Sparsity by Redundancy: Solving $L_1$ with SGD

We propose to minimize a generic differentiable loss function with $L_1$ penalty with a redundant reparametrization and straightforward stochastic gradient descent. Our proposal is the direct generalization of a series of previous ideas that the $L_1$ penalty may be equivalent to a differentiable reparametrization with weight decay. We prove that the proposed method, \textit{spred}, is an exact solver of $L_1$ 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 $L_1$-penalty have been unsuccessful. Conceptually, our result bridges the gap between the sparsity in deep learning and conventional statistical learning.