Parallel stochastic gradient algorithms for large-scale matrix completion

This paper develops Jellyfish , an algorithm for solving data-processing problems with matrix-valued decision variables regularized to have low rank. Particular examples of problems solvable by Jellyfish include matrix completion problems and least-squares problems regularized by the nuclear norm or γ _2 -norm. Jellyfish implements a projected incremental gradient method with a biased, random ordering of the increments. This biased ordering allows for a parallel implementation that admits a speed-up nearly proportional to the number of processors. On large-scale matrix completion tasks, Jellyfish is orders of magnitude more efficient than existing codes. For example, on the Netflix Prize data set, prior art computes rating predictions in approximately 4 h, while Jellyfish solves the same problem in under 3 min on a 12 core workstation.