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

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 $O(\sqrt{d^3B_{\star}^2T_{\star} K})$, 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 $O(\frac{d^3B_{\star}^4}{c_{\min}^2\text{\rm gap}_{\min} }\ln^5\frac{dB_{\star} K}{c_{\min}})$, where $\text{\rm gap}_{\min}$ 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 $O(d^{3.5}B_{\star}\sqrt{K})$ 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).