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

Part of Advances in Neural Information Processing Systems 34 pre-proceedings (NeurIPS 2021)

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Tiancheng Jin, Longbo Huang, Haipeng Luo


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