Title: Differentiable Inverse Optimal Control via Gap-Aware Rayleigh–Ritz and Schur-Complement Reduction
Primary area: optimization
Keywords: Differentiable Inverse Optimal Control; Rayleigh–Ritz subspace iteration; spectral gap regularization; Schur complement; Soft-Ritz; eigenvalue minimization; gradient stability; imitation learning; robotics; end-to-end learning

Abstract: Inverse Optimal Control (IOC) is interpretable but typically non-differentiable and numerically fragile near spectral degeneracy, limiting its use in end-to-end learning. We present a Differentiable IOC (DIOC) layer that accumulates sequential optimality constraints into one symmetric matrix and reads the solution from its null space. We show that hard eigenselection causes gradient blow-up, and introduce Soft-Ritz—a Rayleigh–Ritz subspace method with entropic soft selection—paired with a hinge penalty that maintains a boundary spectral gap; this combination delivers bounded, well-conditioned gradients. On linear and nonlinear robotic systems, DIOC converges more smoothly, achieves lower final error, and is markedly more stable than direct eigendecomposition; ablations confirm the gap hinge is necessary and soft-only targets are insufficient. Unifying classical interpretability with modern differentiability, DIOC serves as a practical, auditable module for perception-to-control pipelines, model-based RL warm starts, and safety-critical robotics.