We study modelfree reinforcement learning (RL) algorithms for infinitehorizon averagereward 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 modelfree RL algorithms is relatively inadequate for the averagereward setting. In this paper, we consider both the online setting and the setting with access to a simulator. We develop computationally efficient modelfree algorithms that achieve sharper guarantees on regret/sample complexity compared with existing results. In the online setting, we design an algorithm, UCBAVG, based on an optimistic variant of variancereduced Qlearning. We show that UCBAVG achieves a regret bound $\widetilde{O}(S^5A^2sp(h^*)\sqrt{T})$ after $T$ steps, where $S\times A$ is the size of stateaction space, and $sp(h^*)$ the span of the optimal bias function. Our result provides the first computationally efficient modelfree 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 UCBAVG to develop a modelfree 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 nearoptimal 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 worstcase 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 referenceadvantage decomposition for variance in optimistic Qlearning. 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 valuedifference 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 goodquality reference value function with $O(SA)$ space complexity.
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A Simple Rewardfree Approach to Constrained Reinforcement 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, rewardfree 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 rewardfree RL and constrained RL. Particularly, we propose a simple metaalgorithm such that given any rewardfree RL oracle, the approachability and constrained RL problems can be directly solved with negligible overheads in sample complexity. Utilizing the existing rewardfree 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 twoplayer Markov games, and gives a new result for constrained RL with linear function approximation.
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 Award ID(s):
 2107304
 NSFPAR ID:
 10471865
 Publisher / Repository:
 PMLR
 Date Published:
 Journal Name:
 International Conference on Machine Learning
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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