ProxSGD: Training Structured Neural Networks under Regularization and Constraints

In this paper, we consider the problem of training structured neural networks (NN) with nonsmooth regularization (e.g. `1-norm) and constraints (e.g. interval constraints). We formulate training as a constrained nonsmooth nonconvex optimization problem, and propose a convergent proximal-type stochastic gradient descent (ProxSGD) algorithm. We show that under properly selected learning rates, with probability 1, every limit point of the sequence generated by the proposed Prox-SGD algorithm is a stationary point. Finally, to support the theoretical analysis and demonstrate the flexibility of ProxSGD, we show by extensive numerical tests how ProxSGD can be used to train either sparse or binary neural networks through an adequate selection of the regularization function and constraint set.