New analysis of linear convergence of gradient-type methods via unifying error bound conditions

  • Hui ZhangEmail author
Full Length Paper Series A


This paper reveals that a common and central role, played in many error bound (EB) conditions and a variety of gradient-type methods, is a residual measure operator. On one hand, by linking this operator with other optimality measures, we define a group of abstract EB conditions, and then analyze the interplay between them; on the other hand, by using this operator as an ascent direction, we propose an abstract gradient-type method, and then derive EB conditions that are necessary and sufficient for its linear convergence. The former provides a unified framework that not only allows us to find new connections between many existing EB conditions, but also paves a way to construct new ones. The latter allows us to claim the weakest conditions guaranteeing linear convergence for a number of fundamental algorithms, including the gradient method, the proximal point algorithm, and the forward–backward splitting algorithm. In addition, we show linear convergence for the proximal alternating linearized minimization algorithm under a group of equivalent EB conditions, which are strictly weaker than the traditional strongly convex condition. Moreover, by defining a new EB condition, we show Q-linear convergence of Nesterov’s accelerated forward–backward algorithm without strong convexity. Finally, we verify EB conditions for a class of dual objective functions.


Residual measure operator Error bound Gradient descent Linear convergence Proximal point algorithm Forward–backward splitting algorithm Proximal alternating linearized minimization Nesterov’s acceleration Dual objective function 

Mathematics Subject Classification

90C25 90C60 65K10 49M29 



I am grateful to the anonymous referees, the associate editor, and the coeditor Prof. Adrian S. Lewis for many useful comments, which allowed me to significantly improve the original presentation. I would like to thank Prof. Zaiwen Wen for his invitation and hospitality during my visit to BeiJing International Center for Mathematical Research, and to thank Prof. Dmitriy Drusvyatskiy for a careful reading of an early draft of this manuscript, and for valuable comments and suggestions. I also thank Profs. Chao Ding, Bin Dong, Lei Guo, Yongjin Liu, Deren Han, Mark Schmidt, Anthony Man-Cho So, and Wotao Yin for their time and many helpful discussions with me. Further thanks due to my cousin Boya Ouyang who helped me with my English writing, and to PhD students Ke Guo, Wei Peng, Ziyang Yuan, Xiaoya Zhang, who looked over the manuscript and corrected several typos. While visiting Chinese Academy of Sciences, I was particularly fortunate to be acquainted with Prof. Florian Jarre, who carefully read and polished this paper. This work is supported by the National Science Foundation of China (Nos. 11501569 and 61571008).


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© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational University of Defense TechnologyChangshaPeople’s Republic of China

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