Improved Regret Bound for Safe Reinforcement Learning via Tighter Cost Pessimism and Reward Optimism

• URL: http://arxiv.org/abs/2410.10158v1
• Date: Mon, 14 Oct 2024 04:51:06 GMT
• Title: Improved Regret Bound for Safe Reinforcement Learning via Tighter Cost Pessimism and Reward Optimism
• Authors: Kihyun Yu, Duksang Lee, William Overman, Dabeen Lee,
• Abstract summary: We propose a model-based algorithm based on novel cost and reward function estimators. Our algorithm achieves a regret upper bound of $widetildemathcalO((bar C - bar C_b)-1H2.5 SsqrtAK)$ where $bar C$ is the cost budget for an episode.
• Score: 1.4999444543328293
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper studies the safe reinforcement learning problem formulated as an episodic finite-horizon tabular constrained Markov decision process with an unknown transition kernel and stochastic reward and cost functions. We propose a model-based algorithm based on novel cost and reward function estimators that provide tighter cost pessimism and reward optimism. While guaranteeing no constraint violation in every episode, our algorithm achieves a regret upper bound of $\widetilde{\mathcal{O}}((\bar C - \bar C_b)^{-1}H^{2.5} S\sqrt{AK})$ where $\bar C$ is the cost budget for an episode, $\bar C_b$ is the expected cost under a safe baseline policy over an episode, $H$ is the horizon, and $S$, $A$ and $K$ are the number of states, actions, and episodes, respectively. This improves upon the best-known regret upper bound, and when $\bar C- \bar C_b=\Omega(H)$, it nearly matches the regret lower bound of $\Omega(H^{1.5}\sqrt{SAK})$. We deduce our cost and reward function estimators via a Bellman-type law of total variance to obtain tight bounds on the expected sum of the variances of value function estimates. This leads to a tighter dependence on the horizon in the function estimators. We also present numerical results to demonstrate the computational effectiveness of our proposed framework.

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