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

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.

@article{liao2025_2404.05185,
  title={ Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size },
  author={ Huafu Liao and Alpár R. Mészáros and Chenchen Mou and Chao Zhou },
  journal={arXiv preprint arXiv:2404.05185},
  year={ 2025 }
}