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1. Model-Based Multi-Agent RL in Zero-Sum Markov Games with Near-Optimal Sample Complexity

   Zhang, Kaiqing; Kakade, Sham; Basar, Tamer; Yang, Lin (January 2020, Advances in neural information processing systems)

   null (Ed.)

   Model-based reinforcement learning (RL), which finds an optimal policy using an empirical model, has long been recognized as one of the cornerstones of RL. It is especially suitable for multi-agent RL (MARL), as it naturally decouples the learning and the planning phases, and avoids the non-stationarity problem when all agents are improving their policies simultaneously using samples. Though intuitive and widely-used, the sample complexity of model-based MARL algorithms has been investigated relatively much less often. In this paper, we aim to address the fundamental open question about the sample complexity of model-based MARL. We study arguably the most basic MARL setting: two-player discounted zero-sum Markov games, given only access to a generative model of state transition. We show that model-based MARL achieves a near optimal sample complexity for finding the Nash equilibrium (NE) \emph{value} up to some additive error. We also show that this method is near-minimax optimal with a tight dependence on the horizon and the number of states. Our results justify the efficiency of this simple model-based approach in the multi-agent RL setting. 

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2. Sharper Model-free Reinforcement Learning for Average-reward Markov Decision Processes

   Zhang, Zihan; Xie, Qiaomin (July 2023, Conference on Learning Theory)

   We study model-free reinforcement learning (RL) algorithms for infinite-horizon average-reward Markov decision process (MDP), which is more appropriate for applications that involve continuing operations not divided into episodes. In contrast to episodic/discounted MDPs, theoretical understanding of model-free RL algorithms is relatively inadequate for the average-reward setting. In this paper, we consider both the online setting and the setting with access to a simulator. We develop computationally efficient model-free algorithms that achieve sharper guarantees on regret/sample complexity compared with existing results. In the online setting, we design an algorithm, UCB-AVG, based on an optimistic variant of variance-reduced Q-learning. We show that UCB-AVG achieves a regret bound $$\widetilde{O}(S^5A^2sp(h^*)\sqrt{T})$$ after $$T$$ steps, where $$S\times A$$ is the size of state-action space, and $sp(h^*)$ the span of the optimal bias function. Our result provides the first computationally efficient model-free algorithm that achieves the optimal dependence in $$T$$ (up to log factors) for weakly communicating MDPs, which is necessary for low regret. In contrast, prior results either are suboptimal in $$T$$ or require strong assumptions of ergodicity or uniformly mixing of MDPs. In the simulator setting, we adapt the idea of UCB-AVG to develop a model-free algorithm that finds an $$\epsilon$$-optimal policy with sample complexity $$\widetilde{O}(SAsp^2(h^*)\epsilon^{-2} + S^2Asp(h^*)\epsilon^{-1}).$$ This sample complexity is near-optimal for weakly communicating MDPs, in view of the minimax lower bound $$\Omega(SAsp(^*)\epsilon^{-2})$$. Existing work mainly focuses on ergodic MDPs and the results typically depend on $$t_{mix},$$ the worst-case mixing time induced by a policy. We remark that the diameter $$D$$ and mixing time $$t_{mix}$$ are both lower bounded by $sp(h^*)$, and $$t_{mix}$$ can be arbitrarily large for certain MDPs. On the technical side, our approach integrates two key ideas: learning an $$\gamma$$-discounted MDP as an approximation, and leveraging reference-advantage decomposition for variance in optimistic Q-learning. As recognized in prior work, a naive approximation by discounted MDPs results in suboptimal guarantees. A distinguishing feature of our method is maintaining estimates of value-difference between state pairs to provide a sharper bound on the variance of reference advantage. We also crucially use a careful choice of the discounted factor $$\gamma$$ to balance approximation error due to discounting and the statistical learning error, and we are able to maintain a good-quality reference value function with $O(SA)$ space complexity. 

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3. Sample-efficient robust multi-agent reinforcement learning in the face of environmental uncertainty

   Shi, Laixi; Mazumdar, Eric; Chi, Yuejie; Wierman, Adam (July 2024, Proceedings of the 41st International Conference on Machine Learning)

   To overcome the sim-to-real gap in reinforcement learning (RL), learned policies must maintain robustness against environmental uncertainties. While robust RL has been widely studied in single-agent regimes, in multi-agent environments, the problem remains understudied-- despite the fact that the problems posed by environmental uncertainties are often exacerbated by strategic interactions. This work focuses on learning in distributionally robust Markov games (RMGs), a robust variant of standard Markov games, wherein each agent aims to learn a policy that maximizes its own worst-case performance when the deployed environment deviates within its own prescribed uncertainty set. This results in a set of robust equilibrium strategies for all agents that align with classic notions of game-theoretic equilibria. Assuming a non-adaptive sampling mechanism from a generative model, we propose a sample-efficient model-based algorithm (DRNVI) with finite-sample complexity guarantees for learning robust variants of various notions of game-theoretic equilibria. We also establish an information-theoretic lower bound for solving RMGs, which confirms the near-optimal sample complexity of DR-NVI with respect to problem-dependent factors such as the size of the state space, the target accuracy, and the horizon length. 

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4. Breaking the sample complexity barrier to regret-optimal model-free reinforcement learning

   https://doi.org/10.1093/imaiai/iaac034  

   Li, Gen; Shi, Laixi; Chen, Yuxin; Chi, Yuejie (February 2023, Information and Inference: A Journal of the IMA)

   Abstract Achieving sample efficiency in online episodic reinforcement learning (RL) requires optimally balancing exploration and exploitation. When it comes to a finite-horizon episodic Markov decision process with $$S$$ states, $$A$$ actions and horizon length $$H$$, substantial progress has been achieved toward characterizing the minimax-optimal regret, which scales on the order of $$\sqrt{H^2SAT}$$ (modulo log factors) with $$T$$ the total number of samples. While several competing solution paradigms have been proposed to minimize regret, they are either memory-inefficient, or fall short of optimality unless the sample size exceeds an enormous threshold (e.g. $$S^6A^4 \,\mathrm{poly}(H)$$ for existing model-free methods). To overcome such a large sample size barrier to efficient RL, we design a novel model-free algorithm, with space complexity $O(SAH)$, that achieves near-optimal regret as soon as the sample size exceeds the order of $$SA\,\mathrm{poly}(H)$$. In terms of this sample size requirement (also referred to the initial burn-in cost), our method improves—by at least a factor of $S^5A^3$—upon any prior memory-efficient algorithm that is asymptotically regret-optimal. Leveraging the recently introduced variance reduction strategy (also called reference-advantage decomposition), the proposed algorithm employs an early-settled reference update rule, with the aid of two Q-learning sequences with upper and lower confidence bounds. The design principle of our early-settled variance reduction method might be of independent interest to other RL settings that involve intricate exploration–exploitation trade-offs. 

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5. A Simple Reward-free Approach to Constrained Reinforcement Learning

   Sobhan Miryoosefi, Chi Jin (June 2022, International Conference on Machine Learning)

   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. 

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