Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions

doi:10.1287/moor.2019.1047

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	Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions
	Zhang, Hui 1; Dai, Yu-Hong 2; Guo, Lei 3; Peng, Wei 1
	2021-02-01
发表期刊	MATHEMATICS OF OPERATIONS RESEARCH
ISSN	0364-765X
卷号	46 期号:1 页码:61-81
摘要	We introduce a unified algorithmic framework, called the proximal-like incremental aggregated gradient (PLIAG) method, for minimizing the sum of a convex function that consists of additive relatively smooth convex components and a proper lower semicontinuous convex regularization function over an abstract feasible set whose geometry can be captured by using the domain of a Legendre function. The PLIAG method includes many existing algorithms in the literature as special cases, such as the proximal gradient method, the Bregman proximal gradient method (also called the NoLips algorithm), the incremental aggregated gradient method, the incremental aggregated proximal method, and the proximal incremental aggregated gradient method. It also includes some novel interesting iteration schemes. First, we show that the PLIAG method is globally sublinearly convergent without requiring a growth condition, which extends the sublinear convergence result for the proximal gradient algorithm to incremental aggregated-type first-order methods. Then, by embedding a so-called Bregman distance growth condition into a descent-type lemma to construct a special Lyapunov function, we show that the PLIAG method is globally linearly convergent in terms of both function values and Bregman distances to the optimal solution set, provided that the step size is not greater than some positive constant. The convergence results derived in this paper are all established beyond the standard assumptions in the literature (i.e., without requiring the strong convexity and the Lipschitz gradient continuity of the smooth part of the objective). When specialized to many existing algorithms, our results recover or supplement their convergence results under strictly weaker conditions.
关键词	incremental aggregated gradient linear convergence Lipschitz-like/convexity relative smoothness Bregman distance growth
DOI	10.1287/moor.2019.1047
收录类别	SCI
语种	英语
资助项目	National Science Foundation of China[61601488] ; National Science Foundation of China[11401379] ; National Science Foundation of China[11631013] ; National Science Foundation of China[11971480] ; National Science Foundation of China[11826204] ; National Science Foundation of China[11771287] ; National Science Foundation of China[71632007] ; Key Project of the Chinese National Programs for Fundamental Research and Development[2015CB856002] ; Fundamental Research Funds for the Central Universities
WOS研究方向	Operations Research & Management Science ; Mathematics
WOS类目	Operations Research & Management Science ; Mathematics, Applied
WOS记录号	WOS:000615980400003
出版者	INFORMS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/58228
专题	中国科学院数学与系统科学研究院
通讯作者	Guo, Lei
作者单位	1.Natl Univ Def Technol, Dept Math, Changsha 410073, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, State Key Lab Sci & Engn Comp, Beijing 100190, Peoples R China 3.East China Univ Sci & Technol, Sch Business, Shanghai 200237, Peoples R China
推荐引用方式 GB/T 7714	Zhang, Hui,Dai, Yu-Hong,Guo, Lei,et al. Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions[J]. MATHEMATICS OF OPERATIONS RESEARCH,2021,46(1):61-81.
APA	Zhang, Hui,Dai, Yu-Hong,Guo, Lei,&Peng, Wei.(2021).Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions.MATHEMATICS OF OPERATIONS RESEARCH,46(1),61-81.
MLA	Zhang, Hui,et al."Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions".MATHEMATICS OF OPERATIONS RESEARCH 46.1(2021):61-81.