This paper focuses on the problem of determining as large a region as possible where a function exceeds a given threshold with high probability. We assume that we only have access to a noise-corrupted version of the function and that function evaluations are costly. To select the next query point, we propose maximizing the expected volume of the domain identified as above the threshold as predicted by a Gaussian process, robustified by a variance term. We also give asymptotic guarantees on the exploration effect of the algorithm, regardless of the prior misspecification. We show by various numerical examples that our approach also outperforms existing techniques in the literature in practice.

Strong worst-case performance bounds for episodic reinforcement learning exist but fortunately in practice RL algorithms perform much better than such bounds would predict. Algorithms and theory that provide strong problemdependent bounds could help illuminate the key features of what makes a RL problem hard and reduce the barrier to using RL algorithms in practice. As a step towards this we derive an algorithm and analysis for finite horizon discrete MDPs with state-of-the-art worst-case regret bounds and substantially tighter bounds if the RL environment has special features but without apriori knowledge of the environment from the algorithm. As a result of our analysis, we also help address an open learning theory question (Jiang & Agarwal, 2018) about episodic MDPs with a constant upper-bound on the sum of rewards, providing a regret bound function of the number of episodes with no dependence on the horizon.

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Policy optimization methods are popular reinforcement learning algorithms, because their incremental and on-policy nature makes them more stable than the value-based counterparts. However, the same properties also make them slow to converge and sample inefficient, as the on-policy requirement precludes data reuse and the incremental updates couple large iteration complexity into the sample complexity. These characteristics have been observed in experiments as well as in theory in the recent work of Agarwal et al. (2020a), which provides a policy optimization method PC-PG that can robustly find near optimal polices for approximately linear Markov decision processes but suffers from an extremely poor sample complexity compared with value-based techniques.In this paper, we propose a new algorithm, COPOE, that overcomes the sample complexity issue of PC-PG while retaining its robustness to model misspecification. Compared with PC-PG, COPOE makes several important algorithmic enhancements, such as enabling data reuse, and uses more refined analysis techniques, which we expect to be more broadly applicable to designing new reinforcement learning algorithms. The result is an improvement in sample complexity from O(1/ǫ 11 ) for PC-PG to O(1/ǫ 3 ) for COPOE, nearly bridging the gap with value-based techniques.

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