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The entropic fictitious play (EFP) is a recently proposed algorithm thatminimizes the sum of a convex functional and entropy in the space of measures-- such an objective naturally arises in the optimization of a two-layer neuralnetwork in the mean-field regime. In this work, we provide a conciseprimal-dual analysis of EFP in the setting where the learning problem exhibitsa finite-sum structure. We establish quantitative global convergence guaranteesfor both the continuous-time and discrete-time dynamics based on properties ofa proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, ourprimal-dual framework entails a memory-efficient particle-based implementationof the EFP update, and also suggests a connection to gradient boosting methods.We illustrate the efficiency of our novel implementation in experimentsincluding neural network optimization and image synthesis.

Primal and Dual Analysis of Entropic Fictitious Play for Finite-sum Problems