How Infinitely Wide Neural Networks Benefit from Multi-task Learning -- an Exact Macroscopic Characterization

In practice, multi-task learning (through learning features shared among tasks) is an essential property of deep neural networks (NNs). While infinite-width limits of NNs can provide a good intuition for their generalization behavior, the well-known infinite-width limits of NNs in the literature (e.g., neural tangent kernels) assume specific settings in which wide ReLU-NNs behave like shallow Gaussian Processes with a fixed kernel. Consequently, in such settings, these NNs lose their ability to benefit from multi-task learning in the infinite-width limit. In contrast, we prove that optimizing wide ReLU neural networks with at least one hidden layer using L2-regularization on the parameters enforces multi-task learning due to representation-learning - also in the limiting regime where the network width tends to infinity. We present an exact quantitative characterization of this infinite width limit in an appropriate function space that neatly describes multi-task learning.