#### ML/Statistics Seminar Series

**Date/Time**: Wed. Sep 6 2017, 3:00 pm – 4:00 pm

**Location**: Advisory Boardroom, #402 Groseclose

**Speaker**: Jason Lee

**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 saddle points and 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 of 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 Ph.D. 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.