  We study the stochastic shortest path problem with adversarial costs and
known transition, and show that the minimax regret is
$\widetilde{O}(\sqrt{DT^\star K})$ and $\widetilde{O}(\sqrt{DT^\star SA K})$
for the full-information setting and the bandit feedback setting respectively,
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 results significantly improve upon the
existing work of (Rosenberg and Mansour, 2020) which only considers the
full-information setting and achieves suboptimal regret. Our work is also the
first to consider bandit feedback with adversarial costs.
  Our algorithms are built on top of the Online Mirror Descent framework with a
variety of new techniques that might be of independent interest, including an
improved multi-scale expert algorithm, a reduction from general stochastic
shortest path to a special loop-free case, a skewed occupancy measure space,
and a novel correction term added to the cost estimators. Interestingly, the
last two elements reduce the variance of the learner via positive bias and the
variance of the optimal policy via negative bias respectively, and having them
simultaneously is critical for obtaining the optimal high-probability bound in
the bandit feedback setting.
