  We consider regression problems with binary weights. Such optimization
problems are ubiquitous in quantized learning models and digital communication
systems. A natural approach is to optimize the corresponding Lagrangian using
variants of the gradient ascent-descent method. Such maximin techniques are
still poorly understood even in the concave-convex case. The non-convex binary
constraints may lead to spurious local minima. Interestingly, we prove that
this approach is optimal in linear regression with low noise conditions as well
as robust regression with a small number of outliers. Practically, the method
also performs well in regression with cross entropy loss, as well as non-convex
multi-layer neural networks. Taken together our approach highlights the
potential of saddle-point optimization for learning constrained models.
