Gavin Zhang

Gavin Zhang, Hong-Ming Chiu, Richard Y. Zhang

The matrix completion problem seeks to recover a $d\times d$ ground truth matrix of low rank $r\ll d$ from observations of its individual elements. Real-world matrix completion is often a huge-scale optimization problem, with $d$ so large that even the simplest full-dimension vector operations with $O(d)$ 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 $O(κ\log(1/ε))$ iterations to get $ε$-close to ground truth matrix with condition number $κ$. 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 $κ$. For a symmetric ground truth and the Root Mean Square Error (RMSE) loss, we prove that the preconditioned SGD converges to $ε$-accuracy in $O(\log(1/ε))$ iterations, with a rapid linear convergence rate as if the ground truth were perfectly conditioned with $κ=1$. 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.]

Gavin Zhang, Hong-Ming Chiu, Richard Y. Zhang

Non-convex gradient descent is a common approach for estimating a low-rank $n\times n$ ground truth matrix from noisy measurements, because it has per-iteration costs as low as $O(n)$ time, and is in theory capable of converging to a minimax optimal estimate. However, the practitioner is often constrained to just tens to hundreds of iterations, and the slow and/or inconsistent convergence of non-convex gradient descent can prevent a high-quality estimate from being obtained. Recently, the technique of preconditioning was shown to be highly effective at accelerating the local convergence of non-convex gradient descent when the measurements are noiseless. In this paper, we describe how preconditioning should be done for noisy measurements to accelerate local convergence to minimax optimality. For the symmetric matrix sensing problem, our proposed preconditioned method is guaranteed to locally converge to minimax error at a linear rate that is immune to ill-conditioning and/or over-parameterization. Using our proposed preconditioned method, we perform a 60 megapixel medical image denoising task, and observe significantly reduced noise levels compared to previous approaches.