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Closing the Gap on the Sample Complexity of 1-Identification

Zitian Li, Wang Chi Cheung

TL;DR

The paper studies 1-identification under fixed confidence, addressing when to declare None or output a qualified arm with high probability. It introduces a robust lower bound for positive instances via an optimization framework and designs a parallel bracket-based SEE algorithm (PSEEB) that achieves δ-PAC guarantees with non-asymptotic upper bounds. For positive instances, the proposed bounds scale as a sum of a term driven by the smallest gap to μ0 and a multi-arm complexity term H(j) modulated by log factors, while negative instances follow a complementary logarithmic-bound regime. The results together close much of the gap on the sample complexity of 1-identification, especially for the case with multiple qualified arms, and provide guidance for efficient pure-exploration strategies in this setting.

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 $μ_0$, 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 $1-δ$, while making expected total pulling times $\mathbb{E}τ$ 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 $\mathbb{E}τ$, 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 $\mathbb{E}τ$ when there are multiple qualified arms, which is an open problem left by history literature.

Closing the Gap on the Sample Complexity of 1-Identification

TL;DR

The paper studies 1-identification under fixed confidence, addressing when to declare None or output a qualified arm with high probability. It introduces a robust lower bound for positive instances via an optimization framework and designs a parallel bracket-based SEE algorithm (PSEEB) that achieves δ-PAC guarantees with non-asymptotic upper bounds. For positive instances, the proposed bounds scale as a sum of a term driven by the smallest gap to μ0 and a multi-arm complexity term H(j) modulated by log factors, while negative instances follow a complementary logarithmic-bound regime. The results together close much of the gap on the sample complexity of 1-identification, especially for the case with multiple qualified arms, and provide guidance for efficient pure-exploration strategies in this setting.

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 , 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 , while making expected total pulling times 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 , 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 when there are multiple qualified arms, which is an open problem left by history literature.
Paper Structure (29 sections, 37 theorems, 195 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 29 sections, 37 theorems, 195 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model & Problem Formulation
  3. Lit Review on 1-identification
  4. Main Contribution
  5. Literature Review on Other Related Works
  6. Lower Bound of Sample Complexity, for a Positive Instance
  7. Upper Bound on the Complexity of 1-identification
  8. Algorithm
  9. Performance Guarantee of Algorithm \ref{['alg:Parallel-SEE-on-Bracket']}
  10. Detailed Literature Review
  11. Simplification of History Result
  12. Theorem 2 in poiani2025nonasymptoticanalysisstickytrackandstop
  13. Theorem 3 in kano2017Good
  14. Thereom 2 in jourdan2023anytime
  15. Theorem 8 in katz2020true
  16. ...and 14 more sections

Key Result

Theorem 1

For any $\delta$-PAC alg and any reward vector $\{\mu_a\}_{a=1}^K\in [0, 1]^K$ satisfying that $\mu_1\geq \cdots\geq \mu_m > \mu_0\geq \mu_{m+1}\geq \cdots\geq \mu_K$, any tolerance level $\delta$ such that $\delta<\min\{\frac{1}{10^{-8}}, \frac{1}{64m^2}\}$, we can find a permutation $\sigma:[K]\ri

Figures (1)

  • Figure 1: Overall Idea of Full Proof of Theorem \ref{['theorem:delta-pac']} and \ref{['theorem:Parallel-SEE-Etau-upper-bound']}

Theorems & Definitions (75)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 3
  • Lemma 4
  • Theorem 5
  • proof : Sketch Proof of Theorem \ref{['theorem:lower-bound-positive-case']}
  • Lemma 6
  • ...and 65 more