Mean-Field Neural ODEs via Relaxed Optimal Control
This provides theoretical guarantees for neural ODE training, addressing a foundational problem in machine learning with broad implications for model robustness and efficiency.
The paper tackles the analysis of neural ODEs trained with stochastic gradient algorithms by connecting control theory, deep learning, and statistical sampling, deriving explicit convergence rates for gradient flow and generalization error that are dimension-independent.
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.