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

arXiv: https://arxiv.org/abs/2404.05185 · HTML: https://mlsanity.com/paper/2404.05185/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 43 of its 78 references that are in the map.
- First posted on arXiv 8 Apr 2024.

## Builds on (most cited first, 30 of 43)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [On the Convergence of Model Free Learning in Mean Field Games](https://mlsanity.com/paper/1907.02633/) (2019)
- [Deep neural networks, generic universal interpolation, and controlled ODEs](https://mlsanity.com/paper/1908.07838/) (2019)
- [Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis](https://mlsanity.com/paper/1812.04300/) (2018)
- [Uniform-in-time propagation of chaos for mean field Langevin dynamics](https://mlsanity.com/paper/2212.03050/) (2022)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications](https://mlsanity.com/paper/1812.05916/) (2018)
- [Mean-Field Neural ODEs via Relaxed Optimal Control](https://mlsanity.com/paper/1912.05475/) (2019)
- [Scaling Properties of Deep Residual Networks](https://mlsanity.com/paper/2105.12245/) (2021)
- [Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics](https://mlsanity.com/paper/1604.04057/) (2016)
- [Limit theory for controlled McKean-Vlasov dynamics](https://mlsanity.com/paper/1609.08064/) (2016)
- [Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity](https://mlsanity.com/paper/2101.12362/) (2021)
- [Rate of convergence for particle approximation of PDEs in Wasserstein space](https://mlsanity.com/paper/2103.00837/) (2021)
- [Large Sample Mean-Field Stochastic Optimization](https://mlsanity.com/paper/1906.08894/) (2019)
- [Extended mean field control problem: a propagation of chaos result](https://mlsanity.com/paper/2006.12996/) (2020)
- [On the Optimal Rate for the Convergence Problem in Mean Field Control](https://mlsanity.com/paper/2305.08423/) (2023)
- [EM Algorithm and Stochastic Control in Economics](https://mlsanity.com/paper/1611.01767/) (2016)

## Abstract (arXiv)

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.