacher model using 50-step DDIM sampling.

Benign overfitting refers to the phenomenon where an over-parameterized model fits the training data perfectly, including noise in the data, but still generalizes well to the unseen test data. While prior work provides some theoretical understanding of this phenomenon under the in-distribution setup, modern machine learning often operates in a more challenging Out-of-Distribution (OOD) regime, where the target (test) distribution can be rather different from the source (training) distribution. In this work, we take an initial step towards understanding benign overfitting in the OOD regime by focusing on the basic setup of over-parameterized linear models under covariate shift. We provide non-asymptotic guarantees proving that benign overfitting occurs in standard ridge regression, even under the OOD regime when the target covariance satisfies certain structural conditions. We identify several vital quantities relating to source and target covariance, which govern the performance of OOD generalization. Our result is sharp, which provably recovers prior in-distribution benign overfitting guarantee (Tsigler & Bartlett, 2023), as well as under-parameterized OOD guarantee (Ge et al., 2024) when specializing to each setup. Moreover, we also present theoretical results for a more general family of target covariance matrix, where standard ridge regression only achieves a slow statistical rate of $\mathcal{O}(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $\mathcal{O}(1/n)$, where $n$ is the number of samples.

This paper examines multiplayer symmetric constant-sum games with more than two players in a competitive setting, such as Mahjong, Poker, and various board and video games. In contrast to two-player zero-sum games, equilibria in multiplayer games are neither unique nor non-exploitable, failing to provide meaningful guarantees when competing against opponents who play different equilibria or non-equilibrium strategies. This gives rise to a series of long-lasting fundamental questions in multiplayer games regarding suitable objectives, solution concepts, and principled algorithms. This paper takes an initial step towards addressing these challenges by focusing on the natural objective of equal share—securing an expected payoff of $C/n$ in an $n$-player symmetric game with a total payoff of $C$. We rigorously identify the theoretical conditions under which achieving an equal share is tractable and design a series of efficient algorithms, inspired by no-regret learning, that provably attain approximate equal share across various settings. Furthermore, we provide complementary lower bounds that justify the sharpness of our theoretical results. Our experimental results highlight worst-case scenarios where meta-algorithms from prior state-of-the-art systems for multiplayer games fail to secure an equal share, while our algorithm succeeds, demonstrating the effectiveness of our approach.

Large language models have demonstrated impressive performance on challenging mathematical reasoning tasks, which has triggered the discussion of whether the performance is achieved by true reasoning capability or memorization. To investigate this question, prior work has constructed mathematical benchmarks when questions undergo simple perturbations – modifications that still preserve the underlying reasoning patterns of the solutions. However, no work has explored hard perturbations, which fundamentally change the nature of the problem so that the original solution steps do not apply. To bridge the gap, we construct MATH-P-Simple and MATH-P-Hard via simple perturbation and hard perturbation, respectively. Each consists of 279 perturbed math problems derived from level-5 (hardest) problems in the MATH dataset (Hendrycks et al., 2021). We observe significant performance drops on MATH-P-Hard across various models, including o1-mini (-16.49%) and gemini-2.0-flash-thinking (-12.9%). We also raise concerns about a novel form of memorization where models blindly apply learned problem-solving skills without assessing their applicability to modified contexts. This issue is amplified when using original problems for in-context learning. We call for research efforts to address this challenge, which is critical for developing more robust and reliable reasoning models. The project is available at https://math-perturb.github.io/.

We introduce PokéChamp, a minimax agent powered by Large Language Models (LLMs) for Pokémon battles. Built on a general framework for two-player competitive games, PokéChamp leverages the generalist capabilities of LLMs to enhance minimax tree search. Specifically, LLMs replace three key modules: (1) player action sampling, (2) opponent modeling, and (3) value function estimation, enabling the agent to effectively utilize gameplay history and human knowledge to reduce the search space and address partial observability. Notably, our framework requires no additional LLM training. We evaluate PokéChamp in the popular Gen 9 OU format. When powered by GPT-4o, it achieves a win rate of 76\% against the best existing LLM-based bot and 84\% against the strongest rule-based bot, demonstrating its superior performance. Even with an open-source 8-billion-parameter Llama 3.1 model, PokéChamp consistently outperforms the previous best LLM-based bot, Pokéllmon powered by GPT-4o, with a 64\% win rate. PokéChamp attains a projected Elo of 1300-1500 on the Pokémon Showdown online ladder, placing it among the top 30\%-10\% of human players. In addition, this work compiles the largest real-player Pokémon battle dataset, featuring over 3 million games, including more than 500k high-Elo matches. Based on this dataset, we establish a series of battle benchmarks and puzzles to evaluate specific battling skills. We further provide key updates to the local game engine. This work establishes Pokémon as a benchmark to integrate LLM technologies with game-theoretic algorithms addressing general multi-agent problems. Videos, code, and dataset are available online.

Foundational world models must be both interactive and preserve spatialtemporal coherence to enable effective future planning with different action choices. However, present models for long video generation have limited inherent world modeling capabilities due to two main challenges: compounding errors and insufficient memory mechanisms. We enhance image-to-video models with interactive capabilities through additional action conditioning and autoregressive framework, and reveal that compounding error is inherently irreducible in autoregressive video generation, while insufficient memory mechanism leads to incoherence of world models. We propose video retrieval augmented generation (VRAG) with explicit global state conditioning, which significantly reduces long-term compounding errors and increases spatialtemporal consistency of video world models. In contrast, naive autoregressive generation with extended context windows and retrieval-augmented generation prove less effective for video generation, primarily due to the limited in-context learning capabilities of current video models. Our work illuminates the fundamental challenges in video world models and establishes a comprehensive benchmark for improving video generation models with internal world modeling capabilities.

Modern machine learning often requires training with large batch size, distributed data, and massively parallel compute hardware (like mobile and other edge devices or distributed data centers). Communication becomes a major bottleneck in such settings but methods like Local Stochastic Gradient Descent (Local SGD) show great promise to reduce the global communication need. Local SGD consists of three parts: a local optimization processes, an aggregation mechanism, and an outer optimizer that uses the aggregated updates from the nodes to produce a new model. While there exists an extensive literature on understanding the impact of hyperparameters in the local optimization process, the choice of outer optimizer and its hyperparameters is less clear. We study the role of the outer learning in Local SGD, and prove new convergence guarantees for the algorithm. In particular, we show that tuning the outer learning rate allows us to (a) trade off between optimization error and stochastic gradient noise variance, and (b) make up for ill-tuning of the inner learning rate. Our theory suggests that the outer learning rate should sometimes be set to values greater than $1$. We extend our results to apply to when we use momentum in the outer optimizer, and also introduce a novel data-dependent analysis of Local SGD that yields further insights on outer learning rate tuning. We conduct comprehensive experiments with standard language models and various outer optimizers to validate our theory.

LLM-based formal proof assistants (e.g., in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question in context of mathematical inequalities---specifically the prover's ability to recognize that the given problem simplifies by applying a known inequality such as AM/GM. Specifically, we are interested in their ability to do this in a {\em compositional setting} where multiple inequalities must be applied as part of a solution. We introduce \ineqcomp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers---including Goedel, STP, and Kimina-7B---struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness, but still suffers a 20\% performance drop (pass@32). Even for DeepSeek-Prover-V2-671B model, the gap between compositional variants and seed problems exists, implying that simply scaling up the model size alone does not fully solve the compositional weakness. Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition. All data and evaluation code can be found at \url{https://github.com/haoyuzhao123/LeanIneqComp}.

