The Pontryagin Maximum Principle for Training Convolutional Neural Networks

A novel batch sequential quadratic Hamiltonian (bSQH) algorithm for training convolutional neural networks (CNNs) with $L^0$-based regularization is presented. This methodology is based on a discrete-time Pontryagin maximum principle (PMP). It uses forward and backward sweeps together with the layerwise approximate maximization of an augmented Hamiltonian function, where the augmentation parameter is chosen adaptively. A technique for determining this augmentation parameter is proposed, and the loss-reduction and convergence properties of the bSQH algorithm are analysed theoretically and validated numerically. Results of numerical experiments in the context of image classification with a sparsity enforcing $L^0$-based regularizer demonstrate the effectiveness of the proposed method in full-batch and mini-batch modes.

Sebastian Hofmann, Alfio Borzì · 2025