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Coordinate descent algorithms

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Abstract

Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. This paper describes the fundamentals of the coordinate descent approach, together with variants and extensions and their convergence properties, mostly with reference to convex objectives. We pay particular attention to a certain problem structure that arises frequently in machine learning applications, showing that efficient implementations of accelerated coordinate descent algorithms are possible for problems of this type. We also present some parallel variants and discuss their convergence properties under several models of parallel execution.

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Acknowledgments

I thank Ji Liu for the pleasure of collaborating with him on this topic over the past two years. I am grateful to the editors and referees of the paper, whose expert and constructive comments led to numerous improvements.

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Correspondence to Stephen J. Wright.

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The author was supported by NSF Awards DMS-1216318 and IIS-1447449, ONR Award N00014-13-1-0129, AFOSR Award FA9550-13-1-0138, and Subcontract 3F-30222 from Argonne National Laboratory.

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Wright, S.J. Coordinate descent algorithms. Math. Program. 151, 3–34 (2015). https://doi.org/10.1007/s10107-015-0892-3

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