Convergence guarantees for a class of non-convex and non-smooth optimization problems

Koulik Khamaru; Martin J. Wainwright

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arXiv · 1804.09629

Convergence guarantees for a class of non-convex and non-smooth optimization problems

Koulik Khamaru·Martin J. Wainwright

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Abstract

We consider the problem of finding critical points of functions that are non-convex and non-smooth. Studying a fairly broad class of such problems, we analyze the behavior of three gradient-based methods (gradient descent, proximal update, and Frank-Wolfe update). For each of these methods, we establish rates of convergence for general problems, and also prove faster rates for continuous sub-analytic functions. We also show that our algorithms can escape strict saddle points for a class of non-smooth functions, thereby generalizing known results for smooth functions. Our analysis leads to a simplification of the popular CCCP algorithm, used for optimizing functions that can be written as a difference of two convex functions. Our simplified algorithm retains all the convergence properties of CCCP, along with a significantly lower cost per iteration. We illustrate our methods and theory via applications to the problems of best subset selection, robust estimation, mixture density estimation, and shape-from-shading reconstruction.

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BibTeX RIS

Koulik Khamaru, Martin J. Wainwright. 2018-04-25. Convergence guarantees for a class of non-convex and non-smooth optimization problems. https://arxiv.org/abs/1804.09629

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