Learning from an Exploring Demonstrator: Optimal Reward Estimation for Bandits

TL;DR

This paper introduces the inverse bandit problem, estimating rewards from a demonstrator's exploration process, and develops optimal estimators that overcome traditional identifiability issues, supported by theoretical bounds and simulations.

Contribution

It proposes leveraging exploration behavior in inverse bandit problems, providing a theoretical framework and efficient estimators that achieve optimal reward estimation tradeoffs.

Findings

Establishes a fundamental information-theoretic lower bound.

Develops reward estimators for upper-confidence-based algorithms.

Demonstrates consistent reward estimation free of identifiability issues.

Abstract

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.