RapidDOC: Riccati-Preconditioned Implicit Differentiation for Differentiable Optimal Control under Operator Mismatch
Abstract
Differentiable optimal control embeds constrained optimization problems in data-driven pipelines by differentiating their optimality conditions and propagating task-loss gradients to model or objective parameters. A key observation is that forward optimization and backward sensitivity computation involve related stage-structured linear systems, making reuse of forward factorization information natural. However, direct reuse is generally invalid, because differences in linearization, barrier terms, regularization, and curvature can create a mismatch between the retained forward operator and the target operator defining the derivative. We therefore propose RapidDOC, a factorization-reuse method that leaves the target adjoint equation unchanged. The retained cache produces a candidate adjoint and, when its target residual exceeds the prescribed tolerance, serves as a fixed right preconditioner for GMRES. A cache-action error operator characterizes directional exactness, the preconditioned iteration, and the relation between target residual and outer-gradient error. An ablation on three constrained robotic OCPs shows that the proposed method achieves lower backward time than both target-system refactorization and a no-reuse iterative method, demonstrating the computational benefit of reusing the retained forward factorizations. Across open- and closed-loop parameter-learning experiments, the proposed method achieves learning performance comparable to the baselines, while yielding the lowest open-loop backward time on all three tasks and reducing total closed-loop solver-call time by 40.5-83.1% relative to the fastest competing method. Finally, deploying the learned controller on a physical vehicle reduces the median figure-eight tracking RMSE from 22.75 to 6.09 cm, demonstrating its potential for hardware deployment.
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BibTeX
@misc{zhu2026rapiddoc,
title = {{RapidDOC: Riccati-Preconditioned Implicit Differentiation for Differentiable Optimal Control under Operator Mismatch}},
author = {Xiangda Zhu and Yuankun Chen and Zifei Nie and Yi Chang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41324},
eprint = {2610.41324},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41324},
}
RIS
TY - UNPB
TI - RapidDOC: Riccati-Preconditioned Implicit Differentiation for Differentiable Optimal Control under Operator Mismatch
AU - Xiangda Zhu
AU - Yuankun Chen
AU - Zifei Nie
AU - Yi Chang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41324
UR - https://arxiv.org/abs/2610.41324
AB - Differentiable optimal control embeds constrained optimization problems in data-driven pipelines by differentiating their optimality conditions and propagating task-loss gradients to model or objective parameters. A key observation is that forward optimization and backward sensitivity computation involve related stage-structured linear systems, making reuse of forward factorization information natural. However, direct reuse is generally invalid, because differences in linearization, barrier terms, regularization, and curvature can create a mismatch between the retained forward operator and the target operator defining the derivative. We therefore propose RapidDOC, a factorization-reuse method that leaves the target adjoint equation unchanged. The retained cache produces a candidate adjoint and, when its target residual exceeds the prescribed tolerance, serves as a fixed right preconditioner for GMRES. A cache-action error operator characterizes directional exactness, the preconditioned iteration, and the relation between target residual and outer-gradient error. An ablation on three constrained robotic OCPs shows that the proposed method achieves lower backward time than both target-system refactorization and a no-reuse iterative method, demonstrating the computational benefit of reusing the retained forward factorizations. Across open- and closed-loop parameter-learning experiments, the proposed method achieves learning performance comparable to the baselines, while yielding the lowest open-loop backward time on all three tasks and reducing total closed-loop solver-call time by 40.5-83.1% relative to the fastest competing method. Finally, deploying the learned controller on a physical vehicle reduces the median figure-eight tracking RMSE from 22.75 to 6.09 cm, demonstrating its potential for hardware deployment.
ER -