Alternating Differentiation for Optimization Layers

• URL: http://arxiv.org/abs/2210.01802v2
• Date: Mon, 24 Apr 2023 06:33:10 GMT
• Title: Alternating Differentiation for Optimization Layers
• Authors: Haixiang Sun, Ye Shi, Jingya Wang, Hoang Duong Tuan, H. Vincent Poor, and Dacheng Tao
• Abstract summary: We develop a new framework, named Alternating Differentiation (Alt-Diff), that differentiates optimization problems. We show that Alt-Diff substantially decreases the dimensions of the Jacobian matrix especially for optimization with large-scale constraints. We also propose to truncate Alt-Diff to further accelerate the computational speed.
• Score: 133.2668019610731
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The idea of embedding optimization problems into deep neural networks as optimization layers to encode constraints and inductive priors has taken hold in recent years. Most existing methods focus on implicitly differentiating Karush-Kuhn-Tucker (KKT) conditions in a way that requires expensive computations on the Jacobian matrix, which can be slow and memory-intensive. In this paper, we developed a new framework, named Alternating Differentiation (Alt-Diff), that differentiates optimization problems (here, specifically in the form of convex optimization problems with polyhedral constraints) in a fast and recursive way. Alt-Diff decouples the differentiation procedure into a primal update and a dual update in an alternating way. Accordingly, Alt-Diff substantially decreases the dimensions of the Jacobian matrix especially for optimization with large-scale constraints and thus increases the computational speed of implicit differentiation. We show that the gradients obtained by Alt-Diff are consistent with those obtained by differentiating KKT conditions. In addition, we propose to truncate Alt-Diff to further accelerate the computational speed. Under some standard assumptions, we show that the truncation error of gradients is upper bounded by the same order of variables' estimation error. Therefore, Alt-Diff can be truncated to further increase computational speed without sacrificing much accuracy. A series of comprehensive experiments validate the superiority of Alt-Diff.

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