Differentiable optimization has received a significant amount of attention due to its foundational role in the domain of machine learning based on neural networks. The existing methods leverages the optimality conditions and implicit function theorem to obtain the Jacobian matrix of the output, which increases the computational cost and limits the application of differentiable optimization. In addition, some non-differentiable constraints lead to more challenges when using prior differentiable optimization layers. This paper proposes a differentiable layer, named Differentiable Frank-Wolfe Layer (DFWLayer), by rolling out the Frank-Wolfe method, a well-known optimization algorithm which can solve constrained optimization problems without projections and Hessian matrix computations, thus leading to a efficient way of dealing with large-scale problems. Theoretically, we establish a bound on the suboptimality gap of the DFWLayer in the context of l1-norm constraints. Experimental assessments demonstrate that the DFWLayer not only attains competitive accuracy in solutions and gradients but also consistently adheres to constraints. Moreover, it surpasses the baselines in both forward and backward computational speeds.

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.