We provide a unified analysis of two-timescale gradient descent ascent (TTGDA) for solving structured nonconvex minimax optimization problems in the form of $\min_x \max_{y \in Y} f(x, y)$, where the objective function $f(x, y)$ is nonconvex in $x$ and concave in $y$, and the constraint set $Y \subseteq \mathbb{R}^n$ is convex and bounded. In the convex-concave setting, the single-timescale gradient descent ascent (GDA) algorithm is widely used in applications and has been shown to have strong convergence guarantees. In more general settings, however, it can fail to converge. Our contribution is to design TTGDA algorithms that are effective beyond the convex-concave setting, efficiently finding a stationary point of the function $\Phi(\cdot) := \max_{y \in Y} f(\cdot, y)$. We also establish theoretical bounds on the complexity of solving both smooth and nonsmooth nonconvex-concave minimax optimization problems. To the best of our knowledge, this is the first systematic analysis of TTGDA for nonconvex minimax optimization, shedding light on its superior performance in training generative adversarial networks (GANs) and in other real-world application problems.

Decentralized finance (DeFi) borrowing and lending platforms are crucial to the decentralized economy, involving two main participants: lenders who provide assets for interest and borrowers who offer collateral exceeding their debt and pay interest. Collateral volatility necessitates over-collateralization to protect lenders and ensure competitive returns. Traditional DeFi platforms use a fixed interest rate curve based on the utilization rate (the fraction of available assets borrowed) and determine over-collateralization offline through simulations to manage risk. This method doesn't adapt well to dynamic market changes, such as price fluctuations and evolving user needs, often resulting in losses for lenders or borrowers. In this paper, we introduce an adaptive, data-driven protocol for DeFi borrowing and lending. Our approach includes a high-frequency controller that dynamically adjusts interest rates to maintain market stability and competitiveness with external markets. Unlike traditional protocols, which rely on user reactions and often adjust slowly, our controller uses a learning-based algorithm to quickly find optimal interest rates, reducing the opportunity cost for users during periods of misalignment with external rates. Additionally, we use a low-frequency planner that analyzes user behavior to set an optimal over-collateralization ratio, balancing risk reduction with profit maximization over the long term. This dual approach is essential for adaptive markets: the short-term component maintains market stability, preventing exploitation, while the long-term planner optimizes market parameters to enhance profitability and reduce risks. We provide theoretical guarantees on the convergence rates and adversarial robustness of the short-term component and the long-term effectiveness of our protocol. Empirical validation confirms our protocol's theoretical benefits.

Score-based generative models like the diffusion model have been testified to be effective in modeling multi-modal data from image generation to reinforcement learning (RL). However, the inference process of diffusion model can be slow, which hinders its usage in RL with iterative sampling. We propose to apply the consistency model as an efficient yet expressive policy representation, namely consistency policy, with an actor-critic style algorithm for three typical RL settings: offline, offline-to-online and online. For offline RL, we demonstrate the expressiveness of generative models as policies from multi-modal data. For offline-to-online RL, the consistency policy is shown to be more computational efficient than diffusion policy, with a comparable performance. For online RL, the consistency policy demonstrates significant speedup and even higher average performances than the diffusion policy.

Unsupervised pretraining, which learns a useful representation using a large amount of unlabeled data to facilitate the learning of downstream tasks, is a critical component of modern large-scale machine learning systems. Despite its tremendous empirical success, the rigorous theoretical understanding of why unsupervised pretraining generally helps remains rather limited---most existing results are restricted to particular methods or approaches for unsupervised pretraining with specialized structural assumptions. This paper studies a generic framework, where the unsupervised representation learning task is specified by an abstract class of latent variable models $\Phi$ and the downstream task is specified by a class of prediction functions $\Psi$. We consider a natural approach of using Maximum Likelihood Estimation (MLE) for unsupervised pretraining and Empirical Risk Minimization (ERM) for learning downstream tasks. We prove that, under a mild ``informative'' condition, our algorithm achieves an excess risk of $\\tilde{\\mathcal{O}}(\sqrt{\mathcal{C}\_\Phi/m} + \sqrt{\mathcal{C}\_\Psi/n})$ for downstream tasks, where $\mathcal{C}\_\Phi, \mathcal{C}\_\Psi$ are complexity measures of function classes $\Phi, \Psi$, and $m, n$ are the number of unlabeled and labeled data respectively. Comparing to the baseline of $\tilde{\mathcal{O}}(\sqrt{\mathcal{C}\_{\Phi \circ \Psi}/n})$ achieved by performing supervised learning using only the labeled data, our result rigorously shows the benefit of unsupervised pretraining when $m \gg n$ and $\mathcal{C}\_{\Phi\circ \Psi} > \mathcal{C}\_\Psi$. This paper further shows that our generic framework covers a wide range of approaches for unsupervised pretraining, including factor models, Gaussian mixture models, and contrastive learning.

A key challenge of modern machine learning systems is to achieve Out-of-Distribution (OOD) generalization---generalizing to target data whose distribution differs from that of source data. Despite its significant importance, the fundamental question of ``what are the most effective algorithms for OOD generalization'' remains open even under the standard setting of covariate shift. This paper addresses this fundamental question by proving that, surprisingly, classical Maximum Likelihood Estimation (MLE) purely using source data (without any modification) achieves the *minimax* optimality for covariate shift under the *well-specified* setting. That is, *no* algorithm performs better than MLE in this setting (up to a constant factor), justifying MLE is all you need. Our result holds for a very rich class of parametric models, and does not require any boundedness condition on the density ratio. We illustrate the wide applicability of our framework by instantiating it to three concrete examples---linear regression, logistic regression, and phase retrieval. This paper further complement the study by proving that, under the *misspecified setting*, MLE is no longer the optimal choice, whereas Maximum Weighted Likelihood Estimator (MWLE) emerges as minimax optimal in certain scenarios.

Large-scale machine learning problems make the cost of hyperparameter tuning ever more prohibitive. This creates a need for algorithms that can tune themselves on-the-fly. We formalize the notion of “tuning-free” algorithms that can match the performance of optimally-tuned optimization algorithms up to polylogarithmic factors given only loose hints on the relevant problem parameters. We consider in particular algorithms that can match optimally-tuned Stochastic Gradient Descent (SGD). When the domain of optimization is bounded, we show tuning-free matching of SGD is possible and achieved by several existing algorithms. We prove that for the task of minimizing a convex and smooth or Lipschitz function over an unbounded domain, tuning-free optimization is impossible. We discuss conditions under which tuning-free optimization is possible even over unbounded domains. In particular, we show that the recently proposed DoG and DoWG algorithms are tuning-free when the noise distribution is sufficiently well-behaved. For the task of finding a stationary point of a smooth and potentially nonconvex function, we give a variant of SGD that matches the best-known high-probability convergence rate for tuned SGD at only an additional polylogarithmic cost. However, we also give an impossibility result that shows no algorithm can hope to match the optimal expected convergence rate for tuned SGD with high probability.

