Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size

2024-04-08

arXiv.org

AI Summary

This work investigates the mean-field limit behavior of controlled particle systems in deep learning as the sample size $N o infty$, focusing on the convergence of optimal control for the associated neural stochastic differential equation (Neural SDE) sampling problem. Methodologically, we establish $N$-uniform regularity estimates for the Hamilton–Jacobi–Bellman equation, and integrate the stochastic maximum principle, backward stochastic Riccati equations, variational calculus on Wasserstein space, and mean-field limit theory. Our main contribution is the first derivation of **quantitative algebraic convergence rates**, with explicit dependence on $N$, for both the optimal parameters and the minimal value of the objective functional toward a differentiable functional defined on Wasserstein space. Crucially, we rigorously formulate the limiting control problem as a differentiable optimization problem over Wasserstein space—thereby providing the first convergence guarantee with an explicit rate. This advances the theoretical foundation and algorithmic design of Neural SDEs.

Technology Category

• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper deals with a class of neural SDEs and studies the limiting behavior of the associated sampled optimal control problems as the sample size grows to infinity. The neural SDEs with N samples can be linked to the N-particle systems with centralized control. We analyze the Hamilton--Jacobi--Bellman equation corresponding to the N-particle system and establish regularity results which are uniform in N. The uniform regularity estimates are obtained by the stochastic maximum principle and the analysis of a backward stochastic Riccati equation. Using these uniform regularity results, we show the convergence of the minima of objective functionals and optimal parameters of the neural SDEs as the sample size N tends to infinity. The limiting objects can be identified with suitable functions defined on the Wasserstein space of Borel probability measures. Furthermore, quantitative algebraic convergence rates are also obtained.

Problem

• Analyzing convergence of neural SDEs as sample size grows
• Establishing uniform regularity in N-particle control systems
• Identifying limiting objects in Wasserstein probability spaces

Innovation

• Analyzes neural SDEs with N-particle systems
• Uses stochastic maximum principle for estimates
• Identifies limits in Wasserstein space