Learning from non-irreducible Markov chains

TL;DR

This paper investigates learning from data generated by non-irreducible Markov chains, establishing uniform convergence and generalization bounds under ergodicity assumptions, extending prior results beyond irreducible chains.

Contribution

It introduces uniform convergence and learnability results for machine learning with data from non-irreducible Markov chains under ergodicity conditions.

Findings

Established uniform convergence of sample error.

Proved learnability of the approximate sample error minimization algorithm.

Derived generalization bounds for non-irreducible Markov chain data.

Abstract

Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong correlation between the marginals of the data-generating process, suggesting that the i.i.d. assumption is not always justified. This problem has been already considered in the context of Markov chains satisfying the Doeblin condition. This condition, among other things, implies that the chain is not singular in its behavior, i.e. it is irreducible. In this article, we focus on the case when the training data set is drawn from a not necessarily irreducible Markov chain. Under the assumption that the chain is uniformly ergodic with respect to the $\mathrm{L}^1$-Wasserstein distance, and certain regularity assumptions on the hypothesis class and the state space of the chain, we first obtain a uniform convergence result for the corresponding sample error, and then we conclude learnability of the approximate sample error minimization algorithm and find its generalization bounds. At the end, a relative uniform convergence result for the sample error is also discussed.