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Primal Averaging: A New Gradient Evaluation Step to Attain the Optimal Individual Convergence
Tao, Wei1; Pan, Zhisong1; Wu, Gaowei2; Tao, Qing2,3
Corresponding AuthorTao, Qing(
AbstractMany well-known first-order gradient methods have been extended to cope with large-scale composite problems, which often arise as a regularized empirical risk minimization in machine learning. However, their optimal convergence is attained only in terms of the weighted average of past iterative solutions. How to make the individual convergence of stochastic gradient descent (SGD) optimal, especially for strongly convex problems has now become a challenging problem in the machine learning community. On the other hand, Nesterov's recent weighted averaging strategy succeeds in achieving the optimal individual convergence of dual averaging (DA) but it fails in the basic mirror descent (MD). In this paper, a new primal averaging (PA) gradient operation step is presented, in which the gradient evaluation is imposed on the weighted average of all past iterative solutions. We prove that simply modifying the gradient operation step in MD by PA strategy suffices to recover the optimal individual rate for general convex problems. Along this line, the optimal individual rate of convergence for strongly convex problems can also be achieved by imposing the strong convexity on the gradient operation step. Furthermore, we extend PA-MD to solve regularized nonsmooth learning problems in the stochastic setting, which reveals that PA strategy is a simple yet effective extra step toward the optimal individual convergence of SGD. Several real experiments on sparse learning and SVM problems verify the correctness of our theoretical analysis.
KeywordConvergence Convex functions Machine learning Optimization methods Linear programming Cybernetics Individual convergence machine learning mirror descent (MD) methods regularized learning problems stochastic gradient descent (SGD) stochastic optimization
Indexed BySCI
Funding ProjectNSFC[61673394] ; NSFC[61673394]
Funding OrganizationNSFC
WOS Research AreaAutomation & Control Systems ; Computer Science
WOS SubjectAutomation & Control Systems ; Computer Science, Artificial Intelligence ; Computer Science, Cybernetics
WOS IDWOS:000506849800036
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Cited Times:2[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Corresponding AuthorTao, Qing
Affiliation1.Army Engn Univ PLA, Command & Control Engn Coll, Nanjing 210007, Peoples R China
2.Chinese Acad Sci, Inst Automat, Beijing 100190, Peoples R China
3.Army Acad Artillery & Air Def, Dept Comp Sci, Hefei 230031, Peoples R China
Corresponding Author AffilicationInstitute of Automation, Chinese Academy of Sciences
Recommended Citation
GB/T 7714
Tao, Wei,Pan, Zhisong,Wu, Gaowei,et al. Primal Averaging: A New Gradient Evaluation Step to Attain the Optimal Individual Convergence[J]. IEEE TRANSACTIONS ON CYBERNETICS,2020,50(2):835-845.
APA Tao, Wei,Pan, Zhisong,Wu, Gaowei,&Tao, Qing.(2020).Primal Averaging: A New Gradient Evaluation Step to Attain the Optimal Individual Convergence.IEEE TRANSACTIONS ON CYBERNETICS,50(2),835-845.
MLA Tao, Wei,et al."Primal Averaging: A New Gradient Evaluation Step to Attain the Optimal Individual Convergence".IEEE TRANSACTIONS ON CYBERNETICS 50.2(2020):835-845.
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