证明经验均值在局部格里文科-坎泰利设定下是最优的
The Empirical Mean is Minimax Optimal for Local Glivenko-Cantelli
- 研究任意估计器在局部收敛下的学习能力
- 发现若排除病态情况，无法超越经验均值的泛化性能
- 仅当允许无限维奇点时，才能学习更广类分布
我们重新审视了近期提出的局部格里文科-坎泰利设定，该设定研究经验均值估计器（EME）在分布依赖下的统一收敛速率。本文探讨了允许使用任意估计器的推广情形，核心问题为：是否能学习到更大类别的测度？能否获得更优的风险衰减速率？我们给出了完整答案，结果均为否定——只要禁止利用某些无限维病态结构，就无法超越经验均值的表现。然而，一旦允许此类病态利用，则可实现严格更大的可学习测度类。
原文摘要 · Abstract (English)
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.
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