Mean-Field Limits for Two-Layer Neural Networks Trained with Consensus-Based Optimization

Abstract: We study two-layer neural networks and train these with a particle-based method called consensus-based optimization (CBO). We compare the performance of CBO against Adam on two test cases and demonstrate how a hybrid approach, combining CBO with Adam, provides faster convergence than CBO. In the context of multi-task learning, we recast CBO into a formulation that offers less memory overhead. The CBO method allows for a mean-field limit formulation, which we couple with the mean-field limit of the neural network. To this end, we first reformulate CBO within the optimal transport framework. Finally, in the limit of infinitely many particles, we define the corresponding dynamics on the Wasserstein-over-Wasserstein space and show that the variance decreases monotonically.