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 due to the new challenges arising from its game-theoretical nature. 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.
This paper considers minimax optimization $\min_x \max_y f(x, y)$ in the challenging setting where $f$ can be both nonconvex in $x$ and nonconcave in $y$. Though such optimization problems arise in many machine learning paradigms including training generative adversarial networks (GANs) and adversarially robust models, many fundamental issues remain in theory, such as the absence of efficiently computable optimality notions, and cyclic or diverging behavior of existing algorithms. Our framework sprouts from the practical consideration that under a computational budget, the max-player can not fully maximize $f(x,\cdot)$ since nonconcave maximization is NP-hard in general. So, we propose a new algorithm for the min-player to play against smooth algorithms deployed by the adversary (i.e., the max-player) instead of against full maximization. Our algorithm is guaranteed to make monotonic progress (thus having no limit cycles), and to find an appropriate "stationary point" in a polynomial number of iterations. Our framework covers practical settings where the smooth algorithms deployed by the adversary are multi-step stochastic gradient ascent, and its accelerated version. We further provide complementing experiments that confirm our theoretical findings and demonstrate the effectiveness of the proposed approach in practice.
In this paper we propose a novel observer to solve the problem of visual simultaneous localization and mapping, using the information of only the bearing vectors of landmarks observed from a single monocular camera and body-fixed velocities. The system state evolves on the manifold $SE(3)\times \mathbb{R}^{3n}$, on which we design dynamic extensions carefully in order to generate an invariant foliation, such that the problem is reformulated into online parameter identification. Then, following the recently introduced parameter estimation-based observer, we provide a novel and simple solution to address the problem. A notable merit is that the proposed observer guarantees almost global asymptotic stability requiring neither persistent excitation nor uniform complete observability, which, however, are widely adopted in the existing works.
This paper considers batch Reinforcement Learning (RL) with general value function approximation. Our study investigates the minimal assumptions to reliably estimate/minimize Bellman error, and characterizes the generalization performance by (local) Rademacher complexities of general function classes, which makes initial steps in bridging the gap between statistical learning theory and batch RL. Concretely, we view the Bellman error as a surrogate loss for the optimality gap, and prove the followings: (1) In double sampling regime, the excess risk of Empirical Risk Minimizer (ERM) is bounded by the Rademacher complexity of the function class. (2) In the single sampling regime, sample-efficient risk minimization is not possible without further assumptions, regardless of algorithms. However, with completeness assumptions, the excess risk of FQI and a minimax style algorithm can be again bounded by the Rademacher complexity of the corresponding function classes. (3) Fast statistical rates can be achieved by using tools of local Rademacher complexity. Our analysis covers a wide range of function classes, including finite classes, linear spaces, kernel spaces, sparse linear features, etc.
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 vastly open how to learn the Stackelberg equilibrium -- an asymmetric analog of the Nash equilibrium -- in general-sum games efficiently from samples. This paper initiates the theoretical study of sample-efficient learning of the Stackelberg equilibrium in two-player turn-based general-sum games. We identify a fundamental gap between the exact value of the Stackelberg equilibrium and its estimated version using finite samples, which can not be closed information-theoretically regardless of the algorithm. We then establish a positive result on sample-efficient learning of Stackelberg equilibrium with value optimal up to the gap identified above. We show that our sample complexity is tight with matching upper and lower bounds. Finally, we extend our learning results to the setting where the follower plays in a Markov Decision Process (MDP), and the setting where the leader and the follower act simultaneously.
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 \citep{zanette2020learning}. To the best of our knowledge, this is the first theoretical result that characterizes the benefits of multi-task representation learning for exploration in RL with function approximation.
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.
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.
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 $\tilde{\mathcal{O}}(\delta_{\mathcal{F}} H^2 \sqrt{T})$ regret, where $\delta_{\mathcal{F}}$ characterizes the intrinsic complexity of the function class $\mathcal{F}$, $H$ 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.