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

2026-02-03

AI Summary

This work investigates the relative difficulty of best-arm identification under fixed-budget (FB) and fixed-confidence (FC) settings, proposing a meta-algorithm named FC2FB that converts any FC algorithm into an FB algorithm. The method provides the first constructive proof that, in terms of sample complexity, the FB setting is no harder than the FC setting—differing only by a logarithmic factor. By integrating state-of-the-art FC algorithms, FC2FB achieves superior empirical performance across various FB tasks, demonstrating that the optimal FC sample complexity serves as a tight upper bound for FB complexity. This result bridges the theoretical gap between the two settings and offers a practical pathway to leverage advances in FC algorithms for FB problems.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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.

Problem

• Best-Arm Identification
• Fixed-Budget
• Fixed-Confidence
• Sample Complexity
• Bandits

Innovation

• best-arm identification
• fixed-budget
• fixed-confidence
• sample complexity
• meta-algorithm