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There is now a CONTENT FREEZE for Mercury while we switch to a new platform. It began on Friday, March 10 at 6pm and will end on Wednesday, March 15 at noon. No new content can be created during this time, but all material in the system as of the beginning of the freeze will be migrated to the new platform, including users and groups. Functionally the new site is identical to the old one. webteam@gatech.edu
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TITLE: Matrix Completion, Saddlepoints, and Gradient Descent
ABSTRACT:
Matrix completion is a fundamental machine learning problem with wide applications in collaborative filtering and recommender systems. Typically, matrix completion are solved by non-convex optimization procedures, which are empirically extremely successful. We prove that the symmetric matrix completion problem has no spurious local minima, meaning all local minima are also global. Thus the matrix completion objective has only saddlepoints an global minima. Next, we show that saddlepoints are easy to avoid for even Gradient Descent -- arguably the simplest optimization procedure. We prove that with probability 1, randomly initialized Gradient Descent converges to a local minimizer. The same result holds for a large class of optimization algorithms including proximal point, mirror descent, and coordinate descent.
BIO: Jason Lee is an assistant professor in Data Sciences and Operations at the University of Southern California. Prior to that, he was a postdoctoral researcher at UC Berkeley working with Michael Jordan. Jason received his PhD at Stanford University advised by Trevor Hastie and Jonathan Taylor. His research interests are in statistics, machine learning, and optimization. Lately, he has worked on high dimensional statistical inference, analysis of non-convex optimization algorithms, and theory for deep learning.
