Dynamics of Learning with Restricted Training Sets II: Tests and Applications

TL;DR

This paper extends a general theory of supervised learning dynamics in neural networks to various learning rules, providing exact solutions for Hebbian learning and good approximations for others, with practical schemes for simplified analysis.

Contribution

It applies a unified theoretical framework to different learning rules, deriving exact and approximate solutions, and introduces simplified schemes for analyzing learning dynamics.

Findings

Exact results for Hebbian learning dynamics.

Good agreement between theory and simulations for Perceptron and AdaTron.

A simple diffusion equation effectively models learning dynamics.

Abstract

We apply a general theory describing the dynamics of supervised learning in layered neural networks in the regime where the size p of the training set is proportional to the number of inputs N, as developed in a previous paper, to several choices of learning rules. In the case of (on-line and batch) Hebbian learning, where a direct exact solution is possible, we show that our theory provides exact results at any time in many different verifiable cases. For non-Hebbian learning rules, such as Perceptron and AdaTron, we find very good agreement between the predictions of our theory and numerical simulations. Finally, we derive three approximation schemes aimed at eliminating the need to solve a functional saddle-point equation at each time step, and assess their performance. The simplest of these schemes leads to a fully explicit and relatively simple non-linear diffusion equation for the joint field distribution, which already describes the learning dynamics surprisingly well over a wide range of parameters.