A chain rule for the expected suprema of Gaussian processes

TL;DR

This paper introduces a chain rule that bounds the expected maximum of Gaussian processes indexed by composite classes, aiding in analyzing complex models like multi-layer neural networks.

Contribution

It provides a novel bound for the expected supremum of Gaussian processes over composite classes, linking properties of the index set and function class.

Findings

Bound applicable to nonlinear transformations and learning algorithms

Useful for analyzing multi-layer models

Enhances understanding of Gaussian process behavior in complex settings

Abstract

The expected supremum of a Gaussian process indexed by the image of an index set under a function class is bounded in terms of separate properties of the index set and the function class. The bound is relevant to the estimation of nonlinear transformations or the analysis of learning algorithms whenever hypotheses are chosen from composite classes, as is the case for multi-layer models.