Rademacher complexity for Markov chains : Applications to kernel smoothing and Metropolis-Hasting

TL;DR

This paper extends Rademacher complexity to Markov chains using renewal theory, enabling new bounds and convergence rates for kernel density estimation and concentration inequalities for Metropolis-Hastings algorithms.

Contribution

It introduces the concept of block Rademacher complexity for Markov chains, providing a framework to analyze empirical processes and derive bounds for dependent data.

Findings

Established bounds on block Rademacher complexity for VC-type classes.

Derived convergence rates for kernel density estimators of stationary measures.

Provided concentration inequalities for Metropolis-Hastings algorithms.

Abstract

Following the seminal approach by Talagrand, the concept of Rademacher complexity for independent sequences of random variables is extended to Markov chains. The proposed notion of "block Rademacher complexity" (of a class of functions) follows from renewal theory and allows to control the expected values of suprema (over the class of functions) of empirical processes based on Harris Markov chains as well as the excess probability. For classes of Vapnik-Chervonenkis type, bounds on the "block Rademacher complexity" are established. These bounds depend essentially on the sample size and the probability tails of the regeneration times. The proposed approach is employed to obtain convergence rates for the kernel density estimator of the stationary measure and to derive concentration inequalities for the Metropolis-Hasting algorithm.