A weak convergence approach to large deviations for stochastic approximations

TL;DR

This paper establishes a large deviation principle for stochastic approximation algorithms with state-dependent Markovian noise, providing new insights into their rare deviation behaviors using a weak convergence approach.

Contribution

It generalizes previous large deviation results for stochastic approximations and introduces a new representation of the rate function involving Markov transition kernels.

Findings

Proves a large deviation principle for stochastic approximations with Markovian noise.

Provides a new action functional representation for the rate function.

Includes examples like stochastic gradient descent and Wang-Landau algorithm.

Abstract

The theory of stochastic approximations form the theoretical foundation for studying convergence properties of many popular recursive learning algorithms in statistics, machine learning and statistical physics. Large deviations for stochastic approximations provide asymptotic estimates of the probability that the learning algorithm deviates from its expected path, given by a limit ODE, and the large deviation rate function gives insights to the most likely way that such deviations occur. In this paper we prove a large deviation principle for general stochastic approximations with state-dependent Markovian noise and decreasing step size. Using the weak convergence approach to large deviations, we generalize previous results for stochastic approximations and identify the appropriate scaling sequence for the large deviation principle. We also give a new representation for the rate function, in which the rate function is expressed as an action functional involving the family of Markov transition kernels. Examples of learning algorithms that are covered by the large deviation principle include stochastic gradient descent, persistent contrastive divergence and the Wang-Landau algorithm.