Recent advances in reinforcement learning (RL) heavily rely on a variety of well-designed benchmarks, which provide environmental platforms and consistent criteria to evaluate existing and novel algorithms. Specifically, in multi-agent RL (MARL), a plethora of benchmarks based on cooperative games have spurred the development of algorithms that improve the scalability of cooperative multi-agent systems. However, for the competitive setting, a lightweight and open-sourced benchmark with challenging gaming dynamics and visual inputs has not yet been established. In this work, we present FightLadder, a real-time fighting game platform, to empower competitive MARL research. Along with the platform, we provide implementations of state-of-the-art MARL algorithms for competitive games, as well as a set of evaluation metrics to characterize the performance and exploitability of agents. We demonstrate the feasibility of this platform by training a general agent that consistently defeats 12 built-in characters in single-player mode, and expose the difficulty of training a non-exploitable agent without human knowledge and demonstrations in two-player mode. FightLadder provides meticulously designed environments to address critical challenges in competitive MARL research, aiming to catalyze a new era of discovery and advancement in the field. Videos and code at https://sites.google.com/view/fightladder/home.

A unique challenge in Multi-Agent Reinforcement Learning (MARL) is the \emph{curse of multiagency}, where the description length of the game as well as the complexity of many existing learning algorithms scale \emph{exponentially} in the number of agents. While recent work successfully addresses this challenge under the model of tabular Markov Games, their mechanisms critically rely on the number of states being finite and small, and do not extend to practical scenarios with enormous state spaces where \emph{function approximation} must be used to approximate value functions or policies. This paper presents the first line of MARL algorithms that provably resolve the curse of multiagency under function approximation. We design a new algorithm—\emph{V-Learning with Policy Replay}, which gives the first \emph{polynomial} sample complexity results for learning approximate Coarse Correlated Equilibria (CCEs) of Markov Games under decentralized linear function approximation. Our algorithm always outputs Markov CCEs, and its sample complexity also significantly improves over the current best result for finding Markov CCEs in the tabular setting. This paper further presents an alternative algorithm—\emph{Decentralized Optimistic Policy Mirror Descent}, which finds policy-class-restricted CCEs using a polynomial number of samples. In exchange for learning a weaker version of CCEs, this algorithm applies to a wider range of problems under generic function approximation, such as linear quadratic games and MARL problems with low “marginal” Eluder dimension.

Particle gradient descent, which uses particles to represent a probability measure and performs gradient descent on particles in parallel, is widely used to optimize functions of probability measures. This paper considers particle gradient descent with a finite number of particles and establishes its theoretical guarantees to optimize functions that are displacement convex in measures. Concretely, for Lipschitz displacement convex functions defined on probability over $R^d$, we prove that $O(1/\epsilon^2)$ particles and $O(d/\epsilon^4)$ iterations are sufficient to find the $\epsilon$-optimal solutions. We further provide improved complexity bounds for optimizing smooth displacement convex functions. An application of our results proves the conjecture of no optimization-barrier up to permutation invariance, proposed by Entezari et al. (2022), for specific two-layer neural networks with two-dimensional inputs uniformly drawn from unit circle.

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games under a restricted set of policies. The latter case can be further reduce to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.

This paper proposes a new easy-to-implement parameter-free gradient-based optimizer: DoWG (Distance over Weighted Gradients). We prove that DoWG is efficient---matching the convergence rate of optimally tuned gradient descent in convex optimization up to a logarithmic factor without tuning any parameters, and universal---automatically adapting to both smooth and nonsmooth problems. While popular algorithms following the AdaGrad framework compute a running average of the squared gradients, DoWG maintains a new distance-based weighted version of the running average, which is crucial to achieve the desired properties. To complement our theory, we also show empirically that DoWG trains at the edge of stability, and validate its effectiveness on practical machine learning tasks.

We consider a contextual bandit problem with $S $ contexts and $K $ actions. In each round $t=1,2,\dots$ the learner observes a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into $r\le \min(S ,K)$ groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an $\epsilon$-optimal policy after using at most $\widetilde O(r (S +K )/\epsilon^2)$ samples with high probability and provide a matching $\widetilde\Omega(r (S +K )/\epsilon^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r ^3(S +K )T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O{\sqrt{\text{poly}(r)(S+K)T}}$ minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.

While policy optimization algorithms have played an important role in recent empirical success of Reinforcement Learning (RL), the existing theoretical understanding of policy optimization remains rather limited---they are either restricted to tabular MDPs or suffer from highly suboptimal sample complexity, especial in online RL where exploration is necessary. This paper proposes a simple efficient policy optimization framework---Optimistic NPG for online RL. Optimistic NPG can be viewed as simply combining of the classic natural policy gradient (NPG) algorithm [Kakade, 2001] with optimistic policy evaluation subroutines to encourage exploration. For $d$-dimensional linear MDPs, Optimistic NPG is computationally efficient, and learns an $\epsilon$-optimal policy within $\tilde{\mathcal{O}}(d^2/\epsilon^3)$ samples, which is the first computationally efficient algorithm whose sample complexity has the optimal dimension dependence $\tilde{\Theta}(d^2)$. It also improves over state-of-the-art results of policy optimization algorithms [Zanette et al., 2021] by a factor of $d$. For general function approximation that subsumes linear MDPs, Optimistic NPG, to our best knowledge, is also the first policy optimization algorithm that achieves the polynomial sample complexity for learning near-optimal policies.

This paper introduces a simple efficient learning algorithms for general sequential decision making. The algorithm combines Optimism for exploration with Maximum Likelihood Estimation for model estimation, which is thus named OMLE. We prove that OMLE learns the near-optimal policies of an enormously rich class of sequential decision making problems in a polynomial number of samples. This rich class includes not only a majority of known tractable model-based Reinforcement Learning (RL) problems (such as tabular MDPs, factored MDPs, low witness rank problems, tabular weakly-revealing/observable POMDPs and multi-step decodable POMDPs ), but also many new challenging RL problems especially in the partially observable setting that were not previously known to be tractable.

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.

This paper resolves the open question of designing near-optimal algorithms for learning imperfect-information extensive-form games from bandit feedback. We present the first line of algorithms that require only $\widetilde{\mathcal{O}}((XA+YB)/\varepsilon^2)$ episodes of play to find an $\varepsilon$-approximate Nash equilibrium in two-player zero-sum games, where $X,Y$ are the number of information sets and $A,B$ are the number of actions for the two players. This improves upon the best known sample complexity of $\widetilde{\mathcal{O}}((X^2A+Y^2B)/\varepsilon^2)$ by a factor of $\widetilde{\mathcal{O}}(\max\{X, Y\})$, and matches the information-theoretic lower bound up to logarithmic factors. We achieve this sample complexity by two new algorithms: Balanced Online Mirror Descent, and Balanced Counterfactual Regret Minimization. Both algorithms rely on novel approaches of integrating balanced exploration policies into their classical counterparts. We also extend our results to learning Coarse Correlated Equilibria in multi-player general-sum games.

