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.

翻译：本文研究一类神经随机微分方程，并探讨当样本量趋于无穷时，相关抽样最优控制问题的极限行为。具有N个样本的神经随机微分方程可关联到具有集中控制的N粒子系统。我们分析了对应于N粒子系统的Hamilton–Jacobi–Bellman方程，并建立了关于N一致的正则性结果。该一致正则性估计通过随机最大值原理以及对一个倒向随机Riccati方程的分析获得。利用这些一致正则性结果，我们证明了当样本量N趋于无穷时，神经随机微分方程目标泛函的最小值及最优参数的收敛性。极限对象可识别为定义在Borel概率测度Wasserstein空间上的适当函数。此外，本文还得到了定量的代数收敛速率。