  We develop fast algorithms and robust software for convex optimization of
two-layer neural networks with ReLU activation functions. Our work leverages a
convex reformulation of the standard weight-decay penalized training problem as
a set of group-$\ell_1$-regularized data-local models, where locality is
enforced by polyhedral cone constraints. In the special case of
zero-regularization, we show that this problem is exactly equivalent to
unconstrained optimization of a convex "gated ReLU" network. For problems with
non-zero regularization, we show that convex gated ReLU models obtain
data-dependent approximation bounds for the ReLU training problem. To optimize
the convex reformulations, we develop an accelerated proximal gradient method
and a practical augmented Lagrangian solver. We show that these approaches are
faster than standard training heuristics for the non-convex problem, such as
SGD, and outperform commercial interior-point solvers. Experimentally, we
verify our theoretical results, explore the group-$\ell_1$ regularization path,
and scale convex optimization for neural networks to image classification on
MNIST and CIFAR-10.
