用于可微分求解器层的残差控制道格拉斯-拉赫福德分裂
Residual-Controlled Douglas--Rachford Splitting for Differentiable Solver Layers
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中文总结 AI 辅助
针对可微分求解器层固定深度展开的权衡问题，提出残差控制道格拉斯-拉赫福德分裂（RCDRS），理论证明其性质，实验显示其提升解质量等性能。
中文摘要 AI 辅助
可微分求解器层将约束优化嵌入端到端学习系统中，但固定深度的展开必须在解质量、可行性和计算预算之间进行权衡。我们提出残差控制道格拉斯-拉赫福德分裂（Residual-Controlled Douglas--Rachford Splitting，RCDRS），这是一种用于锥形线性规划的可微分求解器层。RCDRS 将展开的求解器视为反馈控制的动力系统，其中因果控制器在保留道格拉斯-拉赫福德分裂的投影分裂结构的同时，自适应调整松弛参数和目标驱动参数。理论上，我们证明每个固定的可容许块仍是平均松弛的 DRS 算子，且存在有限步的不动点残差界。我们进一步分析受保护的时变展开作为极限平均算子的可和扰动，并从最终分裂状态恢复原始-对偶诊断信息。在混合锥形基准测试和工程应用上的实验表明，RCDRS 提升了解质量、可行性及下游决策性能。代码可在该 https URL 获取。
英文摘要
Differentiable solver layers embed constrained optimization into end-to-end learning systems, but fixed-depth unrolling must trade off solution quality, feasibility, and computational budget. We propose Residual-Controlled Douglas--Rachford Splitting (RCDRS), a differentiable solver layer for conic linear programs. RCDRS treats an unrolled solver as a feedback-controlled dynamical system, where a causal controller adapts the relaxation and objective-drive parameters while preserving the projection-splitting structure of Douglas--Rachford splitting. Theoretically, we show that each fixed admissible block remains an averaged relaxed DRS operator and admits finite-step fixed-point residual bounds. We further analyze safeguarded time-varying rollouts as summable perturbations of a limiting averaged operator, and recover terminal primal-dual diagnostics from the final splitting state. Experiments on mixed-cone benchmarks and engineering applications show that RCDRS improves solution quality, feasibility and downstream decision performance. The code is available at https://anonymous.4open.science/r/RC-DRS-180C/.
