Exponential Concentration in Stochastic Approximation

TL;DR

This paper establishes exponential concentration bounds for stochastic approximation algorithms, providing a new perspective that complements traditional asymptotic normality results, and applies to several algorithms including stochastic gradient descent.

Contribution

It extends geometric ergodicity techniques to stochastic approximation, deriving exponential tail bounds and convergence rates for multiple algorithms.

Findings

Proves exponential concentration bounds for stochastic approximation.

Demonstrates linear and faster convergence rates for specific algorithms.

Extends Markov chain ergodicity results to stochastic approximation context.

Abstract

We analyze the behavior of stochastic approximation algorithms where iterates, in expectation, progress towards an objective at each step. When progress is proportional to the step size of the algorithm, we prove exponential concentration bounds. These tail-bounds contrast asymptotic normality results, which are more frequently associated with stochastic approximation. The methods that we develop rely on a geometric ergodicity proof. This extends a result on Markov chains due to Hajek (1982) to the area of stochastic approximation algorithms. We apply our results to several different Stochastic Approximation algorithms, specifically Projected Stochastic Gradient Descent, Kiefer-Wolfowitz and Stochastic Frank-Wolfe algorithms. When applicable, our results prove faster $O(1/t)$ and linear convergence rates for Projected Stochastic Gradient Descent with a non-vanishing gradient.