[Paper Review] A Measure Theoretical Approach to the Mean-field Maximum Principle for Training NeurODEs

Benoît Bonnet, Cristina Cipriani+2 more|arXiv (Cornell University)|Jul 19, 2021

Sparse and Compressive Sensing Techniques28 references4 citations

TL;DR

This paper introduces a measure-theoretical framework for training Neural ODEs via mean-field optimal control with L² regularization, deriving a novel mean-field maximum principle (PMP) that guarantees a unique, Lipschitz continuous control solution. The uniqueness enables rigorous quantification of generalization error, providing a theoretical explanation for the double descent phenomenon in overparameterized models.

ABSTRACT

In this paper we consider a measure-theoretical formulation of the training of NeurODEs in the form of a mean-field optimal control with $L^2$-regularization of the control. We derive first order optimality conditions for the NeurODE training problem in the form of a mean-field maximum principle, and show that it admits a unique control solution, which is Lipschitz continuous in time. As a consequence of this uniqueness property, the mean-field maximum principle also provides a strong quantitative generalization error for finite sample approximations. Our derivation of the mean-field maximum principle is much simpler than the ones currently available in the literature for mean-field optimal control problems, and is based on a generalized Lagrange multiplier theorem on convex sets of spaces of measures. The latter is also new, and can be considered as a result of independent interest.

Motivation & Objective

• To provide a rigorous mathematical foundation for training Neural ODEs using mean-field optimal control with L² regularization.
• To derive first-order optimality conditions for NeurODE training via a novel mean-field maximum principle.
• To establish uniqueness and regularity of the optimal control, ensuring strong quantitative generalization error bounds.
• To offer a simplified derivation of the mean-field PMP using a generalized Lagrange multiplier theorem on convex sets of measures.
• To explain the double descent phenomenon through rigorous error quantification in finite-sample approximations.

Proposed method

• Formulates NeurODE training as a mean-field optimal control problem with L² regularization on the control in the space of probability measures.
• Applies a generalized Lagrange multiplier theorem on convex subsets of Banach spaces of measures to derive necessary optimality conditions.
• Derives the mean-field PMP in two settings: continuous and measurable controls, using both Lagrangian and Hamiltonian approaches.
• Establishes well-posedness of the PMP via continuity equations and characteristic flows in measure spaces.
• Uses a fixed-point argument and implicit function theorem in Banach spaces to prove existence and regularity of solutions.
• Validates theoretical results through numerical experiments on synthetic and benchmark datasets.

Experimental results

Research questions

• RQ1 How can the training of Neural ODEs be rigorously formulated as a mean-field optimal control problem with L² regularization?
• RQ2 What are the first-order optimality conditions for such a problem, and do they admit a unique solution?
• RQ3 Can the uniqueness of the optimal control be leveraged to derive quantitative generalization error bounds?
• RQ4 How does the proposed mean-field PMP compare to existing approaches in terms of simplicity and generality?
• RQ5 Can the derived framework explain the double descent phenomenon in overparameterized models?

Key findings

• The mean-field maximum principle for NeurODE training admits a unique control solution that is Lipschitz continuous in time.
• The uniqueness of the optimal control enables a strong quantitative generalization error bound for finite-sample approximations.
• The paper provides a simplified derivation of the mean-field PMP using a new generalized Lagrange multiplier theorem on convex sets of measures, which is of independent interest.
• The theoretical framework rigorously justifies the double descent phenomenon by showing that generalization error decreases even when model parameters exceed training sample size.
• Numerical experiments confirm the theoretical predictions, demonstrating stable training and consistent error behavior across overparameterized regimes.