  In this paper we focus on the problem of finding the optimal weights of the
shallowest of neural networks consisting of a single Rectified Linear Unit
(ReLU). These functions are of the form $\mathbf{x}\rightarrow
\max(0,\langle\mathbf{w},\mathbf{x}\rangle)$ with $\mathbf{w}\in\mathbb{R}^d$
denoting the weight vector. We focus on a planted model where the inputs are
chosen i.i.d. from a Gaussian distribution and the labels are generated
according to a planted weight vector. We first show that mini-batch stochastic
gradient descent when suitably initialized, converges at a geometric rate to
the planted model with a number of samples that is optimal up to numerical
constants. Next we focus on a parallel implementation where in each iteration
the mini-batch gradient is calculated in a distributed manner across multiple
processors and then broadcast to a master or all other processors. To reduce
the communication cost in this setting we utilize a Quanitzed Stochastic
Gradient Scheme (QSGD) where the partial gradients are quantized. Perhaps
unexpectedly, we show that QSGD maintains the fast convergence of SGD to a
globally optimal model while significantly reducing the communication cost. We
further corroborate our numerical findings via various experiments including
distributed implementations over Amazon EC2.
