文献类型：期刊文献

英文题名：Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions

作者：Zhang, Hui[1];Dai, Yu-Hong[2];Guo, Lei[3];Peng, Wei[1]

机构：[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

年份：2021

卷号：46

期号：1

起止页码：61

外文期刊名：MATHEMATICS OF OPERATIONS RESEARCH

收录：;EI(收录号：20210909984020);WOS:【SCI-EXPANDED(收录号:WOS:000615980400003)】；

基金：This work was supported by the National Science Foundation of China [Grants 61601488, 11401379, 11631013, 11971480, 11826204, 11771287, and 71632007], the Key Project of the Chinese National Programs for Fundamental Research and Development [Grant 2015CB856002], and the Fundamental Research Funds for the Central Universities.

语种：英文

外文关键词：incremental aggregated gradient; linear convergence; Lipschitz-like/convexity; relative smoothness; Bregman distance growth

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.