On the Convergence of Stochastic Gradient Descent with Low-Rank Projections for Convex Low-Rank Matrix Problems

We revisit the use of Stochastic Gradient Descent (SGD) for solving convex optimization problems that serve as highly popular convex relaxations for many important low-rank matrix recovery problems such as \textit{matrix completion}, \textit{phase retrieval}, and more. The computational limitation of applying SGD to solving these relaxations in large-scale is the need to compute a potentially high-rank singular value decomposition (SVD) on each iteration in order to enforce the low-rank-promoting constraint. We begin by considering a simple and natural sufficient condition so that these relaxations indeed admit low-rank solutions. This condition is also necessary for a certain notion of low-rank-robustness to hold. Our main result shows that under this condition which involves the eigenvalues of the gradient vector at optimal points, SGD with mini-batches, when initialized with a "warm-start" point, produces iterates that are low-rank with high probability, and hence only a low-rank SVD computation is required on each iteration. This suggests that SGD may indeed be practically applicable to solving large-scale convex relaxations of low-rank matrix recovery problems. Our theoretical results are accompanied with supporting preliminary empirical evidence. As a side benefit, our analysis is quite simple and short.