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Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completion

Abstract

The matrix completion problem seeks to recover a d×dd\times d ground truth matrix of low rank rdr\ll d from observations of its individual elements. Real-world matrix completion is often a huge-scale optimization problem, with dd so large that even the simplest full-dimension vector operations with O(d)O(d) time complexity become prohibitively expensive. Stochastic gradient descent (SGD) is one of the few algorithms capable of solving matrix completion on a huge scale, and can also naturally handle streaming data over an evolving ground truth. Unfortunately, SGD experiences a dramatic slow-down when the underlying ground truth is ill-conditioned; it requires at least O(κlog(1/ϵ))O(\kappa\log(1/\epsilon)) iterations to get ϵ\epsilon-close to ground truth matrix with condition number κ\kappa. In this paper, we propose a preconditioned version of SGD that preserves all the favorable practical qualities of SGD for huge-scale online optimization while also making it agnostic to κ\kappa. For a symmetric ground truth and the Root Mean Square Error (RMSE) loss, we prove that the preconditioned SGD converges to ϵ\epsilon-accuracy in O(log(1/ϵ))O(\log(1/\epsilon)) iterations, with a rapid linear convergence rate as if the ground truth were perfectly conditioned with κ=1\kappa=1. In our experiments, we observe a similar acceleration for item-item collaborative filtering on the MovieLens25M dataset via a pair-wise ranking loss, with 100 million training pairs and 10 million testing pairs. [See supporting code at https://github.com/Hong-Ming/ScaledSGD.]

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