Deviation inequalities for contractive infinite memory processes

TL;DR

This paper develops deviation and moment inequalities for a broad class of infinite memory processes with applications to stochastic gradient Langevin dynamics, extending previous Markov and discrete memory models.

Contribution

It introduces a new class of infinite memory processes and derives deviation inequalities using martingale methods, generalizing prior models and including practical applications.

Findings

Established deviation inequalities for Lipschitz functions of infinite memory processes.

Extended previous Markov and discrete memory models to more general processes.

Applied results to stochastic gradient Langevin dynamic algorithms.

Abstract

In this paper, we introduce a class of processes that contains many natural examples. The interesting feature of such type processes lays on its infinite memory that allows it to record a quite ancient history. Then, using the martingale decomposition method, we establish some deviation and moment inequalities for separately Lipschitz functions of such a process, under various moment conditions on some dominating random variables. Our results generalize the Markov models of Dedecker and Fan [Stochastic Process. Appl., 2015] and a recent paper by Chazottes et al. [Ann. Appl. Probab., 2023] for the special case of a specific class of infinite memory models with discrete values. An application to stochastic gradient Langevin dynamic algorithm is also discussed.