# Non-convex entropic mean-field optimization via Best Response flow arXiv: https://arxiv.org/abs/2505.22760 ยท HTML: https://mlsanity.com/paper/2505.22760/ ## Key facts - Cited by 2 papers in the map (data through 14 Oct 2026). - Builds on 28 of its 47 references that are in the map. - First posted on arXiv 28 May 2025. ## Builds on (most cited first, 28 of 28) - [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018) - [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018) - [On the Global Convergence Rates of Softmax Policy Gradient Methods](https://mlsanity.com/paper/2005.06392/) (2020) - [Mean Field Analysis of Neural Networks: A Central Limit Theorem](https://mlsanity.com/paper/1808.09372/) (2018) - [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018) - [Independent Policy Gradient Methods for Competitive Reinforcement Learning](https://mlsanity.com/paper/2101.04233/) (2021) - [Model-Based Multi-Agent RL in Zero-Sum Markov Games with Near-Optimal Sample Complexity](https://mlsanity.com/paper/2007.07461/) (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) - [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019) - [A mean-field analysis of two-player zero-sum games](https://mlsanity.com/paper/2002.06277/) (2020) - [Faster Last-iterate Convergence of Policy Optimization in Zero-Sum Markov Games](https://mlsanity.com/paper/2210.01050/) (2022) - [A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces](https://mlsanity.com/paper/2310.02951/) (2023) - [Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regime](https://mlsanity.com/paper/2201.07296/) (2022) - [Global optimality of softmax policy gradient with single hidden layer neural networks in the mean-field regime](https://mlsanity.com/paper/2010.11858/) (2020) - [An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Games](https://mlsanity.com/paper/2211.01280/) (2022) - [Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective](https://mlsanity.com/paper/2212.08791/) (2022) - [Entropic Fictitious Play for Mean Field Optimization Problem](https://mlsanity.com/paper/2202.05841/) (2022) - [Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems](https://mlsanity.com/paper/2312.01127/) (2023) - [Gradient Flows for Regularized Stochastic Control Problems](https://mlsanity.com/paper/2006.05956/) (2020) - [Game on Random Environment, Mean-field Langevin System and Neural Networks](https://mlsanity.com/paper/2004.02457/) (2020) - [Polyak-L ojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes](https://mlsanity.com/paper/2206.02774/) (2022) - [Primal and Dual Analysis of Entropic Fictitious Play for Finite-sum Problems](https://mlsanity.com/paper/2303.02957/) (2023) - [A Fisher-Rao gradient flow for entropic mean-field min-max games](https://mlsanity.com/paper/2405.15834/) (2024) - [Entropic mean-field min-max problems via Best Response flow](https://mlsanity.com/paper/2306.03033/) (2023) - [Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives](https://mlsanity.com/paper/2501.17122/) (2025) - [Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games](https://mlsanity.com/paper/2505.07642/) (2025) - [Convex Regularization and Convergence of Policy Gradient Flows under Safety Constraints](https://mlsanity.com/paper/2411.19193/) (2024) ## Built on by (most cited first, 2 of 2) - [Mean-Field Generalisation Bounds for Learning Controls in Stochastic Environments](https://mlsanity.com/paper/2508.16001/) (2025) - [On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization](https://mlsanity.com/paper/2602.15094/) (2026) ## Abstract (arXiv) We study the problem of minimizing non-convex functionals on the space of probability measures, regularized by the relative entropy (KL divergence) with respect to a fixed reference measure, as well as the corresponding problem of solving entropy-regularized non-convex-non-concave min-max problems. We utilize the Best Response flow (also known in the literature as the fictitious play flow) and study how its convergence is influenced by the relation between the degree of non-convexity of the functional under consideration, the regularization parameter and the tail behaviour of the reference measure. In particular, we demonstrate how to choose the regularizer, given the non-convex functional, so that the Best Response operator becomes a contraction with respect to the L1-Wasserstein distance, which ensures the existence of its unique fixed point that is then shown to be the unique global minimizer for our optimization problem. This extends recent results where the Best Response flow was applied to solve convex optimization problems regularized by the relative entropy with respect to arbitrary reference measures, and with arbitrary values of the regularization parameter. Our results explain precisely how the assumption of convexity can be relaxed, at the expense of making a specific choice of the regularizer. Additionally, we demonstrate how these results can be applied in reinforcement learning in the context of policy optimization for Markov Decision Processes and Markov games with softmax parametrized policies in the mean-field regime.