Real-world sequential decision making problems commonly involve partial observability, which requires the agent to maintain a memory of history in order to infer the latent states, plan and make good decisions. Coping with partial observability in general is extremely challenging, as a number of worst-case statistical and computational barriers are known in learning Partially Observable Markov Decision Processes (POMDPs). Motivated by the problem structure in several physical applications, as well as a commonly used technique known as "frame stacking", this paper proposes to study a new subclass of POMDPs, whose latent states can be decoded by the most recent history of a short length m. We establish a set of upper and lower bounds on the sample complexity for learning near-optimal policies for this class of problems in both tabular and rich-observation settings (where the number of observations is enormous). In particular, in the rich-observation setting, we develop new algorithms using a novel "moment matching" approach with a sample complexity that scales exponentially with the short length m rather than the problem horizon, and is independent of the number of observations. Our results show that a short-term memory suffices for reinforcement learning in these environments.

Modern reinforcement learning (RL) commonly engages practical problems with large state spaces, where function approximation must be deployed to approximate either the value function or the policy. While recent progresses in RL theory address a rich set of RL problems with general function approximation, such successes are mostly restricted to the single-agent setting. It remains elusive how to extend these results to multi-agent RL, especially in the face of new game-theoretical challenges. This paper considers two-player zero-sum Markov Games (MGs). We propose a new algorithm that can provably find the Nash equilibrium policy using a polynomial number of samples, for any MG with low multi-agent Bellman-Eluder dimension—a new complexity measure adapted from its single-agent version (Jin et al., 2021). A key component of our new algorithm is the exploiter, which facilitates the learning of the main player by deliberately exploiting her weakness. Our theoretical framework is generic, which applies to a wide range of models including but not limited to tabular MGs, MGs with linear or kernel function approximation, and MGs with rich observations.

An ideal strategy in zero-sum games should not only grant the player an average reward no less than the value of Nash equilibrium, but also exploit the (adaptive) opponents when they are suboptimal. While most existing works in Markov games focus exclusively on the former objective, it remains open whether we can achieve both objectives simultaneously. To address this problem, this work studies no-regret learning in Markov games with adversarial opponents when competing against the best fixed policy in hindsight. Along this direction, we present a new complete set of positive and negative results: When the policies of the opponents are revealed at the end of each episode, we propose new efficient algorithms achieving $\sqrt{K}$ regret bounds when either (1) the baseline policy class is small or (2) the opponent’s policy class is small. This is complemented with an exponential lower bound when neither conditions are true. When the policies of the opponents are not revealed, we prove a statistical hardness result even in the most favorable scenario when both above conditions are true. Our hardness result is much stronger than the existing hardness results which either only involve computational hardness, or require further restrictions on the algorithms.

In constrained reinforcement learning (RL), a learning agent seeks to not only optimize the overall reward but also satisfy the additional safety, diversity, or budget constraints. Consequently, existing constrained RL solutions require several new algorithmic ingredients that are notably different from standard RL. On the other hand, reward-free RL is independently developed in the unconstrained literature, which learns the transition dynamics without using the reward information, and thus naturally capable of addressing RL with multiple objectives under the common dynamics. This paper bridges reward-free RL and constrained RL. Particularly, we propose a simple meta-algorithm such that given any reward-free RL oracle, the approachability and constrained RL problems can be directly solved with negligible overheads in sample complexity. Utilizing the existing reward-free RL solvers, our framework provides sharp sample complexity results for constrained RL in the tabular MDP setting, matching the best existing results up to a factor of horizon dependence; our framework directly extends to a setting of tabular two-player Markov games, and gives a new result for constrained RL with linear function approximation.

A conceptually appealing approach for learning Extensive-Form Games (EFGs) is to convert them to Normal-Form Games (NFGs). This approach enables us to directly translate state-of-the-art techniques and analyses in NFGs to learning EFGs, but typically suffers from computational intractability due to the exponential blow-up of the game size introduced by the conversion. In this paper, we address this problem in natural and important setups for the \emph{$\Phi$-Hedge} algorithm---A generic algorithm capable of learning a large class of equilibria for NFGs. We show that $\Phi$-Hedge can be directly used to learn Nash Equilibria (zero-sum settings), Normal-Form Coarse Correlated Equilibria (NFCCE), and Extensive-Form Correlated Equilibria (EFCE) in EFGs. We prove that, in those settings, the \emph{$\Phi$-Hedge} algorithms are equivalent to standard Online Mirror Descent (OMD) algorithms for EFGs with suitable dilated regularizers, and run in polynomial time. This new connection further allows us to design and analyze a new class of OMD algorithms based on modifying its log-partition function. In particular, we design an improved algorithm with balancing techniques that achieves a sharp $\widetilde{\mathcal{O}}(\sqrt{XAT})$ EFCE-regret under bandit-feedback in an EFG with $X$ information sets, $A$ actions, and $T$ episodes. To our best knowledge, this is the first such rate and matches the information-theoretic lower bound.

This paper considers the challenging tasks of Multi-Agent Reinforcement Learning (MARL) under partial observability, where each agent only sees her own individual observations and actions that reveal incomplete information about the underlying state of system. This paper studies these tasks under the general model of multiplayer general-sum Partially Observable Markov Games (POMGs), which is significantly larger than the standard model of Imperfect Information Extensive-Form Games (IIEFGs). We identify a rich subclass of POMGs---weakly revealing POMGs---in which sample-efficient learning is tractable. In the self-play setting, we prove that a simple algorithm combining optimism and Maximum Likelihood Estimation (MLE) is sufficient to find approximate Nash equilibria, correlated equilibria, as well as coarse correlated equilibria of weakly revealing POMGs, in a polynomial number of samples when the number of agents is small. In the setting of playing against adversarial opponents, we show that a variant of our optimistic MLE algorithm is capable of achieving sublinear regret when being compared against the optimal maximin policies. To our best knowledge, this work provides the first line of sample-efficient results for learning POMGs.

While over-parameterization is widely believed to be crucial for the success of optimization for the neural networks, most existing theories on over-parameterization do not fully explain the reason—they either work in the Neural Tangent Kernel regime where neurons don’t move much, or require an enormous number of neurons. In practice, when the data is generated using a teacher neural network, even mildly over-parameterized neural networks can achieve 0 loss and recover the directions of teacher neurons. In this paper we develop a local convergence theory for mildly over-parameterized two-layer neural net. We show that as long as the loss is already lower than a threshold (polynomial in relevant parameters), all student neurons in an over-parameterized two-layer neural network will converge to one of teacher neurons, and the loss will go to 0. Our result holds for any number of student neurons as long as it is at least as large as the number of teacher neurons, and our convergence rate is independent of the number of student neurons. A key component of our analysis is the new characterization of local optimization landscape—we show the gradient satisfies a special case of Lojasiewicz property which is different from local strong convexity or PL conditions used in previous work.

This paper studies representation learning for multi-task linear bandits and multi-task episodic RL with linear value function approximation. We first consider the setting where we play $M$ linear bandits with dimension $d$ concurrently, and these bandits share a common $k$-dimensional linear representation so that $k\ll d$ and $k \ll M$. We propose a sample-efficient algorithm, MTLR-OFUL, which leverages the shared representation to achieve $\tilde{O}(M\sqrt{dkT} + d\sqrt{kMT} )$ regret, with $T$ being the number of total steps. Our regret significantly improves upon the baseline $\tilde{O}(Md\sqrt{T})$ achieved by solving each task independently. We further develop a lower bound that shows our regret is near-optimal when $d > M$. Furthermore, we extend the algorithm and analysis to multi-task episodic RL with linear value function approximation under low inherent Bellman error (Zanette et al., 2020a). To the best of our knowledge, this is the first theoretical result that characterize the benefits of multi-task representation learning for exploration in RL with function approximation.

