Curvature-Weighted Gradient Diversity: A Noise Measure for Geometry-Adaptive SGD Schedules

• URL: http://arxiv.org/abs/2606.30455v1
• Date: Mon, 29 Jun 2026 15:22:32 GMT
• Title: Curvature-Weighted Gradient Diversity: A Noise Measure for Geometry-Adaptive SGD Schedules
• Abstract summary: We introduce Curvature-Weighted Diversity (CWGD), a geometry-aware gradient measure that weights per gradient by the inverse square root of the Hessian.<n>CWGD-Cosine consistently achieves approximately 20% lower final optimization error than standard cosine.<n>These results establish CWGD as a principled geometry-aware measure of optimization noise and motivate future extensions to more general learning problems.
• Score: 1.1602089225841632
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The standard convergence analysis of mini-batch stochastic gradient descent (SGD) models gradient noise using a single variance term that treats all parameter directions equally, ignoring the fact that noise in high-curvature directions has less impact because learning rates are already constrained there. We introduce Curvature-Weighted Gradient Diversity (CWGD), a geometry-aware measure that weights per-sample gradient diversity by the inverse square root of the Hessian, providing a tighter proxy for the effective optimization noise. For strongly convex quadratic objectives with diagonal Hessians and isotropic noise, we prove that a CWGD-modulated cosine learning-rate schedule can reduce the asymptotic optimization error floor by up to a factor of two compared with standard cosine annealing. We implement this idea as CWGD-Cosine using a Hutchinson-based diagonal Hessian estimator that is exact for quadratic objectives. Across a range of condition numbers, batch sizes, and noise structures, CWGD-Cosine consistently achieves approximately 20% lower final optimization error than standard cosine annealing while incurring negligible overhead in the quadratic setting. We also identify and correct a degenerate curvature estimator, analyze the robustness of the proposed estimator, and explicitly discuss the limitations of the method, including Hessian staleness in non-convex optimization. These results establish CWGD as a principled geometry-aware measure of optimization noise and motivate future extensions to more general learning problems.

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