In this paper, we consider the problem of training neural networks (NN). To promote a NN with specific structures, we explicitly take into consideration the nonsmooth regularization (such as L1-norm) and constraints (such as interval constraint). This is formulated as a constrained nonsmooth nonconvex optimization problem, and we propose a convergent proximal-type stochastic gradient descent (Prox-SGD) algorithm. We show that under properly selected learning rates, momentum eventually resembles the unknown real gradient and thus is crucial in analyzing the convergence. We establish that with probability 1, every limit point of the sequence generated by the proposed Prox-SGD is a stationary point. Then the Prox-SGD is tailored to train a sparse neural network and a binary neural network, and the theoretical analysis is also supported by extensive numerical tests.
We propose a convergent proximal-type stochastic gradient descent algorithm for constrained nonsmooth nonconvex optimization problems This paper proposes Prox-SGD, a theoretical framework for stochastic optimization algorithms shown to converge asymptotically to stationarity for smooth non-convvex loss + convex constraint/regularizer. The paper proposes a new gradient-based stochastic optimization algorithm with gradient averaging by adapting theory for proximal algorithms to the non-convex setting.