### Scaled stochastic gradient descent for low-rank matrix completion

#### Authors

B. Mishra and R. Sepulchre

#### Abstract

The paper looks at a scaled variant of the stochastic gradient descent algorithm for the matrix completion problem. Specifically, we propose a novel matrix-scaling of the partial derivatives that acts as an efficient preconditioning for the standard stochastic gradient descent algorithm. This proposed matrix-scaling provides a trade-off between local and global second order information. It also resolves the issue of scale invariance that exist in matrix factorization models. The overall computational complexity is linear with the number of known entries, thereby extending to a large-scale setup. Numerical comparisons show that the proposed algorithm competes favorably with state-of-the-art algorithms on various different benchmarks.

#### Downloads

**Status:**Accepted to the 55th IEEE Conference on Decision and Control. A different version is accepted to the*internal*Amazon machine learning conference (AMLC) 2016. AMLC is a platform for internal Amazon researchers to present their work.**Paper:**[Publisher’s copy] [arXiv:1603.04989].**Matlab code:**ScaledSGD_17March2016.zip.**Entry:**March 17, 2016: The code is online.

**Note:**The Matlab codes are not tuned to large-scale problem instances, mostly because of for loops. Nevertheless, they show the behavior of algorithms against the number of iterations or epochs. A discussion on the precise computational complexity of the algorithm is in the paper.