Quantile Multi-Armed Bandits with 1-bit Feedback

TL;DR

This paper introduces a method for identifying the best arm based on quantile rewards in a bandit setting with severe communication constraints, demonstrating minimal impact of 1-bit feedback on sample complexity.

Contribution

It proposes a novel algorithm using noisy binary search for quantile identification under 1-bit feedback constraints, with matching upper and lower bounds on sample complexity.

Findings

The algorithm achieves near-optimal sample complexity bounds.

1-bit feedback minimally affects the sample complexity scaling.

Lower bounds hold even without communication constraints.

Abstract

In this paper, we study a variant of best-arm identification involving elements of risk sensitivity and communication constraints. Specifically, the goal of the learner is to identify the arm with the highest quantile reward, while the communication from an agent (who observes rewards) and the learner (who chooses actions) is restricted to only one bit of feedback per arm pull. We propose an algorithm that utilizes noisy binary search as a subroutine, allowing the learner to estimate quantile rewards through 1-bit feedback. We derive an instance-dependent upper bound on the sample complexity of our algorithm and provide an algorithm-independent lower bound for specific instances, with the two matching to within logarithmic factors under mild conditions, or even to within constant factors in certain low error probability scaling regimes. The lower bound is applicable even in the absence of communication constraints, and thus we conclude that restricting to 1-bit feedback has a minimal impact on the scaling of the sample complexity.