Deviation inequalities for stochastic approximation by averaging

TL;DR

This paper develops deviation inequalities for a class of Markov chains, including stochastic approximation by averaging, using martingale methods, and applies these results to empirical risk minimization.

Contribution

It introduces a new class of Markov chains and derives deviation inequalities applicable to stochastic approximation and empirical risk minimization.

Findings

Derived deviation inequalities under various moment conditions.

Applied inequalities to stochastic approximation by averaging.

Extended results to empirical risk minimization.

Abstract

We introduce a class of Markov chains, that contains the model of stochastic approximation by averaging and non-averaging. Using martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differences.Finally, we apply these inequalities to the stochastic approximation by averaging and empirical risk minimisation.