The integration of constrained optimization models as components in deep networks has led to promising advances on many specialized learning tasks. A central challenge in this setting is backpropagation through the solution of an optimization problem, which often lacks a closed form. One typical strategy is algorithm unrolling, which relies on automatic differentiation through the entire chain of operations executed by an iterative optimization solver. This paper provides theoretical insights into the backward pass of unrolled optimization, showing that it is asymptotically equivalent to the solution of a linear system by a particular iterative method. Several practical pitfalls of unrolling are demonstrated in light of these insights, and a system called Folded Optimization is proposed to construct more efficient backpropagation rules from unrolled solver implementations. Experiments over various end-to-end optimization and learning tasks demonstrate the advantages of this system both computationally, and in terms of flexibility over various optimization problem forms.

翻译：将约束优化模型作为深度网络组件集成，已在许多专门学习任务中取得了令人瞩目的进展。该场景下的核心挑战在于通过优化问题的解进行反向传播，而这类问题通常缺乏闭式解。一种典型策略是算法展开，它依赖于对迭代优化求解器执行的整个操作链进行自动微分。本文为展开优化的反向传播提供了理论洞见，表明其渐近等价于通过特定迭代方法求解线性系统。基于这些见解，展示了展开过程的若干实际陷阱，并提出了一种名为折叠优化的系统，用于从展开求解器实现中构建更高效的反向传播规则。在多种端到端优化与学习任务上的实验表明，该系统在计算效率以及针对不同优化问题形式的灵活性方面均具有优势。