General multilevel adaptations for stochastic approximation algorithms

TL;DR

This paper proves central limit theorems for multilevel stochastic approximation algorithms, extending understanding of their asymptotic behavior under mild conditions in different error decay regimes.

Contribution

It introduces multilevel adaptations for stochastic approximation algorithms and establishes their asymptotic normality under broad assumptions.

Findings

Central limit theorems proven for multilevel schemes

Results cover slow and critical error decay regimes

Analysis under very mild technical assumptions

Abstract

In this article we establish central limit theorems for multilevel Polyak-Ruppert averaged stochastic approximation schemes. We work under very mild technical assumptions and consider the slow regime in wich typical errors decay like $N^{-\delta}$ with $\delta\in(0,\frac 12)$ and the critical regime in which errors decay of order $N^{-1/2}\sqrt{\log N}$ in the runtime $N$ of the algorithm.