On the Asymptotic Convergence and Acceleration of Gradient Methods

doi:10.1007/s10915-021-01685-8

	On the Asymptotic Convergence and Acceleration of Gradient Methods
	Huang, Yakui 1; Dai, Yu-Hong 3,4; Liu, Xin-Wei 1; Zhang, Hongchao 2
刊名	JOURNAL OF SCIENTIFIC COMPUTING
	2022
卷号	90 期号:1 页码:29
关键词	Gradient methods Asymptotic convergence Spectral property Acceleration of gradient methods Barzilai-Borwein method Unconstrained optimization Quadratic optimization
ISSN号	0885-7474
DOI	10.1007/s10915-021-01685-8
英文摘要	We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate the family of gradient methods, we further exploit spectral properties of stepsizes to break the zigzagging pattern. In particular, a new stepsize is derived by imposing finite termination on minimizing two-dimensional strictly convex quadratic function. It is shown that, for the general quadratic function, the proposed stepsize asymptotically converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, based on this spectral property, we propose a periodic gradient method by incorporating the Barzilai-Borwein method. Numerical comparisons with some recent successful gradient methods show that our new method is very promising.
资助项目	National Natural Science Foundation of China[11701137] ; National Natural Science Foundation of China[11631013] ; National Natural Science Foundation of China[12071108] ; National Natural Science Foundation of China[11671116] ; National Natural Science Foundation of China[11991021] ; National Natural Science Foundation of China[12021001] ; Strategic Priority Research Program of Chinese Academy of Sciences[XDA27000000] ; Beijing Academy of Artificial Intelligence (BAAI) ; Natural Science Foundation of Hebei Province[A2021202010] ; China Scholarship Council[201806705007] ; USA National Science Foundation[DMS-1819161] ; USA National Science Foundation[DMS-2110722]
WOS研究方向	Mathematics
语种	英语
出版者	SPRINGER/PLENUM PUBLISHERS
WOS记录号	WOS:000720653400002
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/59571]
专题	中国科学院数学与系统科学研究院
通讯作者	Zhang, Hongchao
作者单位	1.Hebei Univ Technol, Inst Math, Tianjin 300401, Peoples R China 2.Louisiana State Univ, Dept Math, Baton Rouge, LA 70803 USA 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China 4.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Huang, Yakui,Dai, Yu-Hong,Liu, Xin-Wei,et al. On the Asymptotic Convergence and Acceleration of Gradient Methods[J]. JOURNAL OF SCIENTIFIC COMPUTING,2022,90(1):29.
APA	Huang, Yakui,Dai, Yu-Hong,Liu, Xin-Wei,&Zhang, Hongchao.(2022).On the Asymptotic Convergence and Acceleration of Gradient Methods.JOURNAL OF SCIENTIFIC COMPUTING,90(1),29.
MLA	Huang, Yakui,et al."On the Asymptotic Convergence and Acceleration of Gradient Methods".JOURNAL OF SCIENTIFIC COMPUTING 90.1(2022):29.