Scalable Deep Unfolding of Conic Optimizers

2026-06-11

AI Summary

This work proposes an efficient and scalable framework for solving large-scale semidefinite programs (SDPs) that addresses the high memory overhead and numerical instability in backpropagating through positive semidefinite (PSD) cone projections inherent in deep unrolling methods. By introducing a matrix-independent implicit differentiation rule, the memory complexity of backpropagation is reduced from O(n²) to O(n). Numerical stability is ensured through the exact treatment of PSD projection derivatives under repeated eigenvalues via the Daleckiĭ–Krein representation. The framework further integrates learnable hyperparameters, warm-starting, and sequential convex programming (SCP). Extensive experiments demonstrate significant performance gains over state-of-the-art solvers across various SDP and second-order cone problems, achieving speedups of up to 50× overall and exceeding 30× faster than COSMO within SCP inner loops.

Abstract

Deep unfolding (DU) accelerates iterative optimizers by introducing learnable components and training them through unrolled iterations, but extending DU to the large-scale semidefinite programs (SDPs) common in robotics has remained limited. Unrolling a full-update conic solver such as COSMO exposes two obstacles that prior work on learned conic solvers has not: backpropagating through the per-iteration linear-system solve incurs memory quadratic in the problem size once the coefficient matrix is formed explicitly, and backpropagating through the positive semidefinite (PSD) cone projection becomes numerically unstable when eigenvalues coincide. We address the first obstacle with a matrix-free implicit differentiation rule that operates entirely through matrix-vector products, reducing memory from $O(n^2)$ to $O(n)$ and enabling backpropagation at scales where direct factorization runs out of memory. We address the second with a backward rule based on the Dalečkii--Krein representation of the Fréchet derivative, which remains well-defined under repeated eigenvalues. Together these make it possible to learn lightweight hyperparameter policies and warm-starts for a full-update conic solver. We evaluate on nonlinear covariance steering problems solved via sequential convex programming (SCP), as well as standalone SDPs and second-order cone programs ranging from max-cut and Lovász $\vartheta$ SDPs to robust estimation and control problems. The learned policies outperform state-of-the-art solvers across all problems, and can provide up to a 50$\times$ speedup depending on the class. When used as a subroutine in SCP, the learned approach delivers over a 30$\times$ speedup compared to COSMO.

Problem

• Deep Unfolding
• Semidefinite Programming
• Backpropagation
• Memory Complexity
• Numerical Stability

Innovation

• deep unfolding
• conic optimization
• matrix-free implicit differentiation
• Daleckiĭ–Krein derivative
• semidefinite programming