  We consider the fundamental problem of ReLU regression, where the goal is to
output the best fitting ReLU with respect to square loss given access to draws
from some unknown distribution. We give the first efficient, constant-factor
approximation algorithm for this problem assuming the underlying distribution
satisfies some weak concentration and anti-concentration conditions (and
includes, for example, all log-concave distributions). This solves the main
open problem of Goel et al., who proved hardness results for any exact
algorithm for ReLU regression (up to an additive $\epsilon$). Using more
sophisticated techniques, we can improve our results and obtain a
polynomial-time approximation scheme for any subgaussian distribution. Given
the aforementioned hardness results, these guarantees can not be substantially
improved.
  Our main insight is a new characterization of surrogate losses for nonconvex
activations. While prior work had established the existence of convex
surrogates for monotone activations, we show that properties of the underlying
distribution actually induce strong convexity for the loss, allowing us to
relate the global minimum to the activation's Chow parameters.
