基于仿射几何设计新优化算法，对病态问题更鲁棒。
Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis
- 用等仿射法向定义搜索方向，天然适应各向异性曲率。
- 严格凸二次目标下一步收敛，一般光滑目标全局收敛。
- 对仿射缩放不变，适合条件极差的优化问题。
我们提出等仿射法向下降（YAND），一种基于光滑无约束优化的几何框架，其搜索方向由等高线超曲面的等仿射法向定义。该方向在保体积仿射变换下不变，且内在适应各向异性曲率。利用仿射微分几何中的等仿射法向解析表达式，我们证明其在凸性条件下等价于经典切片重心构造。对于严格凸二次目标，等仿射法向与牛顿方向共线，精确线搜索下一歩收敛。针对一般光滑（可能非凸）目标，我们精确刻画了等仿射法向产生严格下降的条件，并提出了基于线搜索的YAND算法。在标准光滑性假设下建立全局收敛性，在强凸性和Polyak-Lojasiewicz条件下实现线性收敛，并在非退化极小值点附近达到二次局部收敛。进一步证明等仿射法向对仿射缩放具有鲁棒性，对任意病态变换保持不敏感。数值实验展示了该方法的几何特性及其在强各向异性缩放下仍具鲁棒性。
原文摘要 · Abstract (English)
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.
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