CAO: Curvature-Adaptive Optimization via Periodic Low-Rank Hessian Sketching

• URL: http://arxiv.org/abs/2511.12548v1
• Date: Sun, 16 Nov 2025 10:57:33 GMT
• Title: CAO: Curvature-Adaptive Optimization via Periodic Low-Rank Hessian Sketching
• Abstract summary: First-order curvatureobs are reliable but slow in sharp, anisotropic regions.<n>We use a curvature-declared method that sketches a Hessian subspace via Hessian products.<n>On CIFAR/100 with ResNet-18/34, the method enters the low-loss threshold (0.75x) substantially earlier.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: First-order optimizers are reliable but slow in sharp, anisotropic regions. We study a curvature-adaptive method that periodically sketches a low-rank Hessian subspace via Hessian--vector products and preconditions gradients only in that subspace, leaving the orthogonal complement first-order. For L-smooth non-convex objectives, we recover the standard O(1/T) stationarity guarantee with a widened stable stepsize range; under a Polyak--Lojasiewicz (PL) condition with bounded residual curvature outside the sketch, the loss contracts at refresh steps. On CIFAR-10/100 with ResNet-18/34, the method enters the low-loss region substantially earlier: measured by epochs to a pre-declared train-loss threshold (0.75), it reaches the threshold 2.95x faster than Adam on CIFAR-100/ResNet-18, while matching final test accuracy. The approach is one-knob: performance is insensitive to the sketch rank k across {1,3,5}, and k=0 yields a principled curvature-free ablation. We release anonymized logs and scripts that regenerate all figures and tables.

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