The Effects of Multi-Task Learning on ReLU Neural Network Functions
This provides theoretical insights into multi-task learning for researchers in neural network theory, though it is incremental as it builds on known single-task results.
The paper tackled the problem of non-uniqueness in solutions to multi-task shallow ReLU neural network learning problems by proving that solutions are almost always unique and equivalent to a minimum-norm interpolation in a Sobolev Hilbert space, revealing a novel connection to kernel methods.
This paper studies the properties of solutions to multi-task shallow ReLU neural network learning problems, wherein the network is trained to fit a dataset with minimal sum of squared weights. Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods. It is known that single-task neural network learning problems are equivalent to a minimum norm interpolation problem in a non-Hilbertian Banach space, and that the solutions of such problems are generally non-unique. In contrast, we prove that the solutions to univariate-input, multi-task neural network interpolation problems are almost always unique, and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space. We also demonstrate a similar phenomenon in the multivariate-input case; specifically, we show that neural network learning problems with large numbers of tasks are approximately equivalent to an $\ell^2$ (Hilbert space) minimization problem over a fixed kernel determined by the optimal neurons.