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

arXiv: https://arxiv.org/abs/2303.02957 · HTML: https://mlsanity.com/paper/2303.02957/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 34 references that are in the map.
- First posted on arXiv 6 Mar 2023.

## Builds on (most cited first, 22 of 22)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Modular Primitives for High-Performance Differentiable Rendering](https://mlsanity.com/paper/2011.03277/) (2020)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks](https://mlsanity.com/paper/2202.08658/) (2022)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Mean Field Analysis of Neural Networks: A Central Limit Theorem](https://mlsanity.com/paper/1808.09372/) (2018)
- [Limitations of Lazy Training of Two-layers Neural Networks](https://mlsanity.com/paper/1906.08899/) (2019)
- [Fictitious Play for Mean Field Games: Continuous Time Analysis and Applications](https://mlsanity.com/paper/2007.03458/) (2020)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Stochastic Particle Gradient Descent for Infinite Ensembles](https://mlsanity.com/paper/1712.05438/) (2017)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [Modern Evolution Strategies for Creativity: Fitting Concrete Images and Abstract Concepts](https://mlsanity.com/paper/2109.08857/) (2021)
- [Learning in Mean Field Games: the Fictitious Play](https://mlsanity.com/paper/1507.06280/) (2015)
- [Uniform-in-time propagation of chaos for mean field Langevin dynamics](https://mlsanity.com/paper/2212.03050/) (2022)
- [Particle Dual Averaging: Optimization of Mean Field Neural Networks with Global Convergence Rate Analysis](https://mlsanity.com/paper/2012.15477/) (2020)
- [Entropic Fictitious Play for Mean Field Optimization Problem](https://mlsanity.com/paper/2202.05841/) (2022)
- [Finite mean field games: fictitious play and convergence to a first order continuous mean field game](https://mlsanity.com/paper/1805.05940/) (2018)
- [Generalized conditional gradient and learning in potential mean field games](https://mlsanity.com/paper/2209.12772/) (2022)

## Built on by (most cited first, 4 of 4)

- [Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction](https://mlsanity.com/paper/2306.07221/) (2023)
- [Provably Neural Active Learning Succeeds via Prioritizing Perplexing Samples](https://mlsanity.com/paper/2406.03944/) (2024)
- [Direct Distributional Optimization for Provable Alignment of Diffusion Models](https://mlsanity.com/paper/2502.02954/) (2025)
- [Non-convex entropic mean-field optimization via Best Response flow](https://mlsanity.com/paper/2505.22760/) (2025)

## Abstract (arXiv)

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.