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# Global Convergence of Stochastic Gradient Descent for Some Non-convex Matrix Problems

*Proceedings of the 32nd International Conference on Machine Learning*, PMLR 37:2332-2341, 2015.

#### Abstract

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(ε^-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 some experiments to illustrate the runtime and convergence of the algorithm.