Volume 35, Issue 4
Extrapush for Convex Smooth Decentralized Optimization Over Directed Networks

Jinshan Zeng & Wotao Yin

J. Comp. Math., 35 (2017), pp. 383-396.

Published online: 2017-08

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  • Abstract

In this note, we extend the algorithms Extra [13] and subgradient-push [10] to a new algorithm ExtraPush for consensus optimization with convex differentiable objective functions over a directed network. When the stationary distribution of the network can be computed in advance, we propose a simplified algorithm called Normalized ExtraPush. Just like Extra, both ExtraPush and Normalized ExtraPush can iterate with a fixed step size. But unlike Extra, they can take a column-stochastic mixing matrix, which is not necessarily doubly stochastic. Therefore, they remove the undirected-network restriction of Extra. Subgradient-push, while also works for directed networks, is slower on the same type of problem because it must use a sequence of diminishing step sizes. We present preliminary analysis for ExtraPush under a bounded sequence assumption. For Normalized ExtraPush, we show that it naturally produces a bounded, linearly convergent sequence provided that the objective function is strongly convex. In our numerical experiments, ExtraPush and Normalized ExtraPush performed similarly well. They are significantly faster than subgradient-push, even when we hand-optimize the step sizes for the latter.

  • Keywords

Decentralized optimization Directed graph Consensus Non-doubly stochastic Extra

  • AMS Subject Headings

90C25 90C30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jinshanzeng@jxnu.edu.cn (Jinshan Zeng)

wotaoyin@math.ucla.edu (Wotao Yin)

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@Article{JCM-35-383, author = {Zeng , Jinshan and Yin , Wotao }, title = {Extrapush for Convex Smooth Decentralized Optimization Over Directed Networks}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {4}, pages = {383--396}, abstract = { In this note, we extend the algorithms Extra [13] and subgradient-push [10] to a new algorithm ExtraPush for consensus optimization with convex differentiable objective functions over a directed network. When the stationary distribution of the network can be computed in advance, we propose a simplified algorithm called Normalized ExtraPush. Just like Extra, both ExtraPush and Normalized ExtraPush can iterate with a fixed step size. But unlike Extra, they can take a column-stochastic mixing matrix, which is not necessarily doubly stochastic. Therefore, they remove the undirected-network restriction of Extra. Subgradient-push, while also works for directed networks, is slower on the same type of problem because it must use a sequence of diminishing step sizes. We present preliminary analysis for ExtraPush under a bounded sequence assumption. For Normalized ExtraPush, we show that it naturally produces a bounded, linearly convergent sequence provided that the objective function is strongly convex. In our numerical experiments, ExtraPush and Normalized ExtraPush performed similarly well. They are significantly faster than subgradient-push, even when we hand-optimize the step sizes for the latter.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1606-m2015-0452}, url = {http://global-sci.org/intro/article_detail/jcm/10022.html} }
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