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 $\opt < 1$ be the population loss of the best-fitting ReLU. We prove: \begin{itemize} \item Finding a ReLU with square-loss $\opt + \epsilon$ 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 --{\em unconditionally}-- that gradient descent cannot converge to the global minimum in polynomial time. \item There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error $O(\opt^{2/3})$. The algorithm uses a novel reduction to noisy halfspace learning with respect to $0/1$ loss. \end{itemize} Prior work due to Soltanolkotabi \cite{soltanolkotabi2017learning} showed that gradient descent {\em can} find the best-fitting ReLU with respect to Gaussian marginals, if the training set is {\em exactly} labeled by a ReLU.