Authors: Alex Oshin, Rahul Vodeb Ghosh, Evangelos A. Theodorou
Organizations: Georgia Institute of Technology
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(n2) 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 ϑ 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× speedup depending on the class. When used as a subroutine in SCP, the learned approach delivers over a 30× speedup compared to COSMO.
Distributed optimization is a highly scalable and structurally transparent technique to solve multi-agent robotics problems; however, such methods often suffer from the need for highly-specialized, problem-specific hyperparameter tunings. In this work, we propose Deep Coordinator, a deep-unfolding framework that learns to dynamically adjust the hyperparameters of ADMM-DDP, a popular distributed solver for robotics tasks, at solve-time in response to optimizer performance. Our architecture consists of unrolling a fixed number of ADMM-DDP iterations into a neural network with learnable functions between layers mapping the optimizer state to the next hyperparameters. To the best of our knowledge, Deep Coordinator is the first deep-unfolding framework to adapt the penalty parameters of a non-convex optimizer at solve-time; we show that the mainstream supervised approach can yield degenerate solutions when training such models, and propose an unsupervised learning scheme. On simulations with fleets of cars and quadrotors, Deep Coordinator produces trajectories of comparable quality 6.18-9.44x faster than conventional solvers. Furthermore, Deep Coordinator retains its performance benefits when deployed to systems up to 8x larger than trained on.
End-to-end deep learning has achieved impressive results but often relies on large labeled datasets, exhibits limited generalization to unseen scenarios, and incurs substantial computational cost. Classical optimization methods, in contrast, are more data-efficient and lightweight but frequently suffer from slow convergence. Learned optimizers aim to bridge this gap, yet existing approaches have focused primarily on first-order methods, while learned second-order optimization has received much less attention. We introduce L-SR1, a learned second-order optimizer inspired by the classical Symmetric Rank-One (SR1) method. At its core, L-SR1 employs a Projection-Guided Secant Mechanism (PGSM) that generates positive semi-definite preconditioners and biases meta-training toward the quasi-Newton secant relation. Through controlled analytic benchmarks, we study stability, generalization across problem dimensions, and search direction quality, and further evaluate L-SR1 on Monocular Human Mesh Recovery (HMR), where it outperforms both classical and learned optimization-based baselines. With a compact model and no reliance on task-specific fine-tuning or annotated data, L-SR1 demonstrates strong generalization and can be integrated into a broad range of iterative optimization problems to accelerate convergence and reduce the required number of iterations.
Trajectory Optimization (TO) solvers exploit known system dynamics to compute locally optimal trajectories through iterative improvements. A downside is that each new problem instance is solved independently; therefore, convergence speed and quality of the solution found depend on the initial trajectory proposed. To improve efficiency, a natural approach is to warm-start TO with initial guesses produced by a learned policy trained on trajectories previously generated by the solver. Diffusion-based policies have recently emerged as expressive imitation learning models, making them promising candidates for this role. Yet, a counterintuitive challenge comes from the local optimality of TO demonstrations: when a policy is rolled out, small non-optimal deviations may push it into situations not represented in the training data, triggering compounding errors over long horizons. In this work, we focus on learning-based warm-starting for gradient-based TO solvers that also provide feedback gains. Exploiting this specificity, we derive a first-order loss for Sobolev learning of diffusion-based policies using both trajectories and feedback gains. Through comprehensive experiments, we demonstrate that the resulting policy avoids compounding errors, and so can learn from very few trajectories to provide initial guesses reducing solving time by 2× to 20×. Incorporating first-order information enables predictions with fewer diffusion steps, reducing inference latency.
Théotime Le Hellard, Franki Nguimatsia Tiofack, Quentin Le Lidec +1