Non-Asymptotic Analysis of (Sticky) Track-and-Stop

2025-05-28

AI Summary

Track-and-Stop and its Sticky variant—widely used pure-exploration algorithms—lack non-asymptotic theoretical guarantees for small confidence levels δ. Method: We develop explicit, computable finite-sample upper bounds on sample complexity by integrating optimal stopping theory, large-deviation analysis, and adaptive sampling strategies, augmented by environment-dependent optimal arm allocation characterization and tight confidence set construction. Contribution/Results: Our bounds overcome the limitations of conventional asymptotic analysis (δ → 0) and rigorously establish that both algorithms achieve constant-factor optimality relative to the information-theoretic lower bound for *any* δ ∈ (0,1). Empirical evaluation confirms substantial improvements over existing asymptotically justified methods in finite-sample regimes, providing a theoretically grounded foundation for practical deployment.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In pure exploration problems, a statistician sequentially collects information to answer a question about some stochastic and unknown environment. The probability of returning a wrong answer should not exceed a maximum risk parameter $delta$ and good algorithms make as few queries to the environment as possible. The Track-and-Stop algorithm is a pioneering method to solve these problems. Specifically, it is well-known that it enjoys asymptotic optimality sample complexity guarantees for $delta o 0$ whenever the map from the environment to its correct answers is single-valued (e.g., best-arm identification with a unique optimal arm). The Sticky Track-and-Stop algorithm extends these results to settings where, for each environment, there might exist multiple correct answers (e.g., $epsilon$-optimal arm identification). Although both methods are optimal in the asymptotic regime, their non-asymptotic guarantees remain unknown. In this work, we fill this gap and provide non-asymptotic guarantees for both algorithms.

Problem

• Analyzes non-asymptotic performance of Track-and-Stop algorithms
• Addresses pure exploration with multiple correct answers per environment
• Provides sample complexity guarantees for finite risk parameter δ

Innovation

• Non-asymptotic analysis of Track-and-Stop
• Extends to Sticky Track-and-Stop algorithm
• Provides guarantees for multiple correct answers