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We consider the best-of-both-worlds problem for learning an episodic MarkovDecision Process through $T$ episodes, with the goal of achieving$\widetilde{\mathcal{O}}(\sqrt{T})$ regret when the losses are adversarial andsimultaneously $\mathcal{O}(\text{polylog}(T))$ regret when the losses are(almost) stochastic. Recent work by [Jin and Luo, 2020] achieves this goal whenthe fixed transition is known, and leaves the case of unknown transition as amajor open question. In this work, we resolve this open problem by using thesame Follow-the-Regularized-Leader ($\text{FTRL}$) framework together with aset of new techniques. Specifically, we first propose a loss-shifting trick inthe $\text{FTRL}$ analysis, which greatly simplifies the approach of [Jin andLuo, 2020] and already improves their results for the known transition case.Then, we extend this idea to the unknown transition case and develop a novelanalysis which upper bounds the transition estimation error by (a fraction of)the regret itself in the stochastic setting, a key property to ensure$\mathcal{O}(\text{polylog}(T))$ regret.

The best of both worlds: stochastic and adversarial episodic MDPs with unknown transition