{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/accelerating-sgd-for-highly-ill-conditioned","title":"Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completion","arxiv_id":"2208.11246","date":"2022-08-24","proceeding":null,"authors":["Gavin Zhang","Hong-Ming Chiu","Richard Y. Zhang"],"abstract":"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 $O(\\kappa\\log(1/\\epsilon))$ iterations to get $\\epsilon$-close to ground truth matrix with condition number $\\kappa$. 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 $\\kappa$. For a symmetric ground truth and the Root Mean Square Error (RMSE) loss, we prove that the preconditioned SGD converges to $\\epsilon$-accuracy in $O(\\log(1/\\epsilon))$ iterations, with a rapid linear convergence rate as if the ground truth were perfectly conditioned with $\\kappa=1$. 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.]","url_abs":"https://arxiv.org/abs/2208.11246v2","url_pdf":"https://arxiv.org/pdf/2208.11246v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"accelerating-sgd-for-highly-ill-conditioned","repo_url":"https://github.com/hong-ming/scaledsgd","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"collaborative-filtering","task_name":"Collaborative Filtering"},{"task_slug":"matrix-completion","task_name":"Matrix Completion"}],"methods":[{"method_slug":"sgd","method_name":"SGD"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2208.11246","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}