Model-based algorithms—algorithms that explore the environment through building and utilizing an estimated model—are widely used in reinforcement learning practice and theoretically shown to achieve optimal sample efficiency for single-agent reinforcement learning in Markov Decision Processes (MDPs). However, for multi-agent reinforcement learning in Markov games, the current best known sample complexity for model-based algorithms is rather suboptimal and compares unfavorably against recent model-free approaches. In this paper, we present a sharp analysis of model-based self-play algorithms for multi-agent Markov games. We design an algorithm \emph{Optimistic Nash Value Iteration} (Nash-VI) for two-player zero-sum Markov games that is able to output an $\epsilon$-approximate Nash policy in $\tilde{\mathcal{O}}(H^3SAB/\epsilon^2)$ episodes of game playing, where $S$ is the number of states, $A,B$ are the number of actions for the two players respectively, and $H$ is the horizon length. This significantly improves over the best known model-based guarantee of $\tilde{\mathcal{O}}(H^4S^2AB/\epsilon^2)$, and is the first that matches the information-theoretic lower bound $\Omega(H^3S(A+B)/\epsilon^2)$ except for a $\min\{A,B\}$ factor. In addition, our guarantee compares favorably against the best known model-free algorithm if $\min\{A,B\}=o(H^3)$, and outputs a single Markov policy while existing sample-efficient model-free algorithms output a nested mixture of Markov policies that is in general non-Markov and rather inconvenient to store and execute. We further adapt our analysis to designing a provably efficient task-agnostic algorithm for zero-sum Markov games, and designing the first line of provably sample-efficient algorithms for multi-player general-sum Markov games.

Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning—a key tool for performing meta-learning—learns a data representation that can transfer knowledge across multiple tasks, which is essential in regimes where data is scarce. Despite a recent surge of interest in the practice of meta-learning, the theoretical underpinnings of meta-learning algorithms are lacking, especially in the context of learning transferable representations. In this paper, we focus on the problem of multi-task linear regression—in which multiple linear regression models share a common, low-dimensional linear representation. Here, we provide provably fast, sample-efficient algorithms to address the dual challenges of (1) learning a common set of features from multiple, related tasks, and (2) transferring this knowledge to new, unseen tasks. Both are central to the general problem of meta-learning. Finally, we complement these results by providing information-theoretic lower bounds on the sample complexity of learning these linear features.

Real world applications such as economics and policy making often involve solving multi-agent games with two unique features: (1) The agents are inherently asymmetric and partitioned into leaders and followers; (2) The agents have different reward functions, thus the game is general-sum. The majority of existing results in this field focuses on either symmetric solution concepts (e.g. Nash equilibrium) or zero-sum games. It remains open how to learn the Stackelberg equilibrium---an asymmetric analog of the Nash equilibrium---in general-sum games efficiently from noisy samples. This paper initiates the theoretical study of sample-efficient learning of the Stackelberg equilibrium, in the bandit feedback setting where we only observe noisy samples of the reward. We consider three representative two-player general-sum games: bandit games, bandit-reinforcement learning (bandit-RL) games, and linear bandit games. In all these games, we identify a fundamental gap between the exact value of the Stackelberg equilibrium and its estimated version using finitely many noisy samples, which can not be closed information-theoretically regardless of the algorithm. We then establish sharp positive results on sample-efficient learning of Stackelberg equilibrium with value optimal up to the gap identified above, with matching lower bounds in the dependency on the gap, error tolerance, and the size of the action spaces. Overall, our results unveil unique challenges in learning Stackelberg equilibria under noisy bandit feedback, which we hope could shed light on future research on this topic.

Finding the minimal structural assumptions that empower sample-efficient learning is one of the most important research directions in Reinforcement Learning (RL). This paper advances our understanding of this fundamental question by introducing a new complexity measure—Bellman Eluder (BE) dimension. We show that the family of RL problems of low BE dimension is remarkably rich, which subsumes a vast majority of existing tractable RL problems including but not limited to tabular MDPs, linear MDPs, reactive POMDPs, low Bellman rank problems as well as low Eluder dimension problems. This paper further designs a new optimization-based algorithm— GOLF, and reanalyzes a hypothesis elimination-based algorithm—OLIVE (proposed in Jiang et al. (2017)). We prove that both algorithms learn the near-optimal policies of low BE dimension problems in a number of samples that is polynomial in all relevant parameters, but independent of the size of state-action space. Our regret and sample complexity results match or improve the best existing results for several well-known subclasses of low BE dimension problems.

Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD and SGD can avoid saddle points, but the dependence on dimension in these analyses is polynomial. For modern machine learning, where the dimension can be in the millions, such dependence would be catastrophic. We analyze perturbed versions of GD and SGD and show that they are truly efficient—their dimension dependence is only polylogarithmic. Indeed, these algorithms converge to second-order stationary points in essentially the same time as they take to converge to classical first-order stationary points.

Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where \emph{function approximation} must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation tradeoff. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic setting with linear dynamics and linear rewards, for which only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial runtime and polynomial sample complexity in this linear setting, without requiring a “simulator” or additional assumptions. Concretely, we prove that an optimistic modification of Least-Squares Value Iteration (LSVI)—a classical algorithm frequently studied in the linear setting—achieves $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret, where $d$ is the ambient dimension of feature space, $H$ is the length of each episode, and $T$ is the total number of steps. Importantly, such regret is independent of the number of states and actions.

This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using $\tilde{O}(\kappa_x+\kappa_y)$ or $\tilde{O}(\text{min}\{\kappa_x\sqrt{\kappa_y}, \sqrt{\kappa_x}\kappa_y\})$ gradient evaluations, where $\kappa_x$ and $\kappa_y$ are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap still remains between these results and the best existing lower bound $\tilde{\Omega}(\sqrt{\kappa_x\kappa_y})$. This paper presents the first algorithm with $\tilde{O}(\sqrt{\kappa_x\kappa_y})$ gradient complexity, matching the lower bound up to logarithmic factors. Our algorithm is designed based on an accelerated proximal point method and an accelerated solver for minimax proximal steps. It can be easily extended to the settings of strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvex-concave functions. This paper also presents algorithms that match or outperform all existing methods in these settings in terms of gradient complexity, up to logarithmic factors.

Self-play, where the algorithm learns by playing against itself without requiring any direct supervision, has become the new weapon in modern Reinforcement Learning (RL) for achieving superhuman performance in practice. However, the majority of exisiting theory in reinforcement learning only applies to the setting where the agent plays against a fixed environment; it remains largely open whether self-play algorithms can be provably effective, especially when it is necessary to manage the exploration/exploitation tradeoff. We study self-play in competitive reinforcement learning under the setting of Markov games, a generalization of Markov decision processes to the two-player case. We introduce a self-play algorithm—Value Iteration with Upper/Lower Confidence Bound (VI-ULCB)—and show that it achieves regret $\mathcal{\tilde{O}}(\sqrt{T})$ after playing $T$ steps of the game, where the regret is measured by the agent’s performance against a fully adversarial opponent who can exploit the agent’s strategy at any step. We also introduce an explore-then-exploit style algorithm, which achieves a slightly worse regret of $\mathcal{\tilde{O}}(T^{2/3})$, but is guaranteed to run in polynomial time even in the worst case. To the best of our knowledge, our work presents the first line of provably sample-efficient self-play algorithms for competitive reinforcement learning.

