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Exploration-free Algorithms for Multi-group Mean Estimation

Ziyi Wei, Huaiyang Zhong, Xiaocheng Li

TL;DR

The paper tackles multi-group mean estimation under a finite budget, showing that the optimal allocation requires $\Theta(T)$ samples per arm and enabling exploration-free strategies. It strengthens subgaussian variance concentration via the Hanson–Wright framework and characterizes strictly subgaussian distributions with sharper guarantees, while developing exploration-free non-adaptive and adaptive algorithms that achieve tighter regret bounds. The framework is extended to contextual bandits, with algorithms that exploit side information and provide provable guarantees on estimation error. Empirical results across Gaussian, Rademacher, and contextual settings corroborate the theoretical gains and demonstrate practical applicability to experimental design and personalized inference.

Abstract

We address the problem of multi-group mean estimation, which seeks to allocate a finite sampling budget across multiple groups to obtain uniformly accurate estimates of their means. Unlike classical multi-armed bandits, whose objective is to minimize regret by identifying and exploiting the best arm, the optimal allocation in this setting requires sampling every group on the order of $Θ(T)$ times. This fundamental distinction makes exploration-free algorithms both natural and effective. Our work makes three contributions. First, we strengthen the existing results on subgaussian variance concentration using the Hanson-Wright inequality and identify a class of strictly subgaussian distributions that yield sharper guarantees. Second, we design exploration-free non-adaptive and adaptive algorithms, and we establish tighter regret bounds than the existing results. Third, we extend the framework to contextual bandit settings, an underexplored direction, and propose algorithms that leverage side information with provable guarantees. Overall, these results position exploration-free allocation as a principled and efficient approach to multi-group mean estimation, with potential applications in experimental design, personalization, and other domains requiring accurate multi-group inference.

Exploration-free Algorithms for Multi-group Mean Estimation

TL;DR

The paper tackles multi-group mean estimation under a finite budget, showing that the optimal allocation requires samples per arm and enabling exploration-free strategies. It strengthens subgaussian variance concentration via the Hanson–Wright framework and characterizes strictly subgaussian distributions with sharper guarantees, while developing exploration-free non-adaptive and adaptive algorithms that achieve tighter regret bounds. The framework is extended to contextual bandits, with algorithms that exploit side information and provide provable guarantees on estimation error. Empirical results across Gaussian, Rademacher, and contextual settings corroborate the theoretical gains and demonstrate practical applicability to experimental design and personalized inference.

Abstract

We address the problem of multi-group mean estimation, which seeks to allocate a finite sampling budget across multiple groups to obtain uniformly accurate estimates of their means. Unlike classical multi-armed bandits, whose objective is to minimize regret by identifying and exploiting the best arm, the optimal allocation in this setting requires sampling every group on the order of times. This fundamental distinction makes exploration-free algorithms both natural and effective. Our work makes three contributions. First, we strengthen the existing results on subgaussian variance concentration using the Hanson-Wright inequality and identify a class of strictly subgaussian distributions that yield sharper guarantees. Second, we design exploration-free non-adaptive and adaptive algorithms, and we establish tighter regret bounds than the existing results. Third, we extend the framework to contextual bandit settings, an underexplored direction, and propose algorithms that leverage side information with provable guarantees. Overall, these results position exploration-free allocation as a principled and efficient approach to multi-group mean estimation, with potential applications in experimental design, personalization, and other domains requiring accurate multi-group inference.

Paper Structure

This paper contains 53 sections, 21 theorems, 107 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review.
  3. Contributions.
  4. Problem Setup
  5. Subgaussian variance concentration
  6. Non-adaptive-Style Algorithm
  7. Analysis of Algorithm \ref{['alg:FETE']}
  8. General Case
  9. Contextual Case
  10. Problem Setting
  11. Algorithm and Analysis
  12. Numerical Experiments
  13. Gaussian alternatives
  14. Rademacher and Gaussian alternatives
  15. Contextual Bandit
  16. ...and 38 more sections

Key Result

Proposition 2.1

Suppose one knows $\sigma_k^2$, then the optimal allocation $\bm{n}^*=(n_1^*,...,n_{K}^*)$ is given by where $q\coloneqq \tfrac{2p}{p+1}$ if $p$ is finite and $q=2$ if $p =\infty$ and the optimal objective value is:

Figures (5)

  • Figure 1: Algorithm \ref{['alg:adaptiveETC']} under two settings.
  • Figure 2: Algorithm \ref{['alg:adaptiveETC']}: Rademacher and Gaussian alternatives in SSG setting.
  • Figure 3: Algorithm \ref{['alg:contextualETC']}: Contextual setting.
  • Figure 4: Algorithm \ref{['alg:adaptiveETC']}: comparison of three regimes with and without $\underline{\sigma}^2$.
  • Figure 5: Algorithm \ref{['alg:adaptiveETC']}: Symmetric Beta.

Theorems & Definitions (38)

  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Lemma 5.2
  • Theorem 5.3
  • Lemma 5.4
  • Theorem 5.5
  • ...and 28 more