  We introduce the "inverse bandit" problem of estimating the rewards of a
multi-armed bandit instance from observing the learning process of a low-regret
demonstrator. Existing approaches to the related problem of inverse
reinforcement learning assume the execution of an optimal policy, and thereby
suffer from an identifiability issue. In contrast, we propose to leverage the
demonstrator's behavior en route to optimality, and in particular, the
exploration phase, for reward estimation. We begin by establishing a general
information-theoretic lower bound under this paradigm that applies to any
demonstrator algorithm, which characterizes a fundamental tradeoff between
reward estimation and the amount of exploration of the demonstrator. Then, we
develop simple and efficient reward estimators for upper-confidence-based
demonstrator algorithms that attain the optimal tradeoff, showing in particular
that consistent reward estimation -- free of identifiability issues -- is
possible under our paradigm. Extensive simulations on both synthetic and
semi-synthetic data corroborate our theoretical results.
