We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly delicate. We trace the difficulty back to a geometric obstacle: On a nonsmooth set, there may be sequences of points along which standard measures of stationarity tend to zero, but whose limit points are not stationary. We name such events apocalypses, as they can cause optimization algorithms to converge to non-stationary points. We illustrate this explicitly for an existing optimization algorithm on bounded-rank matrices. To provably find stationary points, we modify a trust-region method on a standard smooth parameterization of the variety. The method relies on the known fact that second-order stationary points on the parameter space map to stationary points on the variety. Our geometric observations and proposed algorithm generalize beyond bounded-rank matrices. We give a geometric characterization of apocalypses on general constraint sets, which implies that Clarke-regular sets do not admit apocalypses. Such sets include smooth manifolds, manifolds with boundaries, and convex sets. Our trust-region method supports parameterization by any complete Riemannian manifold.
翻译:我们认为,找到一个平稳功能的固定点的问题,应该尽量缩小在各种封闭式矩阵上的位置。结果,这是出乎意料的微妙之处。我们把困难追溯到几何障碍:在非悬浮的一组中,可能有标准定点的顺序,标准定点的定点一般为零,但其限点不是固定的。我们点出这样的事件,因为它们可以导致优化算法接近非静止的点。我们明确地说明这一点,以便在约束式矩阵上对现有优化算法作最优化算法。为了找到固定点,我们修改信任区方法,使多样性的标准平稳参数化。这个方法以已知的事实为依据,即参数空间地图上的第二阶点固定点一般为零点,但这种定点的定点的定点不是固定的。我们的几何观察和拟议的算法一般定点可以使得优化算法与非静止的基点相交汇。我们用几何方法来描述一般约束式的顶点的顶点,这意味着Clacer-正常规组合不会接受孔化。这种设置包括平滑的数、有边界的数、数级支持任何矩阵和等数的参数。