  Policy optimization methods are popular reinforcement learning algorithms in
practice. Recent works have built theoretical foundation for them by proving
$\sqrt{T}$ regret bounds even when the losses are adversarial. Such bounds are
tight in the worst case but often overly pessimistic. In this work, we show
that in tabular Markov decision processes (MDPs), by properly designing the
regularizer, the exploration bonus and the learning rates, one can achieve a
more favorable polylog$(T)$ regret when the losses are stochastic, without
sacrificing the worst-case guarantee in the adversarial regime. To our
knowledge, this is also the first time a gap-dependent polylog$(T)$ regret
bound is shown for policy optimization. Specifically, we achieve this by
leveraging a Tsallis entropy or a Shannon entropy regularizer in the policy
update. Then we show that under known transitions, we can further obtain a
first-order regret bound in the adversarial regime by leveraging the
log-barrier regularizer.
