Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis

TL;DR

Yau's Affine Normal Descent (YAND) is a geometric optimization framework using affine normal directions that adapt to curvature, ensuring convergence and robustness under affine transformations.

Contribution

The paper introduces YAND, a novel affine-invariant optimization method with convergence guarantees and robustness to ill-conditioning, based on affine differential geometry.

Findings

YAND achieves one-step convergence for quadratic objectives.

The method guarantees global convergence under standard smoothness.

YAND is robust to affine scalings and anisotropic curvature.

Abstract

We propose Yau's Affine Normal Descent (YAND), a geometric framework for smooth unconstrained optimization in which search directions are defined by the equi-affine normal of level-set hypersurfaces. The resulting directions are invariant under volume-preserving affine transformations and intrinsically adapt to anisotropic curvature. Using the analytic representation of the affine normal from affine differential geometry, we establish its equivalence with the classical slice-centroid construction under convexity. For strictly convex quadratic objectives, affine-normal directions are collinear with Newton directions, implying one-step convergence under exact line search. For general smooth (possibly nonconvex) objectives, we characterize precisely when affine-normal directions yield strict descent and develop a line-search-based YAND. We establish global convergence under standard smoothness assumptions, linear convergence under strong convexity and Polyak-Lojasiewicz conditions, and quadratic local convergence near nondegenerate minimizers. We further show that affine-normal directions are robust under affine scalings, remaining insensitive to arbitrarily ill-conditioned transformations. Numerical experiments illustrate the geometric behavior of the method and its robustness under strong anisotropic scaling.