Extensions to the Proximal Distance Method of Constrained Optimization.

IF 4.3 3区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS
Journal of Machine Learning Research Pub Date : 2022-01-01
Alfonso Landeros, Oscar Hernan Madrid Padilla, Hua Zhou, Kenneth Lange
{"title":"Extensions to the Proximal Distance Method of Constrained Optimization.","authors":"Alfonso Landeros,&nbsp;Oscar Hernan Madrid Padilla,&nbsp;Hua Zhou,&nbsp;Kenneth Lange","doi":"","DOIUrl":null,"url":null,"abstract":"<p><p>The current paper studies the problem of minimizing a loss <i>f</i>(<b><i>x</i></b>) subject to constraints of the form <b><i>Dx</i></b> ∈ <i>S</i>, where <i>S</i> is a closed set, convex or not, and <i><b>D</b></i> is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method of optimization with the proximal distance principle. The latter is driven by minimization of penalized objectives <math><mrow><mi>f</mi><mo>(</mo><mstyle><mi>x</mi></mstyle><mo>)</mo><mo>+</mo><mfrac><mi>ρ</mi><mn>2</mn></mfrac><mtext>dist</mtext><msup><mrow><mo>(</mo><mstyle><mi>D</mi><mi>x</mi></mstyle><mo>,</mo><mi>S</mi><mo>)</mo></mrow><mn>2</mn></msup></mrow></math> involving large tuning constants <i>ρ</i> and the squared Euclidean distance of <b><i>Dx</i></b> from <i>S</i>. The next iterate <b><i>x</i></b><sub><i>n</i>+1</sub> of the corresponding proximal distance algorithm is constructed from the current iterate <b><i>x</i></b><sub><i>n</i></sub> by minimizing the majorizing surrogate function <math><mrow><mi>f</mi><mo>(</mo><mstyle><mi>x</mi></mstyle><mo>)</mo><mo>+</mo><mfrac><mi>ρ</mi><mn>2</mn></mfrac><msup><mrow><mrow><mo>‖</mo><mrow><mstyle><mi>D</mi><mi>x</mi></mstyle><mo>-</mo><msub><mi>𝒫</mi><mi>S</mi></msub><mrow><mo>(</mo><mrow><mstyle><mi>D</mi></mstyle><msub><mstyle><mi>x</mi></mstyle><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>‖</mo></mrow></mrow><mn>2</mn></msup></mrow></math>. For fixed <i>ρ</i> and a subanalytic loss <i>f</i>(<b><i>x</i></b>) and a subanalytic constraint set <i>S</i>, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare their results to those delivered by the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems. Julia code to replicate all of our experiments can be found at https://github.com/alanderos91/ProximalDistanceAlgorithms.jl.</p>","PeriodicalId":50161,"journal":{"name":"Journal of Machine Learning Research","volume":"23 ","pages":""},"PeriodicalIF":4.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10191389/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Machine Learning Research","FirstCategoryId":"94","ListUrlMain":"","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0

Abstract

The current paper studies the problem of minimizing a loss f(x) subject to constraints of the form DxS, where S is a closed set, convex or not, and D is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method of optimization with the proximal distance principle. The latter is driven by minimization of penalized objectives f(x)+ρ2dist(Dx,S)2 involving large tuning constants ρ and the squared Euclidean distance of Dx from S. The next iterate xn+1 of the corresponding proximal distance algorithm is constructed from the current iterate xn by minimizing the majorizing surrogate function f(x)+ρ2Dx-𝒫S(Dxn)2. For fixed ρ and a subanalytic loss f(x) and a subanalytic constraint set S, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare their results to those delivered by the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems. Julia code to replicate all of our experiments can be found at https://github.com/alanderos91/ProximalDistanceAlgorithms.jl.

约束优化近距离法的扩展。
本文研究了在形式为Dx∈S的约束下最小化损失f(x)的问题,其中S是一个闭集,可以是凸的也可以是非凸的,D是一个融合参数的矩阵。融合约束可以捕获平滑性、稀疏性或更一般的约束模式。为了解决这类问题,我们将优化的Beltrami-Courant惩罚方法与近距离原则结合起来。后者是由最小化惩罚目标f(x)+ρ2dist(Dx,S)2驱动的,涉及大的调谐常数ρ和Dx到S的平方欧几里德距离。相应的近距离算法的下一个迭代xn+1是通过最小化最大化代理函数f(x)+ρ2‖Dx-𝒫S(Dxn)‖2从当前迭代xn构建的。对于固定的ρ和一个亚解析损失f(x)和一个亚解析约束集S,我们证明了收敛到一个平稳点。在更强的假设下,我们给出了收敛率并证明了线性局部收敛。我们还构造了一个最陡下降(SD)变量,以避免昂贵的线性系统求解。为了对我们的算法进行基准测试,我们将它们的结果与乘法器交替方向方法(ADMM)提供的结果进行了比较。我们广泛的数值测试包括度量投影、凸回归、凸聚类、全变差图像去噪以及矩阵到良好条件数的投影问题。这些实验证明了我们最陡峭的变体在高维问题上的优越速度和可接受的精度。Julia复制我们所有实验的代码可以在https://github.com/alanderos91/ProximalDistanceAlgorithms.jl上找到。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Journal of Machine Learning Research
Journal of Machine Learning Research 工程技术-计算机:人工智能
CiteScore
18.80
自引率
0.00%
发文量
2
审稿时长
3 months
期刊介绍: The Journal of Machine Learning Research (JMLR) provides an international forum for the electronic and paper publication of high-quality scholarly articles in all areas of machine learning. All published papers are freely available online. JMLR has a commitment to rigorous yet rapid reviewing. JMLR seeks previously unpublished papers on machine learning that contain: new principled algorithms with sound empirical validation, and with justification of theoretical, psychological, or biological nature; experimental and/or theoretical studies yielding new insight into the design and behavior of learning in intelligent systems; accounts of applications of existing techniques that shed light on the strengths and weaknesses of the methods; formalization of new learning tasks (e.g., in the context of new applications) and of methods for assessing performance on those tasks; development of new analytical frameworks that advance theoretical studies of practical learning methods; computational models of data from natural learning systems at the behavioral or neural level; or extremely well-written surveys of existing work.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信