The Empirical Mean is Minimax Optimal for Local Glivenko-Cantelli

TL;DR

This paper investigates the limits of distribution learning in the Local Glivenko-Cantelli setting, showing that the Empirical Mean is minimax optimal unless infinite-dimensional pathologies are exploited, which then broadens learnable measures.

Contribution

It extends the Local Glivenko-Cantelli framework to arbitrary estimators, establishing the optimality of the Empirical Mean and characterizing conditions for broader learnability.

Findings

Empirical Mean is minimax optimal under certain conditions.

Allowing exploits of infinite-dimensional pathologies increases learnable measures.

No improvement in risk decay rates with other estimators unless pathologies are exploited.

Abstract

We revisit the recently introduced Local Glivenko-Cantelli setting, which studies distribution-dependent uniform convergence rates of the Empirical Mean Estimator (EME). In this work, we investigate generalizations of this setting where arbitrary estimators are allowed rather than just the EME. Can a strictly larger class of measures be learned? Can better risk decay rates be obtained? We provide exhaustive answers to these questions, which are both negative, provided the learner is barred from exploiting some infinite-dimensional pathologies. On the other hand, allowing such exploits does lead to a strictly larger class of learnable measures.