  We make significant progress toward the stochastic shortest path problem with
adversarial costs and unknown transition. Specifically, we develop algorithms
that achieve $\widetilde{O}(\sqrt{S^2ADT_\star K})$ regret for the
full-information setting and $\widetilde{O}(\sqrt{S^3A^2DT_\star K})$ regret
for the bandit feedback setting, where $D$ is the diameter, $T_\star$ is the
expected hitting time of the optimal policy, $S$ is the number of states, $A$
is the number of actions, and $K$ is the number of episodes. Our work strictly
improves (Rosenberg and Mansour, 2020) in the full information setting, extends
(Chen et al., 2020) from known transition to unknown transition, and is also
the first to consider the most challenging combination: bandit feedback with
adversarial costs and unknown transition. To remedy the gap between our upper
bounds and the current best lower bounds constructed via a stochastically
oblivious adversary, we also propose algorithms with near-optimal regret for
this special case.
