Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size
It provides theoretical guarantees for deep learning models by linking finite-sample particle systems to infinite-sample limits, which is incremental but important for understanding scalability.
This paper analyzes the convergence of optimal control problems for neural stochastic differential equations as sample size increases, showing that minima and optimal parameters converge to functions on Wasserstein space with quantitative rates.
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.