Entropic Fictitious Play for Mean Field Optimization Problem

TL;DR

This paper introduces entropic fictitious play, a novel algorithm for optimizing two-layer neural networks in the mean field limit, demonstrating exponential convergence and connecting game theory concepts with neural network training.

Contribution

The paper proposes a new entropic fictitious play algorithm for mean field neural network optimization, with proven exponential convergence and a novel two-loop iteration structure.

Findings

Proves exponential convergence of the entropic fictitious play algorithm.

Demonstrates the algorithm's effectiveness through numerical examples.

Identifies the fixed point as the minimizer in the mean field limit with entropic regularization.

Abstract

We study two-layer neural networks in the mean field limit, where the number of neurons tends to infinity. In this regime, the optimization over the neuron parameters becomes the optimization over the probability measures, and by adding an entropic regularizer, the minimizer of the problem is identified as a fixed point. We propose a novel training algorithm named entropic fictitious play, inspired by the classical fictitious play in game theory for learning Nash equilibriums, to recover this fixed point, and the algorithm exhibits a two-loop iteration structure. Exponential convergence is proved in this paper and we also verify our theoretical results by simple numerical examples.