A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization

2025-12-31 · Source: JMLR · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Theoretical Machine Learning · Depth: Expert, quick

Summary

A study analyzes minimax optimization problems involving over-parameterized two-layer neural networks, specifically those estimating linear functional equations with quadratic objectives. It investigates the convergence of the stochastic gradient descent-ascent (SGDA) algorithm and the representation learning capabilities of these networks. The research establishes convergence within the mean-field regime, where SGDA is shown to correspond to a Wasserstein gradient flow. This flow globally converges to a stationary point of the minimax objective at a sublinear rate of ℮(T⁻¹ + α⁻¹ ). Furthermore, it finds the functional equation solution when the regularizer is strongly convex. The study also reveals that neural network feature representations may deviate from initial representations by ℮(α⁻¹), measured by Wasserstein distance. Applications include policy evaluation, nonparametric instrumental variable regression, and asset pricing.

Key takeaway

For AI scientists developing or analyzing functional minimax optimization problems with over-parameterized neural networks, understanding the mean-field convergence properties of SGDA is crucial. This analysis confirms global convergence at a ℮(T⁻¹ + α⁻¹ ) rate and highlights how strong convexity in the regularizer ensures finding functional equation solutions. You should consider these theoretical guarantees when designing robust optimization strategies and interpreting representation learning dynamics, especially regarding feature deviation.

Key insights

Neural Stochastic Gradient Descent-Ascent converges globally in the mean-field regime for functional minimax optimization, linking to Wasserstein gradient flow.

Principles

• SGDA in mean-field corresponds to Wasserstein gradient flow.
• Strong convexity of regularizer ensures functional equation solution.
• Feature representation deviation is ℮(α⁻¹) by Wasserstein distance.

In practice

• Policy evaluation.
• Nonparametric instrumental variable regression.
• Asset pricing.

Topics

• Minimax Optimization
• Neural Networks
• Stochastic Gradient Descent-Ascent
• Mean-Field Theory
• Wasserstein Gradient Flow
• Representation Learning

Best for: Research Scientist, AI Scientist

Related on AIssential

• Functional Gradient Descent with Adaptive Representations — Adaptive representation in functional gradient descent ensures convergence to global minimizers by dynamically refining gradient approximations.
• Uniform-in-time concentration in two-layer neural networks via transportation inequalities — The study quantifies neural network prediction discrepancies from mean-field limits using transportation inequalities.
• Scalable Mean-Field Variational Inference via Preconditioned Primal-Dual Optimization — Primal-dual optimization with preconditioning offers scalable, robust, and faster convergence for large-scale mean-field variational inference.
• Why SGD is not Brownian Motion: A New Perspective on Stochastic Dynamics — SGD's dynamics are better modeled as deterministic in a fluctuating loss landscape, not Brownian motion.
• Stochastic Gradient Methods: Bias, Stability and Generalization — A new framework analyzes stability and generalization for biased stochastic gradient methods.
• Wasserstein Convergence of ODE-Based Samplers in Decentralized Diffusion Model via Velocity Field Decomposition — The first W₂ convergence guarantee for decentralized diffusion models with ODE-based sampling is established, showing ℮(N⁻¹/²+ε) rate.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by JMLR.