Chaining, Interpolation, and Convexity

TL;DR

This paper improves classical chaining bounds for convex sets by using interpolation techniques, leading to sharper, more computable bounds that are effective in various geometric contexts.

Contribution

It introduces a new approach to chaining bounds leveraging real interpolation, simplifying calculations and enhancing sharpness for convex sets.

Findings

Improved bounds for suprema of random processes on convex sets

Entropy numbers can be computed for thin subsets rather than entire sets

New geometric principles for controlling chaining functionals

Abstract

We show that classical chaining bounds on the suprema of random processes in terms of entropy numbers can be systematically improved when the underlying set is convex: the entropy numbers need not be computed for the entire set, but only for certain "thin" subsets. This phenomenon arises from the observation that real interpolation can be used as a natural chaining mechanism. Unlike the general form of Talagrand's generic chaining method, which is sharp but often difficult to use, the resulting bounds involve only entropy numbers but are nonetheless sharp in many situations in which classical entropy bounds are suboptimal. Such bounds are readily amenable to explicit computations in specific examples, and we discover some old and new geometric principles for the control of chaining functionals as special cases.