Randomness and permutations in coordinate descent methods

• 成果类型: Article
• 署名作者: Gurbuzbalaban, Mert; Ozdaglar, Asuman; Vanli, Nuri Denizcan; Wright, Stephen J.
• 署名单位: Rutgers University System; Rutgers University New Brunswick; Massachusetts Institute of Technology (MIT); University of Wisconsin System; University of Wisconsin Madison; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-019-01438-4
• 发表日期: 2020
• 页码: 349-376
• 关键词: Optimization; CONVERGENCE; EFFICIENCY

摘要

We consider coordinate descent (CD) methods with exact line search on convex quadratic problems. Our main focus is to study the performance of the CD method that use random permutations in each epoch and compare it to the performance of the CD methods that use deterministic orders and random sampling with replacement. We focus on a class of convex quadratic problems with a diagonally dominant Hessian matrix, for which we show that using random permutations instead of random with-replacement sampling improves the performance of the CD method in the worst-case. Furthermore, we prove that as the Hessian matrix becomes more diagonally dominant, the performance improvement attained by using random permutations increases. We also show that for this problem class, using any fixed deterministic order yields a superior performance than using random permutations. We present detailed theoretical analyses with respect to three different convergence criteria that are used in the literature and support our theoretical results with numerical experiments.