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

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Multi-Armed Bandit TheoryPure ExplorationFixed-Confidence
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 δ\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 δ→0\delta \to 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., ϵ\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.

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Overview

Overall Novelty Assessment

The paper contributes non-asymptotic sample complexity guarantees for the Track-and-Stop and Sticky Track-and-Stop algorithms in fixed-confidence best-arm identification. It resides in the Fixed-Confidence Best-Arm Identification leaf, which contains nine papers including the original work. This leaf sits within the broader Best-Arm Identification Frameworks and Objectives branch, representing a well-established and moderately populated research direction. The focus on non-asymptotic analysis addresses a recognized gap in the theoretical understanding of these pioneering adaptive sampling methods.

The taxonomy reveals that fixed-confidence best-arm identification is one of several core problem formulations, with sibling leaves addressing fixed-budget settings, epsilon-optimality with multiple correct answers, and risk-aware objectives. The paper's position in the fixed-confidence leaf places it alongside foundational work on asymptotic optimality and stopping rules. Neighboring branches explore structured settings such as linear bandits and combinatorial pure exploration, as well as algorithmic innovations including sequential hypothesis testing and game-theoretic approaches. The scope note for the leaf explicitly excludes fixed-budget and structural extensions, clarifying that this work focuses on classical confidence-controlled identification without additional constraints.

Among thirty candidates examined through semantic search and citation expansion, the analysis found limited prior work overlap. The non-asymptotic analysis of Track-and-Stop examined ten candidates with zero refutations, suggesting this contribution addresses a relatively underexplored aspect of the algorithm. The Sticky Track-and-Stop analysis also examined ten candidates but identified one refutable match, indicating some existing non-asymptotic work in this area. The novel proof techniques contribution examined ten candidates with no refutations. These statistics reflect a focused literature search rather than exhaustive coverage, and the low refutation counts suggest the non-asymptotic perspective represents a meaningful extension of prior asymptotic results.

Based on the limited search scope of thirty semantically related papers, the work appears to fill a recognized theoretical gap within a moderately crowded research area. The analysis does not cover the full breadth of pure exploration literature, and the single refutation for Sticky Track-and-Stop warrants closer examination to assess the degree of overlap. The contribution's novelty hinges on whether existing non-asymptotic analyses for related algorithms can be straightforwardly adapted or whether the Track-and-Stop framework requires fundamentally new proof machinery.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
30
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: sequential pure exploration in multi-armed bandits. The field is organized around several major branches that reflect different problem formulations and methodological emphases. Best-Arm Identification Frameworks and Objectives encompass foundational settings such as fixed-confidence and fixed-budget identification, where the goal is to identify the optimal arm with minimal sample complexity or maximal correctness probability. Structured and Combinatorial Pure Exploration extends these ideas to scenarios with combinatorial action spaces or additional structure, while Algorithm Design and Optimization Techniques focuses on algorithmic innovations—ranging from adaptive sampling rules to sequential testing methods. Extended Problem Settings and Constraints address non-standard environments (e.g., delayed feedback, limited precision, or risk-aware objectives), and Distributed and Collaborative Pure Exploration considers multi-agent or federated scenarios. Specialized Extensions and Applications capture domain-specific adaptations, from brain-computer interfaces to quantum settings. Within the best-arm identification landscape, a central tension exists between fixed-confidence approaches—which aim to minimize sample complexity while guaranteeing a confidence level—and fixed-budget methods that maximize correctness under a hard budget constraint. Early foundational work such as Pure Exploration Bandits[1] and Best Arm Identification[11] established core algorithmic principles, while more recent efforts like Best Arm Complexity[2] and Fixed Confidence Optimal[6] refine sample complexity bounds and stopping rules. Sticky Track and Stop[0] sits squarely in the fixed-confidence branch, emphasizing adaptive tracking mechanisms that balance exploration and stopping decisions. Its design contrasts with simpler uniform-allocation baselines and shares thematic similarities with lil UCB[22], which also leverages confidence sequences for anytime guarantees. Compared to works addressing cost-aware settings like Cost Aware Identification[5] or extended objectives such as Pure Exploration[18], Sticky Track and Stop[0] focuses on classical best-arm identification with refined algorithmic control, illustrating ongoing efforts to tighten theoretical guarantees and improve practical performance in this well-studied domain.

Claimed Contributions

Non-asymptotic analysis of Track-and-Stop algorithm

The authors provide the first finite-confidence upper bounds on the expected stopping time of the Track-and-Stop (TAS) algorithm for single-answer pure exploration problems. Their analysis characterizes TAS performance in the non-asymptotic regime while recovering asymptotic optimality as the risk parameter approaches zero.

10 retrieved papers
Non-asymptotic analysis of Sticky Track-and-Stop algorithm

The authors establish the first finite-confidence guarantees for the Sticky Track-and-Stop (S-TAS) algorithm, which handles multiple-answer pure exploration problems. This provides the first non-asymptotic analysis for general multiple-answer settings where multiple correct answers may exist for each environment.

10 retrieved papers
Can Refute
Novel proof techniques for analyzing Track-and-Stop methods

The authors develop new proof techniques that differ from prior asymptotic analyses by reasoning about information accumulation through functions of the form of sums of weighted KL divergences, without relying on convergence properties of empirical pull strategies or convexity arguments used in previous work.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Non-asymptotic analysis of Track-and-Stop algorithm

The authors provide the first finite-confidence upper bounds on the expected stopping time of the Track-and-Stop (TAS) algorithm for single-answer pure exploration problems. Their analysis characterizes TAS performance in the non-asymptotic regime while recovering asymptotic optimality as the risk parameter approaches zero.

Contribution

Non-asymptotic analysis of Sticky Track-and-Stop algorithm

The authors establish the first finite-confidence guarantees for the Sticky Track-and-Stop (S-TAS) algorithm, which handles multiple-answer pure exploration problems. This provides the first non-asymptotic analysis for general multiple-answer settings where multiple correct answers may exist for each environment.

Contribution

Novel proof techniques for analyzing Track-and-Stop methods

The authors develop new proof techniques that differ from prior asymptotic analyses by reasoning about information accumulation through functions of the form of sums of weighted KL divergences, without relying on convergence properties of empirical pull strategies or convexity arguments used in previous work.

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