Concentration Inequalities for Suprema of Empirical Processes with Dependent Data via Generic Chaining with Applications to Statistical Learning

TL;DR

The results show that empirical risk minimzaton with dependent data attains a prediction accuracy comparable to that in the i.i.d. setting for a wide range of nonlinear regression models.

Abstract

This paper develops a general concentration inequality for the suprema of empirical processes with dependent data. The concentration inequality is obtained by combining generic chaining with a coupling-based strategy. Our framework accommodates high-dimensional and heavy-tailed (sub-Weibull) data. We demonstrate the usefulness of our result by deriving non-asymptotic predictive performance guarantees for empirical risk minimization in regression problems with dependent data. In particular, we establish an oracle inequality for a broad class of nonlinear regression models and, as a special case, a single-layer neural network model. Our results show that empirical risk minimzaton with dependent data attains a prediction accuracy comparable to that in the i.i.d. setting for a wide range of nonlinear regression models.

Theorems & Definitions (23)

• Theorem 2.1: Concentration
• Proposition 3.1
• Corollary 3.1
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3: Generic Chaining
• Proposition 4.4
• proof : Proof of Proposition \ref{['corollary:erm:reg']}
• proof : Proof of Corollary \ref{['corollary:erm:neural_nets']}
• proof : Proof of Proposition \ref{['prop:coupling']}