Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis
This work addresses optimization problems for researchers and practitioners by introducing a novel geometric method that is robust to affine scalings, though it appears incremental as it builds on classical concepts like Newton directions.
The paper tackles smooth unconstrained optimization by proposing Yau's Affine Normal Descent (YAND), a geometric framework using equi-affine normal directions that are invariant under volume-preserving affine transformations and adapt to anisotropic curvature, achieving results such as one-step convergence for strictly convex quadratic objectives under exact line search and global convergence with linear or quadratic rates under specific conditions.
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.