Model-Free, Regret-Optimal Best Policy Identification in Online CMDPs

• URL: http://arxiv.org/abs/2309.15395v5
• Date: Mon, 15 Apr 2024 03:20:29 GMT
• Title: Model-Free, Regret-Optimal Best Policy Identification in Online CMDPs
• Authors: Zihan Zhou, Honghao Wei, Lei Ying,
• Abstract summary: This paper considers the best policy identification problem in online Constrained Markov Decision Processes (CMDPs) We are interested in algorithms that are model-free, have low regret, and identify an approximately optimal policy with a high probability. Existing model-free algorithms for online CMDPs with sublinear regret and constraint violation do not provide any convergence guarantee to an optimal policy.
• Score: 17.62509045102346
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper considers the best policy identification (BPI) problem in online Constrained Markov Decision Processes (CMDPs). We are interested in algorithms that are model-free, have low regret, and identify an approximately optimal policy with a high probability. Existing model-free algorithms for online CMDPs with sublinear regret and constraint violation do not provide any convergence guarantee to an optimal policy and provide only average performance guarantees when a policy is uniformly sampled at random from all previously used policies. In this paper, we develop a new algorithm, named Pruning-Refinement-Identification (PRI), based on a fundamental structural property of CMDPs proved before, which we call limited stochasticity. The property says for a CMDP with $N$ constraints, there exists an optimal policy with at most $N$ stochastic decisions. The proposed algorithm first identifies at which step and in which state a stochastic decision has to be taken and then fine-tunes the distributions of these stochastic decisions. PRI achieves trio objectives: (i) PRI is a model-free algorithm; and (ii) it outputs an approximately optimal policy with a high probability at the end of learning; and (iii) PRI guarantees $\tilde{\mathcal{O}}(H\sqrt{K})$ regret and constraint violation, which significantly improves the best existing regret bound $\tilde{\mathcal{O}}(H^4 \sqrt{SA}K^{\frac{4}{5}})$ under a model-free algorithm, where $H$ is the length of each episode, $S$ is the number of states, $A$ is the number of actions, and the total number of episodes during learning is $2K+\tilde{\cal O}(K^{0.25}).$ We further present a matching lower via an example that shows under any online learning algorithm, there exists a well-separated CMDP instance such that either the regret or violation has to be $\Omega(H\sqrt{K}),$ which matches the upper bound by a polylogarithmic factor.

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