Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path

• URL: http://arxiv.org/abs/2402.08998v1
• Date: Wed, 14 Feb 2024 07:52:00 GMT
• Title: Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path
• Authors: Qiwei Di, Jiafan He, Dongruo Zhou, Quanquan Gu
• Abstract summary: We study the Shortest Path (SSP) problem with a linear mixture transition kernel. An agent repeatedly interacts with a environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the iteration cost function or an upper bound of the expected length for the optimal policy.
• Score: 80.60592344361073
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the Stochastic Shortest Path (SSP) problem with a linear mixture transition kernel, where an agent repeatedly interacts with a stochastic environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the cost function or an upper bound of the expected length for the optimal policy. In this paper, we propose a new algorithm to eliminate these restrictive assumptions. Our algorithm is based on extended value iteration with a fine-grained variance-aware confidence set, where the variance is estimated recursively from high-order moments. Our algorithm achieves an $\tilde{\mathcal O}(dB_*\sqrt{K})$ regret bound, where $d$ is the dimension of the feature mapping in the linear transition kernel, $B_*$ is the upper bound of the total cumulative cost for the optimal policy, and $K$ is the number of episodes. Our regret upper bound matches the $\Omega(dB_*\sqrt{K})$ lower bound of linear mixture SSPs in Min et al. (2022), which suggests that our algorithm is nearly minimax optimal.

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