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

• URL: http://arxiv.org/abs/2606.14690v1
• Date: Fri, 12 Jun 2026 17:54:26 GMT
• Title: A Complexity Measure for Active Learning in Multi-group Mean Estimation
• Abstract summary: We study a emphmax-risk objective for active learning in a multi-group mean estimation $d$-armed bandits.<n>A learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case uncertainty index.<n>For smooth classes, the $mathrmVLC$ is a reparametrization of a variance--Fisher information, with closed-form values for common families.
• Score: 8.550300650352732
• License: http://creativecommons.org/publicdomain/zero/1.0/
• 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.

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