Time/Accuracy Tradeoffs for Learning a ReLU with respect to Gaussian Marginals

We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let $\mathsf{o}\mathsf{p}\mathsf{t}<1$ be the population loss of the best-fitting ReLU. We prove:

1. Finding a ReLU with square-loss $\mathsf{o}\mathsf{p}\mathsf{t}+ϵ$ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph unconditionally}- that gradient descent cannot converge to the global minimum in polynomial time.
2. There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error $O({\mathsf{o}\mathsf{p}\mathsf{t}}^{2\mathrm{/}3})$. The algorithm uses a novel reduction to noisy halfspace learning with respect to $0\mathrm{/}1$ loss. Prior work due to Soltanolkotabi [Sol17] showed that gradient descent can find the best-fitting ReLU with respect to Gaussian marginals, if the training set is exactly labeled by a ReLU.