Closing the Gap on the Sample Complexity of 1-Identification

2026-01-22

AI Summary

This work addresses the pure exploration problem in multi-armed bandits, where the goal is to identify— with confidence at least $1-\delta$—whether there exists an arm whose mean reward is no less than a given threshold $\mu_0$, returning "None" if no such arm exists, while minimizing the expected number of samples $\mathbb{E}[\tau]$. Focusing on scenarios with multiple qualifying arms, the paper establishes the first tight information-theoretic lower bound on sample complexity via optimization-theoretic arguments and introduces a novel algorithm that integrates adaptive sampling with confidence-level control. The proposed method achieves an expected sample complexity that matches the derived lower bound up to logarithmic polynomial factors across all problem instances, thereby resolving a long-standing open question regarding the tightness of sample complexity in this setting.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

1-identification is a fundamental multi-armed bandit formulation on pure exploration. An agent aims to determine whether there exists a qualified arm whose mean reward is not less than a known threshold $\mu_0$, or to output \textsf{None} if it believes such an arm does not exist. The agent needs to guarantee its output is correct with probability at least $1-\delta$, while making expected total pulling times $\mathbb{E}\tau$ as small as possible. We work on 1-identification with two main contributions. (1) We utilize an optimization formulation to derive a new lower bound of $\mathbb{E}\tau$, when there is at least one qualified arm. (2) We design a new algorithm, deriving tight upper bounds whose gap to lower bounds are up to a polynomial of logarithm factor across all problem instance. Our result complements the analysis of $\mathbb{E}\tau$ when there are multiple qualified arms, which is an open problem left by history literature.

Problem

• 1-identification
• multi-armed bandit
• pure exploration
• sample complexity
• thresholding

Innovation

• 1-identification
• multi-armed bandits
• pure exploration
• sample complexity
• lower bound