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The Frank-Wolfe method and its extensions are well-suited for deliveringsolutions with desirable structural properties, such as sparsity or low-rankstructure. We introduce a new variant of the Frank-Wolfe method that combinesFrank-Wolfe steps and steepest descent steps, as well as a novel modificationof the "Frank-Wolfe gap" to measure convergence in the non-convex case. Wefurther extend this method to incorporate in-face directions for preservingstructured solutions as well as block coordinate steps, and we demonstratecomputational guarantees in terms of the modified Frank-Wolfe gap for all ofthese variants. We are particularly motivated by the application of thismethodology to the training of neural networks with sparse properties, and weapply our block coordinate method to the problem of $\ell_1$ regularized neuralnetwork training. We present the results of several numerical experiments onboth artificial and real datasets demonstrating significant improvements of ourmethod in training sparse neural networks.

Stochastic In-Face Frank-Wolfe Methods for Non-Convex Optimization and Sparse Neural Network Training