An inexact accelerated stochastic ADMM for separable convex optimization

Jianchao Bai (), William W. Hager () and Hongchao Zhang ()
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Jianchao Bai: Northwestern Polytechnical University
William W. Hager: University of Florida
Hongchao Zhang: Louisiana State University

Computational Optimization and Applications, 2022, vol. 81, issue 2, No 5, 479-518

Abstract: Abstract An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Problems having this structure often arise in machine learning and data mining applications. AS-ADMM combines the ideas of both ADMM and the stochastic gradient methods using variance reduction techniques. One of the ADMM subproblems employs a linearization technique while a similar linearization could be introduced for the other subproblem. For a specified choice of the algorithm parameters, it is shown that the objective error and the constraint violation are $$\mathcal {O}(1/k)$$ O ( 1 / k ) relative to the number of outer iterations k. Under a strong convexity assumption, the expected iterate error converges to zero linearly. A linearized variant of AS-ADMM and incremental sampling strategies are also discussed. Numerical experiments with both stochastic and deterministic ADMM algorithms show that AS-ADMM can be particularly effective for structured optimization arising in big data applications.

Keywords: Convex optimization; Separable structure; Accelerated stochastic ADMM; Inexact stochastic ADMM; AS-ADMM; Accelerated gradient method; Complexity; Big data (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10589-021-00338-8

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