Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path

Qiwei Di, Jiafan He, Dongruo Zhou, Quanquan Gu
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:7837-7864, 2023.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-di23a, title = {Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path}, author = {Di, Qiwei and He, Jiafan and Zhou, Dongruo and Gu, Quanquan}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {7837--7864}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/di23a/di23a.pdf}, url = {https://proceedings.mlr.press/v202/di23a.html}, 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.} }
Endnote
%0 Conference Paper %T Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path %A Qiwei Di %A Jiafan He %A Dongruo Zhou %A Quanquan Gu %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-di23a %I PMLR %P 7837--7864 %U https://proceedings.mlr.press/v202/di23a.html %V 202 %X 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.
APA
Di, Q., He, J., Zhou, D. & Gu, Q.. (2023). Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:7837-7864 Available from https://proceedings.mlr.press/v202/di23a.html.

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