  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).
