Residual-Controlled Douglas--Rachford Splitting for Differentiable Solver Layers

Abstract

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/.