Improved No-Regret Algorithms for Stochastic Shortest Path with Linear MDP

• URL: http://arxiv.org/abs/2112.09859v1
• Date: Sat, 18 Dec 2021 06:47:31 GMT
• Title: Improved No-Regret Algorithms for Stochastic Shortest Path with Linear MDP
• Authors: Liyu Chen, Rahul Jain, Haipeng Luo
• Abstract summary: We introduce two new no-regret algorithms for the shortest path (SSP) problem with a linear MDP. Our first algorithm is computationally efficient and achieves a regret bound $widetildeOleft(sqrtd3B_star2T_star Kright)$. Our second algorithm is computationally inefficient but achieves the first "horizon-free" regret bound $widetildeO(d3.5B_starsqrtK)$ with no dependency on $T_star
• Score: 31.62899359543925
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce two new no-regret algorithms for the stochastic shortest path (SSP) problem with a linear MDP that significantly improve over the only existing results of (Vial et al., 2021). Our first algorithm is computationally efficient and achieves a regret bound $\widetilde{O}\left(\sqrt{d^3B_{\star}^2T_{\star} K}\right)$, where $d$ is the dimension of the feature space, $B_{\star}$ and $T_{\star}$ are upper bounds of the expected costs and hitting time of the optimal policy respectively, and $K$ is the number of episodes. The same algorithm with a slight modification also achieves logarithmic regret of order $O\left(\frac{d^3B_{\star}^4}{c_{\min}^2\text{gap}_{\min}}\ln^5\frac{dB_{\star} K}{c_{\min}} \right)$, where $\text{gap}_{\min}$ is the minimum sub-optimality gap and $c_{\min}$ is the minimum cost over all state-action pairs. Our result is obtained by developing a simpler and improved analysis for the finite-horizon approximation of (Cohen et al., 2021) with a smaller approximation error, which might be of independent interest. On the other hand, using variance-aware confidence sets in a global optimization problem, our second algorithm is computationally inefficient but achieves the first "horizon-free" regret bound $\widetilde{O}(d^{3.5}B_{\star}\sqrt{K})$ with no polynomial dependency on $T_{\star}$ or $1/c_{\min}$, almost matching the $\Omega(dB_{\star}\sqrt{K})$ lower bound from (Min et al., 2021).

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