论文标题
非凸优化的加速一阶方法
An accelerated first-order method for non-convex optimization on manifolds
论文作者
论文摘要
我们描述了关于Riemannian歧管的第一个梯度方法，以实现非凸vex情况下的加速率。在Lipschitz在Riemannian梯度和成本函数的Hessian上的假设下，这些方法比常规梯度下降更快地发现了近似的一阶关键点。随机版本还可以找到近似的二阶关键点。算法及其分析在欧几里得案中的现有工作基于现有工作。基本操作包括在当前切线空间中运行欧几里得加速梯度下降法（适当安全地保护非凸度），然后回到歧管并重复。这就需要将成本函数从流动空间提升到切线空间，例如通过Riemannian指数映射来完成。为了获得成功的方法，提升的成本功能（称为回调）必须保留某些Lipschitz属性。作为独立利益的贡献，我们用明确的常数证明了这一效果的精确主张。这些主张受歧管的riemannian曲率影响，这反过来影响了我们优化算法的最坏情况复杂性界限。
We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.
