We introduce a DNN training technique that learns only a fraction of the full parameter set without incurring an accuracy penalty. To do this, our algorithm constrains the total number of weights updated during backpropagation to those with the highest total gradients. The remaining weights are not tracked, and their initial value is regenerated at every access to avoid storing them in memory. This can dramatically reduce the number of off-chip memory accesses during both training and inference, a key component of the energy needs of DNN accelerators. By ensuring that the total weight diffusion remains close to that of baseline unpruned SGD, networks pruned using our technique are able to retain state-of-the-art accuracy across network architectures -- including networks previously identified as difficult to compress, such as Densenet and WRN. With ResNet18 on ImageNet, we observe an 11.7$\times$ weight reduction with no accuracy loss, and up to 24.4$\times$ with a small accuracy impact.