Global Convergence of Stochastic Gradient Descent for Some Non-convex   Matrix Problems
  Stochastic gradient descent (SGD) on a low-rank factorization is commonly employed to speed up matrix problems including matrix completion, subspace tracking, and SDP relaxation. In this paper, we exhibit a step size scheme for SGD on a low-rank least-squares problem, and we prove that, under broad sampling conditions, our method converges globally from a random starting point within $O(\epsilon^{-1} n \log n)$ steps with constant probability for constant-rank problems. Our modification of SGD relates it to stochastic power iteration. We also show experiments to illustrate the runtime and convergence of the algorithm. 
Christopher De Sa, Kunle Olukotun, and Christopher R\'e
cs.LG math.OC stat.ML