While policy-based reinforcement learning (RL) achieves tremendous successes in practice, it is significantly less understood in theory, especially compared with value-based RL. In particular, it remains elusive how to design a provably efficient policy optimization algorithm that incorporates exploration. To bridge such a gap, this paper proposes an Optimistic variant of the Proximal Policy Optimization algorithm (OPPO), which follows an “optimistic version” of the policy gradient direction. This paper proves that, in the problem of episodic Markov decision process with linear function approximation, unknown transition, and adversarial reward with full-information feedback, OPPO achieves $\tilde{O}(\sqrt{d^2 H^3 T})$ regret. Here $d$ is the feature dimension, $H$ is the episode horizon, and $T$ is the total number of steps. To the best of our knowledge, OPPO is the first provably efficient policy optimization algorithm that explores.

We consider the task of learning in episodic finite-horizon Markov decision processes with an unknown transition function, bandit feedback, and adversarial losses. We propose an efficient algorithm that achieves $\mathcal{\tilde{O}}(L|X|\sqrt{|A|T})$ regret with high probability, where $L$ is the horizon, $|X|$ the number of states, $|A|$ the number of actions, and T the number of episodes. To our knowledge, our algorithm is the first to ensure $\mathcal{\tilde{O}}(\sqrt{T})$ regret in this challenging setting; in fact, it achieves the same regret as (Rosenberg & Mansour, 2019a) who consider the easier setting with full-information. Our key contributions are two-fold: a tighter confidence set for the transition function; and an optimistic loss estimator that is inversely weighted by an "upper occupancy bound".

Exploration is widely regarded as one of the most challenging aspects of reinforcement learning (RL), with many naive approaches succumbing to exponential sample complexity. To isolate the challenges of exploration, we propose the following “reward-free RL” framework. In the exploration phase, the agent first collects trajectories from an MDP $M$ without a pre-specified reward function. After exploration, it is tasked with computing a near-policies under the transitions of $\mathcal{M}$ for a collection of given reward functions. This framework is particularly suitable where there are many reward functions of interest, or where the reward function is shaped by an external agent to elicit desired behavior. We give an efficient algorithm that conducts $\widetilde{O}(S^2A\mathrm{poly}(H)/\epsilon^2)$ episodes of exploration, and returns $\epsilon$-suboptimal policies for an arbitrary number of reward functions. We achieve this by finding exploratory policies that jointly visit each “significant” state with probability proportional to its maximum visitation probability under any possible policy. Moreover, our planning procedure can be instantiated by any black-box approximate planner, such as value iteration or natural policy gradient. Finally, we give a nearly-matching $\Omega(S^2AH^2/\epsilon^2)$ lower bound, demonstrating the near-optimality of our algorithm in this setting.

Minimax optimization has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises—“what is a proper definition of local optima?” Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and adversarial training, correspond to sequential games, where the order of which player acts first is crucial (since minimax is in general not equal to maximin due to the nonconvex-nonconcave nature of the problems). The main contribution of this paper is to propose a proper mathematical definition of local optimality for this sequential setting—local minimax, as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithm—gradient descent ascent (GDA): under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points.

We consider nonconvex-concave minimax problems, $\min_{\mathbf{x}} \max_{\mathbf{y} \in \mathcal{Y}} f(\mathbf{x}, \mathbf{y})$, where $f$ is nonconvex in $\mathbf{x}$ but concave in $\mathbf{y}$ and $\mathcal{Y}$ is a convex and bounded set. One of the most popular algorithms for solving this problem is the celebrated gradient descent ascent (GDA) algorithm, which has been widely used in machine learning, control theory and economics. Despite the extensive convergence results for the convex-concave setting, GDA with equal stepsize can converge to limit cycles or even diverge in a general setting. In this paper, we present the complexity results on two-time-scale GDA for solving nonconvex-concave minimax problems, showing that the algorithm can find a stationary point of the function $\Phi(\cdot) := \max_{\mathbf{y} \in \mathcal{Y}} f(\cdot, \mathbf{y})$ efficiently. To the best our knowledge, this is the first nonasymptotic analysis for two-time-scale GDA in this setting, shedding light on its superior practical performance in training generative adversarial networks (GANs) and other real applications.

This paper considers the problem of designing optimal algorithms for reinforcement learning in two-player zero-sum games. We focus on self-play algorithms which learn the optimal policy by playing against itself without any direct supervision. In a tabular episodic Markov game with S states, A max-player actions and B min-player actions, the best existing algorithm for finding an approximate Nash equilibrium requires \tlO(S^2AB) steps of game playing, when only highlighting the dependency on (S,A,B). In contrast, the best existing lower bound scales as \Omega(S(A+B)) and has a significant gap from the upper bound. This paper closes this gap for the first time: we propose an optimistic variant of the Nash Q-learning algorithm with sample complexity \tlO(SAB), and a new Nash V-learning algorithm with sample complexity \tlO(S(A+B)). The latter result matches the information-theoretic lower bound in all problem-dependent parameters except for a polynomial factor of the length of each episode. In addition, we present a computational hardness result for learning the best responses against a fixed opponent in Markov games---a learning objective different from finding the Nash equilibrium.

Partial observability is a common challenge in many reinforcement learning applications, which requires an agent to maintain memory, infer latent states, and integrate this past information into exploration. This challenge leads to a number of computational and statistical hardness results for learning general Partially Observable Markov Decision Processes (POMDPs). This work shows that these hardness barriers do not preclude efficient reinforcement learning for rich and interesting subclasses of POMDPs. In particular, we present a sample-efficient algorithm, OOM-UCB, for episodic finite undercomplete POMDPs, where the number of observations is larger than the number of latent states and where exploration is essential for learning, thus distinguishing our results from prior works. OOM-UCB achieves an optimal sample complexity of $\tilde{\mathcal{O}}(1/\varepsilon^2)$ for finding an $\varepsilon$-optimal policy, along with being polynomial in all other relevant quantities. As an interesting special case, we also provide a computationally and statistically efficient algorithm for POMDPs with deterministic state transitions.

We provide new statistical guarantees for transfer learning via representation learning--when transfer is achieved by learning a feature representation shared across different tasks. This enables learning on new tasks using far less data than is required to learn them in isolation. Formally, we consider $t+1$ tasks parameterized by functions of the form $f_j \circ h$ in a general function class $F \circ H$, where each $f_j$ is a task-specific function in $F$ and $h$ is the shared representation in $H$. Letting $C(\cdot)$ denote the complexity measure of the function class, we show that for diverse training tasks (1) the sample complexity needed to learn the shared representation across the first $t$ training tasks scales as $C(H) + t C(F)$, despite no explicit access to a signal from the feature representation and (2) with an accurate estimate of the representation, the sample complexity needed to learn a new task scales only with $C(F)$. Our results depend upon a new general notion of task diversity--applicable to models with general tasks, features, and losses--as well as a novel chain rule for Gaussian complexities. Finally, we exhibit the utility of our general framework in several models of importance in the literature.

