It has been experimentally observed in recent years that multi-layer artificial neural networks have a surprising ability to generalize, even when trained with far more parameters than observations. Is there a theoretical basis for this? The best available bounds on their metric entropy and associated complexity measures are essentially linear in the number of parameters, which is inadequate to explain this phenomenon. Here we examine the statistical risk (mean squared predictive error) of multi-layer networks with L1 controls on their parameters  and with ramp activation functions (also called lower-rectified linear units). In this setting, the risk is shown to be upper-bounded  by [(L^3 log d)/n]^{1/2}, where d is the input dimension to each  layer, L is the number of layers, and n is the sample size. In this  way, the input dimension can be much larger than the sample size and the estimator can still be accurate, provided the target function has such L1 controls and that the sample size is at least moderately large compared to L^3 log d. The heart of the analysis is the development of a sampling strategy that demonstrates the accuracy of a sparse covering of deep ramp networks. Lower bounds show that the identified risk is minimax optimal, this being so already in the subclass of functions with L = 2.  This is  joint work with Jason Klusowski.