spred: Solving L1 Penalty with SGD

We propose to minimize a generic differentiable objective with L1 constraint using a simple reparametrization and straightforward stochastic gradient descent. Our proposal is the direct generalization of previous ideas that the L1 penalty may be equivalent to a differentiable reparametrization with weight decay. We prove that the proposed method, textitspred, is an exact differentiable solver of L1 and that the reparametrization trick is completely ``benign" for a generic nonconvex function. Practically, we demonstrate the usefulness of the method in (1) training sparse neural networks to perform gene selection tasks, which involves finding relevant features in a very high dimensional space, and (2) neural network compression task, to which previous attempts at applying the L1-penalty have been unsuccessful. Conceptually, our result bridges the gap between the sparsity in deep learning and conventional statistical learning.