A mesh-independent method for second-order potential mean field games - Liu, Kang; Pfeiffer, Laurent (IMA Journal of Numerical Analysis, 2025)
This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.
CRIS-Team Scopus Importer:2025-04-18
Keywords: finite differences; Frank–Wolfe algorithm; mean field games; mesh independence

A Potential Game Perspective in Federated Learning - Liu, Kang; Wang, Ziqi; Zuazua Iriondo, Enrique (2025)
Federated learning (FL) is an emerging paradigm for training machine learning models across distributed clients. Traditionally, in FL settings, a central server assigns training efforts (or strategies) to clients. However, from a market-oriented perspective, clients may independently choose their training efforts based on rational self-interest. To explore this, we propose a potential game framework where each client's payoff is determined by their individual efforts and the rewards provided by the server. The rewards are influenced by the collective efforts of all clients and can be modulated through a reward factor. Our study begins by establishing the existence of Nash equilibria (NEs), followed by an investigation of uniqueness in homogeneous settings. We demonstrate a significant improvement in clients' training efforts at a critical reward factor, identifying it as the optimal choice for the server. Furthermore, we prove the convergence of the best-response algorithm to compute NEs for our FL game. Finally, we apply the training efforts derived from specific NEs to a real-world FL scenario, validating the effectiveness of the identified optimal reward facto
https://cris.fau.de/converis/publicweb/Publication/331321390

Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics - Liu, Kang; Zuazua Iriondo, Enrique (Mathematical Models & Methods in Applied Sciences, 2025)
In this work, we address three non-convex optimization problems associated with the training of shallow neural networks (NNs) for exact and approximate representation, as well as for regression tasks. Through a mean-field approach, we convexify these problems and, applying a representer theorem, prove the absence of relaxation gaps. We establish generalization bounds for the resulting NN solutions, assessing their predictive performance on test datasets, analyzing the impact of key hyperparameters on these bounds, and proposing optimal choices. On the computational side, we examine the discretization of the convexified problems and derive convergence rates. For low-dimensional datasets, these discretized problems are efficiently solvable using the simplex method. For high-dimensional datasets, we propose a sparsification algorithm that, combined with gradient descent for over-parameterized shallow NNs, yields effective solutions to the primal problem
CRIS-Team Scopus Importer:2025-04-11
Keywords: generalization; mean-field relaxation; Neural network; numerical algorithm; representer theorem

Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications - Li, Ziqian; Liu, Kang; Liverani, Lorenzo; Zuazua Iriondo, Enrique (2025)
In this paper, we introduce semi-autonomous neural ordinary differential equations (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach.<br
https://cris.fau.de/converis/publicweb/Publication/326007759
Keywords: neural ODEs; universal approximation; Barron space; transport equations.