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Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path
Qiwei Di · Jiafan He · Dongruo Zhou · Quanquan Gu

Tue Jul 25 02:00 PM -- 04:30 PM (PDT) @ Exhibit Hall 1 #605
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.

Author Information

Qiwei Di (University of California, Los Angeles)
Jiafan He (University of California, Los Angeles)
Dongruo Zhou (UCLA)
Quanquan Gu (University of California, Los Angeles)

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