Sashank J. Reddi

Stochastic Frank-Wolfe methods for nonconvex optimization

We study Frank-Wolfe methods for nonconvex stochastic and finite-sum optimization problems. Frank-Wolfe methods (in the convex case) have gained tremendous recent interest in machine learning and optimization due to their projection-free property and their ability to exploit structured constraints. However, our understanding of these algorithms in the nonconvex setting is fairly limited. In this paper, we propose nonconvex stochastic Frank-Wolfe methods and analyze their convergence properties. Furthermore, for objective functions that decompose into a finite-sum, we leverage ideas from variance reduction for convex optimization to obtain new variance reduced nonconvex Frank-Wolfe methods that have provably faster convergence than the classical Frank-Wolfe method.

Fast incremental method for smooth nonconvex optimization

We analyze a fast incremental aggregated gradient method for optimizing nonconvex problems of the form minΣifi(x). Specifically, we analyze the SAGA algorithm within an Incremental First-order Oracle framework, and show that it converges to a stationary point provably faster than both gradient descent and stochastic gradient descent. We also discuss a Polyaks special class of nonconvex problems for which SAGA converges at a linear rate to the global optimum. Finally, we analyze the practically valuable regularized and minibatch variants of SAGA. To our knowledge, this paper presents the first analysis of fast convergence for an incremental aggregated gradient method for nonconvex problems.