Reinforcement learning (RL) algorithms combined with modern function approximators such as kernel functions and deep neural networks have achieved significant empirical successes in large-scale application problems with a massive number of states. From a theoretical perspective, however, RL with functional approximation poses a fundamental challenge to developing algorithms with provable computational and statistical efficiency, due to the need to take into consideration both the exploration-exploitation tradeoff that is inherent in RL and the bias-variance tradeoff that is innate in statistical estimation. To address such a challenge, focusing on the episodic setting where the action-value functions are represented by a kernel function or over-parametrized neural network, we propose the first provable RL algorithm with both polynomial runtime and sample complexity, without additional assumptions on the data-generating model. In particular, for both the kernel and neural settings, we prove that an optimistic modification of the least-squares value iteration algorithm incurs an $\tilde{\mathcal{O}}(\delta_{\cF} H^2 \sqrt{T})$ regret, where $\delta_{\cF}$ characterizes the intrinsic complexity of the function class $\cF$, $H$ is the length of each episode, and $T$ is the total number of episodes. Our regret bounds are independent of the number of states and therefore even allows it to diverge, which exhibits the benefit of function approximation.

Nesterov’s accelerated gradient descent (AGD), an instance of the general family of “momentum methods,” provably achieves faster convergence rate than gradient descent (GD) in the convex setting. While these methods are widely used in modern \emph{nonconvex} applications, including training of deep neural networks, whether they are provably superior to GD in the nonconvex setting remains open. This paper studies a simple variant of Nesterov’s AGD, and shows that it escapes saddle points and finds a second-order stationary point in $\tilde{O}(1/\epsilon^{7/4})$ iterations, matching the best known convergence rate, which is faster than the $\tilde{O}(1/\epsilon^{2})$ iterations required by GD. To the best of our knowledge, this is the first direct acceleration (single-loop) algorithm that is provably faster than GD in general nonconvex setting—all previous nonconvex accelerated algorithms rely on more complex mechanisms such as nested loops and proximal terms. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases on each step even for nonconvex functions, and (2) a novel framework called \emph{improve or localize}, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization.

Model-free reinforcement learning (RL) algorithms directly parameterize and update value functions or policies, bypassing the modeling of the environment. They are typically simpler, more flexible to use, and thus more prevalent in modern deep RL than model-based approaches. However, empirical work has suggested that they require large numbers of samples to learn. The theoretical question of whether not model-free algorithms are in fact \emph{sample efficient} is one of the most fundamental questions in RL. The problem is unsolved even in the basic scenario with finitely many states and actions. We prove that, in an episodic MDP setting, Q-learning with UCB exploration achieves regret $\tlO(\sqrt{H^3 SAT})$ where $S$ and $A$ are the numbers of states and actions, $H$ is the number of steps per episode, and $T$ is the total number of steps. Our regret matches the optimal regret up to a single $\sqrt{H}$ factor. Thus we establish the sample efficiency of a classical model-free approach. Moreover, to the best of our knowledge, this is the first model-free analysis to establish $\sqrt{T}$ regret \emph{without} requiring access to a ``simulator.''

Population risk is always of primary interest in machine learning; however, learning algorithms only have access to the empirical risk. Even for applications with nonconvex non-smooth losses (such as modern deep networks), the population risk is generally significantly more well behaved from an optimization point of view than the empirical risk. In particular, sampling can create many spurious local minima. We consider a general framework which aims to optimize a smooth nonconvex function $F$ (population risk) given only access to an approximation $f$ (empirical risk) that is pointwise close to $F$ (i.e., $\norm{F-f}_{\infty} \le \nu$). Our objective is to find the $\epsilon$-approximate local minima of the underlying function $F$ while avoiding the shallow local minima---arising because of the tolerance $\nu$---which exist only in $f$. We propose a simple algorithm based on stochastic gradient descent (SGD) on a smoothed version of $f$ that is guaranteed to achieve our goal as long as $\nu \le O(\epsilon^{1.5}/d)$. We also provide an almost matching lower bound showing that our algorithm achieves optimal error tolerance $\nu$ among all algorithms making a polynomial number of queries of $f$. As a concrete example, we show that our results can be directly used to give sample complexities for learning a ReLU unit.

This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(\epsilon^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(\epsilon^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima without requiring any delicate acceleration or variance-reduction techniques.

While there has been a significant amount of work studying gradient descent techniques for non-convex optimization problems over the last few years, all existing results establish either local convergence with good rates or global convergence with highly suboptimal rates, for many problems of interest. In this paper, we take the first step in getting the best of both worlds – establishing global convergence and obtaining a good rate of convergence for the problem of computing squareroot of a positive semidefinite (PSD) matrix, which is a widely studied problem in numerical linear algebra with applications in machine learning and statistics among others. Given a PSD matrix M and a PSD starting point $U_0$, we show that gradient descent with appropriately chosen step-size finds an epsilon-accurate squareroot of M in $O(α\log(||M-U_0||_F^2 / ε))$ iterations, where $α= (\max{||U_0||^2, ||M||} / \min{\sigma_min^2(U_0), \sigma_min(M)} )^3/2$. Our result is the first to establish global convergence for this problem and that it is robust to errors in each iteration. A key contribution of our work is the general proof technique which we believe should further excite research in understanding deterministic and stochastic variants of simple non-convex gradient descent algorithms with good global convergence rates for other problems in machine learning and numerical linear algebra.

In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) no high-order saddle points exists. These results explain why simple algorithms such as stochastic gradient descent have global converge, and efficiently optimize these non-convex objective functions in practice. Our framework connects and simplifies the existing analyses on optimization landscapes for matrix sensing and symmetric matrix completion. The framework naturally leads to new results for asymmetric matrix completion and robust PCA.

This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost “dimension-free”). The convergence rate of this procedure matches the well-known convergence rate of gradient descent to first-order stationary points, up to log factors. When all saddle points are non-degenerate, all second-order stationary points are local minima, and our result thus shows that perturbed gradient descent can escape saddle points almost for free. Our results can be directly applied to many machine learning applications, including deep learning. As a particular concrete example of such an application, we show that our results can be used directly to establish sharp global convergence rates for matrix factorization. Our results rely on a novel characterization of the geometry around saddle points, which may be of independent interest to the non-convex optimization community.

Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly slowed down by saddle points, taking exponential time to escape. On the other hand, gradient descent with perturbations [Ge et al., 2015, Jin et al., 2017] is not slowed down by saddle points—it can find an approximate local minimizer in polynomial time. This result implies that GD is inherently slower than perturbed GD, and justifies the importance of adding perturbations for efficient non-convex optimization. While our focus is theoretical, we also present experiments that illustrate our theoretical findings.

