Fixed Budget is No Harder Than Fixed Confidence in Best-Arm Identification up to Logarithmic Factors

TL;DR

This paper demonstrates that fixed-budget best-arm identification is not significantly harder than fixed-confidence, introducing a meta-algorithm that converts FC algorithms into FB algorithms with comparable sample complexity.

Contribution

The paper introduces FC2FB, a novel meta-algorithm that transforms fixed-confidence algorithms into fixed-budget algorithms, establishing their sample complexities are equivalent up to logarithmic factors.

Findings

FC2FB achieves near-matching sample complexity to FC algorithms.

Fixed-budget and fixed-confidence problems are fundamentally similar up to logarithmic factors.

Applying FC2FB with existing FC algorithms improves FB problem sample efficiency.

Abstract

The best-arm identification (BAI) problem is one of the most fundamental problems in interactive machine learning, which has two flavors: the fixed-budget setting (FB) and the fixed-confidence setting (FC). For $K$-armed bandits with the unique best arm, the optimal sample complexities for both settings have been settled down, and they match up to logarithmic factors. This prompts an interesting research question about the generic, potentially structured BAI problems: Is FB harder than FC or the other way around? In this paper, we show that FB is no harder than FC up to logarithmic factors. We do this constructively: we propose a novel algorithm called FC2FB (fixed confidence to fixed budget), which is a meta algorithm that takes in an FC algorithm $\mathcal{A}$ and turn it into an FB algorithm. We prove that this FC2FB enjoys a sample complexity that matches, up to logarithmic factors, that of the sample complexity of $\mathcal{A}$. This means that the optimal FC sample complexity is an upper bound of the optimal FB sample complexity up to logarithmic factors. Our result not only reveals a fundamental relationship between FB and FC, but also has a significant implication: FC2FB, combined with existing state-of-the-art FC algorithms, leads to improved sample complexity for a number of FB problems.