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We introduce the "inverse bandit" problem of estimating the rewards of amulti-armed bandit instance from observing the learning process of a low-regretdemonstrator. Existing approaches to the related problem of inversereinforcement learning assume the execution of an optimal policy, and therebysuffer from an identifiability issue. In contrast, we propose to leverage thedemonstrator's behavior en route to optimality, and in particular, theexploration phase, for reward estimation. We begin by establishing a generalinformation-theoretic lower bound under this paradigm that applies to anydemonstrator algorithm, which characterizes a fundamental tradeoff betweenreward estimation and the amount of exploration of the demonstrator. Then, wedevelop simple and efficient reward estimators for upper-confidence-baseddemonstrator algorithms that attain the optimal tradeoff, showing in particularthat consistent reward estimation -- free of identifiability issues -- ispossible under our paradigm. Extensive simulations on both synthetic andsemi-synthetic data corroborate our theoretical results.

Learning from an Exploring Demonstrator: Optimal Reward Estimation for Bandits