Hoeffding's Inequality for Markov Chains under Generalized Concentrability Condition

TL;DR

This paper extends Hoeffding's inequality to a broader class of Markov chains using a generalized concentrability condition based on IPM, enabling new non-asymptotic bounds in machine learning.

Contribution

It introduces a flexible framework for Hoeffding's inequality under generalized conditions, applicable beyond traditional ergodic Markov chains.

Findings

Generalization bound for empirical risk minimization with Markovian data

Finite sample guarantee for Ployak-Ruppert averaging of SGD

New regret bound for rested Markovian bandits with general state space

Abstract

This paper studies Hoeffding's inequality for Markov chains under the generalized concentrability condition defined via integral probability metric (IPM). The generalized concentrability condition establishes a framework that interpolates and extends the existing hypotheses of Markov chain Hoeffding-type inequalities. The flexibility of our framework allows Hoeffding's inequality to be applied beyond the ergodic Markov chains in the traditional sense. We demonstrate the utility by applying our framework to several non-asymptotic analyses arising from the field of machine learning, including (i) a generalization bound for empirical risk minimization with Markovian samples, (ii) a finite sample guarantee for Ployak-Ruppert averaging of SGD, and (iii) a new regret bound for rested Markovian bandits with general state space.