Infinite width (finite depth) neural networks benefit from multi-task learning unlike shallow Gaussian Processes -- an exact quantitative macroscopic characterization

We prove in this paper that optimizing wide ReLU neural networks (NNs) with at least one hidden layer using l2-regularization on the parameters enforces multi-task learning due to representation-learning - also in the limit of width to infinity. This is in contrast to multiple other results in the literature, in which idealized settings are assumed and where wide (ReLU)-NNs loose their ability to benefit from multi-task learning in the infinite width limit. We deduce the ability of multi-task learning from proving an exact quantitative macroscopic characterization of the learned NN in an appropriate function space.