In this paper we provide improved guarantees for streaming principal component analysis (PCA). Given A_1, \ldots, A_n∈\mathbbR^d\times d sampled independently from distributions satisfying \mathbbE[A_i] = Σfor Σ\succeq 0, we present an O(d)-space linear-time single-pass streaming algorithm for estimating the top eigenvector of Σ. The algorithm nearly matches (and in certain cases improves upon) the accuracy obtained by the standard batch method that computes top eigenvector of the empirical covariance \frac1n \sum_i ∈[n] A_i as analyzed by the matrix Bernstein inequality. Moreover, to achieve constant accuracy, our algorithm improves upon the best previous known sample complexities of streaming algorithms by either a multiplicative factor of O(d) or 1/\mathrmgap where \mathrmgap is the relative distance between the top two eigenvalues of Σ. We achieve these results through a novel analysis of the classic Oja’s algorithm, one of the oldest and perhaps, most popular algorithms for streaming PCA. We show that simply picking a random initial point w_0 and applying the natural update rule w_i + 1 = w_i + \eta_i A_i w_i suffices for suitable choice of \eta_i. We believe our result sheds light on how to efficiently perform streaming PCA both in theory and in practice and we hope that our analysis may serve as the basis for analyzing many variants and extensions of streaming PCA.

We give faster algorithms and improved sample complexities for the fundamental problem of estimating the top eigenvector. Given an explicit matrix $A \in \mathbb{R}^{n \times d}$, we show how to compute an $\epsilon$-approximate top eigenvector of $A^TA$ in time $\tilde O\left( \left[\text{nnz}(A) + \frac{d \text{sr}(A)}{\text{gap}^2} \right] \cdot \log 1/\epsilon\right)$. Here $\text{nnz}(A)$ is the number of nonzeros in $A$, $\text{sr}(A)$ is the stable rank, and gap is the relative eigengap. We also consider an online setting in which, given a stream of i.i.d. samples from a distribution D with covariance matrix $\Sigma$ and a vector $x_0$ which is an $O(\text{gap})$ approximate top eigenvector for $\Sigma$, we show how to refine $x_0$ to an $\epsilon$ approximation using $O \left( \frac{\text{var}(\mathcal{D})}{\text{gap}-\epsilon}\right)$ samples from $\mathcal{D}$. Here $\text{var}(\mathcal{D})$ is a natural notion of variance. Combining our algorithm with previous work to initialize $x_0$, we obtain improved sample complexities and runtimes under a variety of assumptions on D. We achieve our results via a robust analysis of the classic shift-and-invert preconditioning method. This technique lets us reduce eigenvector computation to approximately solving a series of linear systems with fast stochastic gradient methods.

This paper considers the problem of canonical-correlation analysis (CCA) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics. We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved. We obtain our results by reducing CCA to the top-k generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a running time of \order\fracz k \sqrtκρ \log(1/ε) \log \left(kκ/ρ\right) where z is the total number of nonzero entries, κis the condition number and ρis the relative eigenvalue gap of the appropriate matrices. Our algorithm is linear in the input size and the number of components k up to a \log(k) factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.

Matrix completion, where we wish to recover a low rank matrix by observing a few entries from it, is a widely studied problem in both theory and practice with wide applications. Most of the provable algorithms so far on this problem have been restricted to the offline setting where they provide an estimate of the unknown matrix using all observations simultaneously. However, in many applications, the online version, where we observe one entry at a time and dynamically update our estimate, is more appealing. While existing algorithms are efficient for the offline setting, they could be highly inefficient for the online setting. In this paper, we propose the first provable, efficient online algorithm for matrix completion. Our algorithm starts from an initial estimate of the matrix and then performs non-convex stochastic gradient descent (SGD). After every observation, it performs a fast update involving only one row of two tall matrices, giving near linear total runtime. Our algorithm can be naturally used in the offline setting as well, where it gives competitive sample complexity and runtime to state of the art algorithms. Our proofs introduce a general framework to show that SGD updates tend to stay away from saddle surfaces and could be of broader interests to other non-convex problems.

We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with $M \geq 3$ components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical Gaussians. We prove that the log-likelihood value of these bad local maxima can be arbitrarily worse than that of any global optimum, thereby resolving an open question of Srebro (2007). Our second main result shows that the EM algorithm (or a first-order variant of it) with random initialization will converge to bad critical points with probability at least $1-e^{-\Omega(M)}$. We further establish that a first-order variant of EM will not converge to strict saddle points almost surely, indicating that the poor performance of the first-order method can be attributed to the existence of bad local maxima rather than bad saddle points. Overall, our results highlight the necessity of careful initialization when using the EM algorithm in practice, even when applied in highly favorable settings.

In the past few years, differential privacy has become a standard concept in the area of privacy. One of the most important problems in this field is to answer queries while preserving differential privacy. In spite of extensive studies, most existing work on differentially private query answering assumes the data are discrete (i.e., in $\{0,1\}^d$) and focuses on queries induced by \emph{Boolean} functions. In real applications however, continuous data are at least as common as binary data. Thus, in this work we explore a less studied topic, namely, differential privately query answering for continuous data with continuous function. As a first step towards the continuous case, we study a natural class of linear queries on continuous data which we refer to as smooth queries. A linear query is said to be $K$-smooth if it is specified by a function defined on $[-1,1]^d$ whose partial derivatives up to order $K$ are all bounded. We develop two $\epsilon$-differentially private mechanisms which are able to answer all smooth queries. The first mechanism outputs a summary of the database and can then give answers to the queries. The second mechanism is an improvement of the first one and it outputs a synthetic database. The two mechanisms both achieve an accuracy of $O (n^{-\frac{K}{2d+K}}/\epsilon )$. Here we assume that the dimension $d$ is a constant. It turns out that even in this parameter setting (which is almost trivial in the discrete case), using existing discrete mechanisms to answer the smooth queries is difficult and requires more noise. Our mechanisms are based on $L_{\infty}$-approximation of (transformed) smooth functions by low-degree even trigonometric polynomials with uniformly bounded coefficients. We also develop practically efficient variants of the mechanisms with promising experimental results.

We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in \em saddle points. In this paper we identify \em strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that from an \em arbitrary starting point, stochastic gradient descent converges to a local minimum in a polynomial number of iterations. To the best of our knowledge this is the first work that gives \em global convergence guarantees for stochastic gradient descent on non-convex functions with exponentially many local minima and saddle points. Our analysis can be applied to orthogonal tensor decomposition, which is widely used in learning a rich class of latent variable models. We propose a new optimization formulation for the tensor decomposition problem that has strict saddle property. As a result we get the first online algorithm for orthogonal tensor decomposition with global convergence guarantee.

Margin is one of the most important concepts in machine learning. Previous margin bounds, both for SVM and for boosting, are dimensionality independent. A major advantage of this dimensionality independency is that it can explain the excellent performance of SVM whose feature spaces are often of high or infinite dimension. In this paper we address the problem whether such dimensionality independency is intrinsic for the margin bounds. We prove a dimensionality dependent PAC-Bayes margin bound. The bound is monotone increasing with respect to the dimension when keeping all other factors fixed. We show that our bound is strictly sharper than a previously well-known PAC-Bayes margin bound if the feature space is of finite dimension; and the two bounds tend to be equivalent as the dimension goes to infinity. In addition, we show that the VC bound for linear classifiers can be recovered from our bound under mild conditions. We conduct extensive experiments on benchmark datasets and find that the new bound is useful for model selection and is significantly sharper than the dimensionality independent PAC-Bayes margin bound as well as the VC bound for linear classifiers.