Simultaneously Learning Stochastic and Adversarial Episodic MDPs with Known Transition
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This work studies the problem of learning episodic Markov Decision Processes with known transition and bandit feedback. We develop the first algorithm with a “best-of-both-worlds” guarantee: it achieves O(log T) regret when the losses are stochastic, and simultaneously enjoys worst-case robustness with Õ(√(T)) regret even when the losses are adversarial, where T is the number of episodes. More generally, it achieves Õ(√(C)) regret in an intermediate setting where the losses are corrupted by a total amount of C. Our algorithm is based on the Follow-the-Regularized-Leader method from Zimin and Neu (2013), with a novel hybrid regularizer inspired by recent works of Zimmert et al. (2019a, 2019b) for the special case of multi-armed bandits. Crucially, our regularizer admits a non-diagonal Hessian with a highly complicated inverse. Analyzing such a regularizer and deriving a particular self-bounding regret guarantee is our key technical contribution and might be of independent interest.
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