# Set structured global empirical risk minimizers are rate optimal in general dimensions

## TL;DR

This paper shows that in models with certain set structures, global empirical risk minimizers achieve optimal convergence rates even when traditional entropy-based bounds suggest sub-optimality, resolving longstanding doubts.

## Contribution

It demonstrates that set structures in models enable global ERMs to be rate-optimal in general dimensions despite diverging entropy integrals.

## Key findings

- Global ERMs are rate-optimal in models with set structures.
- Matching upper and lower bounds are established for empirical process over measurable sets.
- The results apply to image estimation, classification, isotonic regression, and density estimation.

## Abstract

Entropy integrals are widely used as a powerful empirical process tool to obtain upper bounds for the rates of convergence of global empirical risk minimizers (ERMs), in standard settings such as density estimation and regression. The upper bound for the convergence rates thus obtained typically matches the minimax lower bound when the entropy integral converges, but admits a strict gap compared to the lower bound when it diverges. Birg\'e and Massart [BM93] provided a striking example showing that such a gap is real with the entropy structure alone: for a variant of the natural H\"older class with low regularity, the global ERM actually converges at the rate predicted by the entropy integral that substantially deviates from the lower bound. The counter-example has spawned a long-standing negative position on the use of global ERMs in the regime where the entropy integral diverges, as they are heuristically believed to converge at a sub-optimal rate in a variety of models. The present paper demonstrates that this gap can be closed if the models admit certain degree of `set structures' in addition to the entropy structure. In other words, the global ERMs in such set structured models will indeed be rate-optimal, matching the lower bound even when the entropy integral diverges. The models with set structures we investigate include (i) image and edge estimation, (ii) binary classification, (iii) multiple isotonic regression, (iv) $s$-concave density estimation, all in general dimensions when the entropy integral diverges. Here set structures are interpreted broadly in the sense that the complexity of the underlying models can be essentially captured by the size of the empirical process over certain class of measurable sets, for which matching upper and lower bounds are obtained to facilitate the derivation of sharp convergence rates for the associated global ERMs.