Parallel Coordinate Descent Newton Method for Efficient L1-Regularized Loss Minimization

Yatao An Bian, Xiong Li, Yuncai Liu, Ming Hsuan Yang

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1 Citation (Scopus)

Abstract

The recent years have witnessed advances in parallel algorithms for large-scale optimization problems. Notwithstanding the demonstrated success, existing algorithms that parallelize over features are usually limited by divergence issues under high parallelism or require data preprocessing to alleviate these problems. In this paper, we propose a Parallel Coordinate Descent algorithm using approximate Newton steps (PCDN) that is guaranteed to converge globally without data preprocessing. The key component of the PCDN algorithm is the high-dimensional line search, which guarantees the global convergence with high parallelism. The PCDN algorithm randomly partitions the feature set into b subsets/bundles of size P, and sequentially processes each bundle by first computing the descent directions for each feature in parallel and then conducting P-dimensional line search to compute the step size. We show that: 1) the PCDN algorithm is guaranteed to converge globally despite increasing parallelism and 2) the PCDN algorithm converges to the specified accuracy ϵ within the limited iteration number of Tϵ, and Tϵ decreases with increasing parallelism. In addition, the data transfer and synchronization cost of the P-dimensional line search can be minimized by maintaining intermediate quantities. For concreteness, the proposed PCDN algorithm is applied to L1-regularized logistic regression and L1-regularized L2-loss support vector machine problems. Experimental evaluations on seven benchmark data sets show that the PCDN algorithm exploits parallelism well and outperforms the state-of-the-art methods.

Original languageEnglish
Article number8661743
Pages (from-to)3233-3245
Number of pages13
JournalIEEE Transactions on Neural Networks and Learning Systems
Volume30
Issue number11
DOIs
Publication statusPublished - 2019 Nov

Bibliographical note

Funding Information:
Manuscript received March 1, 2017; revised January 15, 2018; accepted problem. While this method performs well, no analysis of con-December16,2018.DateofpublicationMarch6,2019;dateofcurrentversion vergence rate is presented. In this paper, by further exploring PlanckETHCenterforLearningSystems.TheworkofX.LiwassupportedOctober29,2019.TheworkofY.A.BianwassupportedinpartbytheMax the idea, we propose a generalized Parallel Coordinate Descent by the NSFC Program under Grant U1636123. (Corresponding author: method using approximate Newton steps (PCDN) for generic Ming-HsuanYang.) L1-optimization problems and present thorough theoretical Zürich,Switzerland(e-mail:yatao.bian@gmail.com).Y.A.BianiswiththeDepartmentofComputerScience,ETHZurich,8092 analysis on the proposed method. X. Li is with the National Computer Network Emergency Response Tech-The contributions and novelty of this paper are summarized nicalTeam,Beijing100029,China(e-mail:li.xiong@foxmail.com). as follows. We present a theoretical analysis on the upper sity,Shanghai200000,China(e-mail:whomliu@sjtu.edu.cn).Y.LiuiswiththeDepartmentofAutomation,ShanghaiJiaoTongUniver- bound of the expected line search step in each iteration. We M.-H. Yang is with the School of Engineering, University of California at analyze the iteration complexity of the proposed PCDN algo-Merced,Merced,CA95344USA(e-mail:mhyang@ucmerced.edu). rithm and show that, for any bundle size P (i.e., parallelism), onlineathttp://ieeexplore.ieee.org.Colorversions ofoneormoreof the figures in this article are available it is guaranteed to converge to a specified accuracy ϵ within Digital Object Identifier 10.1109/TNNLS.2018.2889976 Tϵ iterations. The iteration number Tϵ decreases with the

All Science Journal Classification (ASJC) codes

  • Software
  • Computer Science Applications
  • Computer Networks and Communications
  • Artificial Intelligence

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