aiqbench analysis
This paper provides a rigorous mathematical analysis of neural stochastic differential equations by examining their convergence to infinite-sample limits within Wasserstein space. By establishing uniform regularity estimates for the associated optimal control problems, it offers theoretical foundations relevant to understanding how deep learning models behave as data scales approach infinity. The work bridges advanced probability theory and machine learning optimization, specifically addressing the mathematical underpinnings of large-scale neural network training dynamics.
Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size
arXiv:2404.05185v4 Announce Type: replace-cross 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 the 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 convergence rates are also obtained.
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 the 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 convergence rates are also obtained.
50 pages; to appear in Appl. Math. Optim
Optimization and Control (math.OC); Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML)
arXiv:2404.05185 [math.OC]
(or arXiv:2404.05185v4 [math.OC] for this version)
From: Alpár R. Mészáros [view email] [v1] Mon, 8 Apr 2024 04:22:55 UTC (117 KB) [v2] Fri, 28 Mar 2025 12:47:57 UTC (116 KB) [v3] Wed, 18 Jun 2025 08:55:23 UTC (42 KB) [v4] Tue, 4 Aug 2026 10:13:43 UTC (47 KB)
Full text under the publisher's open licence · 209 words
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