Hybrid SE-PG: Reliable Gradient Estimation in Pre-Convergence Differentiable Optimization

Differentiable optimization enables end-to-end learning through iterative solvers, yet obtaining accurate gradients remains challenging when solvers operate in the pre-convergence regime. Existing approaches face a fundamental trade-off: Implicit Differentiation (IFT) provides accurate gradients only near solver fixed points, while truncated methods improve efficiency at the cost of missing future gradient propagation. We introduce Hybrid Spectral Extrapolation Phantom Gradient (Hybrid SE-PG), a gradient estimation framework for pre-convergence solver states. SE-PG exploits locally predictable gradient evolution to approximate missing gradient propagation, while an adaptive hybrid mechanism integrates IFT as solvers approach convergence. This enables efficient gradient estimation across different solver convergence states without requiring full convergence before gradient computation. We evaluate Hybrid SE-PG on differentiable optimization benchmarks, including classical control tasks and high-dimensional manipulation problems. Gradient fidelity experiments show that SE-PG provides more accurate approximations to fully unrolled gradients under pre-convergence solver states. Training experiments further demonstrate improved optimization efficiency compared with implicit differentiation baselines while maintaining accurate gradient estimation.