Mean-Field Neural ODEs via Relaxed Optimal Control

We develop a framework for the analysis of deep neural networks and neural ODE models that are trained with stochastic gradient algorithms. We do that by identifying the connections between control theory, deep learning and theory of statistical sampling. We derive Pontryagin's optimality principle and study the corresponding gradient flow in the form of Mean-Field Langevin dynamics (MFLD) for solving relaxed data-driven control problems. Subsequently, we study uniform-in-time propagation of chaos of time-discretised MFLD. We derive explicit convergence rate in terms of the learning rate, the number of particles/model parameters and the number of iterations of the gradient algorithm. In addition, we study the error arising when using a finite training data set and thus provide quantitive bounds on the generalisation error. Crucially, the obtained rates are dimension-independent. This is possible by exploiting the regularity of the model with respect to the measure over the parameter space.