Stochastic linearized generalized alternating direction method of multipliers: Expected convergence rates and large deviation properties

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-03-14 DOI:10.1017/s096012952300004x
Jia Hu, T. Guo, Congying Han
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引用次数: 2

Abstract

Alternating direction method of multipliers (ADMM) receives much attention in the field of optimization and computer science, etc. The generalized ADMM (G-ADMM) proposed by Eckstein and Bertsekas incorporates an acceleration factor and is more efficient than the original ADMM. However, G-ADMM is not applicable in some models where the objective function value (or its gradient) is computationally costly or even impossible to compute. In this paper, we consider the two-block separable convex optimization problem with linear constraints, where only noisy estimations of the gradient of the objective function are accessible. Under this setting, we propose a stochastic linearized generalized ADMM (called SLG-ADMM) where two subproblems are approximated by some linearization strategies. And in theory, we analyze the expected convergence rates and large deviation properties of SLG-ADMM. In particular, we show that the worst-case expected convergence rates of SLG-ADMM are $\mathcal{O}\left( {{N}^{-1/2}}\right)$ and $\mathcal{O}\left({\ln N} \cdot {N}^{-1}\right)$ for solving general convex and strongly convex problems, respectively, where N is the iteration number, similarly hereinafter, and with high probability, SLG-ADMM has $\mathcal{O}\left ( \ln N \cdot N^{-1/2} \right ) $ and $\mathcal{O}\left ( \left ( \ln N \right )^{2} \cdot N^{-1} \right ) $ constraint violation bounds and objective error bounds for general convex and strongly convex problems, respectively.
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乘法器的随机线性化广义交替方向法:预期收敛速率和大偏差特性
乘法器的交替方向法(ADMM)在优化和计算机科学等领域受到了广泛的关注。Eckstein和Bertsekas提出的广义ADMM(G-ADMM)包含了加速因子,比原来的ADMM更有效。然而,G-ADMM不适用于目标函数值(或其梯度)计算成本高甚至不可能计算的一些模型。在本文中,我们考虑具有线性约束的两块可分离凸优化问题,其中只有目标函数梯度的噪声估计是可访问的。在这种情况下,我们提出了一种随机线性化的广义ADMM(称为SLG-ADMM),其中两个子问题通过一些线性化策略近似。在理论上,我们分析了SLG-ADMM的预期收敛速度和大偏差特性。特别地,我们证明了SLG-ADMM在求解一般凸和强凸问题时,最坏情况下的预期收敛速度分别为$\mathcal{O}\left({{N}^{-1/2}}\right)$和$\mathcal{O}\left,SLG-ADMM对于一般凸和强凸问题分别具有$\mathcal{O}\left(\ln N\cdot N^{-1/2}\right)$和$\mathical{O}\left(\left(\ ln N\right)^{2}\cdot N ^{-1}\right)$约束违反界和目标误差界。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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