A Complexity Measure for Active Learning in Multi-group Mean Estimation

2026-06-12

AI Summary

This study addresses the optimal allocation of a finite sampling budget for estimating multiple group means, aiming to minimize the worst-case uncertainty—defined as the maximum ratio of group variance to sample size. To tackle this max-risk objective, the authors introduce a local minimax framework that establishes, for the first time, a universal lower bound applicable to any hypothesis class with arbitrary finite variances. They orthogonally decompose problem difficulty into three components: sampling budget, heteroscedasticity index, and variance local curvature (VLC), revealing an intrinsic connection between VLC and variance–Fisher information. By integrating ℓ₁ geometric structure, a hard-instance generator, and random matrix analysis, their method achieves near-optimal performance up to logarithmic factors across broad settings, uncovers systematic gaps in highly heterogeneous instances, and provides closed-form expressions for VLC in common distribution families.

Abstract

We study a \emph{max-risk} objective for active learning in a multi-group mean estimation $d$-armed bandits: a learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case uncertainty index $\max_{k\in[d]}σ_k^2/n_k$, where $σ_k$ is the standard deviation of the distribution of arm $d$, and $n_k$ is the number of times arm $d$ is sampled. We develop a local minimax framework and prove the first general lower bound for this objective, valid for any finite-variance hypothesis class. The bound separates difficulty into three orthogonal factors: a \emph{budget} term, a \emph{heteroscedasticity} index measuring how unevenly the uncertainty is spread across arms, and a model-dependent complexity measure, the \emph{Variance Local Curvature} ($\mathrm{VLC}$), which captures how much information a local change of variance creates inside the hypothesis class. For smooth classes, the $\mathrm{VLC}$ is a reparametrization of a variance--Fisher information, with closed-form values for common families. Benchmarking against the strongest available upper bound shows near-optimality up to logarithmic factors in broad regimes, and pinpoints a systematic gap in highly heterogeneous instances. Our proof introduces two key ingredients: a loss-induced $\ell_1$ geometry on the decision space, and a representation-based instance generator that reduces hard-instance construction to an explicit random matrix calculation.

Problem

• active learning
• multi-group mean estimation
• max-risk
• variance heterogeneity
• minimax complexity

Innovation

• active learning
• local minimax lower bound
• Variance Local Curvature (VLC)
• heteroscedasticity
• multi-group mean estimation