Self-Averaging and On-line Learning

TL;DR

This paper establishes conditions under which the stochastic dynamics of on-line learning can be approximated by deterministic equations in large systems, highlighting the importance of constraints for self-averaging.

Contribution

It provides theoretical conditions for self-averaging in on-line learning, linking stochastic dynamics to deterministic evolution of order parameters.

Findings

Self-averaging occurs under specific constraints on increment magnitudes.

Without constraints, convergence is only in probability.

The deterministic description simplifies analysis of large-scale on-line learning.

Abstract

Conditions are given under which one may prove that the stochastic dynamics of on-line learning can be described by the deterministic evolution of a finite set of order parameters in the thermodynamic limit. A global constraint on the average magnitude of the increments in the stochastic process is necessary to ensure self-averaging. In the absence of such a constraint, convergence may only be in probability.