L-SR1: Learned Symmetric-Rank-One Preconditioning

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 data-efficient and lightweight but frequently suffer from slow or unstable convergence. Learned optimizers aim to bridge this gap, yet existing approaches predominantly focus on first-order methods, leaving learned second-order optimization largely unexplored. We introduce L-SR1, a novel learned second-order optimizer that augments the classical Symmetric Rank-One (SR1) method with a lightweight, trainable preconditioning unit. This unit generates data-driven rank-one updates that are explicitly aligned with the secant condition via a learned projection mechanism, ensuring stable and consistent curvature estimation. Through controlled analytic benchmarks, we systematically analyze the stability, generalization across problem dimensions, and the quality of the resulting search directions. We further evaluate L-SR1 on the real-world task of Monocular Human Mesh Recovery (HMR), where it outperforms existing optimization-based and learned-optimization baselines. With a compact model and no reliance on task-specific fine-tuning or annotated data, L-SR1 demonstrates strong generalization and provides an effective drop-in optimizer for a wide range of iterative optimization problems.