非凸复合函数的镜像凸/凹优化。

IF 4.3 3区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS Journal of Machine Learning Research Pub Date : 2016-01-01
Rina Foygel Barber, Emil Y Sidky
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引用次数: 0

摘要

高维统计中出现的许多优化问题自然分解为若干项的和,其中单个项相对简单,而复合目标函数只能通过迭代算法进行优化。本文研究F(Kx) + G(x)形式的最优化问题,其中K是一个固定的线性变换,而F和G是非凸和/或不可微的函数。特别是,如果任何一项都是非凸的,现有的交替最小化技术可能无法收敛;其他类型的现有方法可能无法处理不可微性。我们提出了MOCCA(镜像凸/凹)算法,这是一种原始/对偶优化方法,在每次迭代中对每个项进行局部凸逼近。受计算机断层扫描(CT)成像中的优化问题的启发,该算法可以处理一系列非凸复合优化问题,并为整体问题近似凸时的收敛性提供理论保证(即一项的任何凹性被另一项的凸性抵消)。实验结果表明,该方法具有较快的收敛速度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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MOCCA: Mirrored Convex/Concave Optimization for Nonconvex Composite Functions.

Many optimization problems arising in high-dimensional statistics decompose naturally into a sum of several terms, where the individual terms are relatively simple but the composite objective function can only be optimized with iterative algorithms. In this paper, we are interested in optimization problems of the form F(Kx) + G(x), where K is a fixed linear transformation, while F and G are functions that may be nonconvex and/or nondifferentiable. In particular, if either of the terms are nonconvex, existing alternating minimization techniques may fail to converge; other types of existing approaches may instead be unable to handle nondifferentiability. We propose the MOCCA (mirrored convex/concave) algorithm, a primal/dual optimization approach that takes a local convex approximation to each term at every iteration. Inspired by optimization problems arising in computed tomography (CT) imaging, this algorithm can handle a range of nonconvex composite optimization problems, and offers theoretical guarantees for convergence when the overall problem is approximately convex (that is, any concavity in one term is balanced out by convexity in the other term). Empirical results show fast convergence for several structured signal recovery problems.

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来源期刊
Journal of Machine Learning Research
Journal of Machine Learning Research 工程技术-计算机:人工智能
CiteScore
18.80
自引率
0.00%
发文量
2
审稿时长
3 months
期刊介绍: The Journal of Machine Learning Research (JMLR) provides an international forum for the electronic and paper publication of high-quality scholarly articles in all areas of machine learning. All published papers are freely available online. JMLR has a commitment to rigorous yet rapid reviewing. JMLR seeks previously unpublished papers on machine learning that contain: new principled algorithms with sound empirical validation, and with justification of theoretical, psychological, or biological nature; experimental and/or theoretical studies yielding new insight into the design and behavior of learning in intelligent systems; accounts of applications of existing techniques that shed light on the strengths and weaknesses of the methods; formalization of new learning tasks (e.g., in the context of new applications) and of methods for assessing performance on those tasks; development of new analytical frameworks that advance theoretical studies of practical learning methods; computational models of data from natural learning systems at the behavioral or neural level; or extremely well-written surveys of existing work.
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