  The entropic fictitious play (EFP) is a recently proposed algorithm that
minimizes 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 neural
network in the mean-field regime. In this work, we provide a concise
primal-dual analysis of EFP in the setting where the learning problem exhibits
a finite-sum structure. We establish quantitative global convergence guarantees
for both the continuous-time and discrete-time dynamics based on properties of
a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our
primal-dual framework entails a memory-efficient particle-based implementation
of the EFP update, and also suggests a connection to gradient boosting methods.
We illustrate the efficiency of our novel implementation in experiments
including neural network optimization and image synthesis.
