论文标题
Riemannian平滑梯度类型算法] {在紧凑型Riemannian Submanifold嵌入在Euclidean Space中
Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space
论文作者
论文摘要
在本文中，我们介绍了一类无概念和非平滑复合综合最小化问题的广义$ε$ -Stationality的概念，这些问题嵌入了euclidean空间中的紧凑型Riemannian Submanifold。为了找到一种广义的$ε$ - 定位点，我们基于莫罗封装技术开发了一个riemannian梯度型方法，其平滑参数的顺序降低，即riemannian平滑梯度和Riemannian平滑梯度。我们证明Riemannian平滑梯度方法具有$ \ Mathcal {O}（ε^{ - 3}）$的迭代复杂性，用于驱动通用$ε$ - 稳定点。据我们所知，这是歧管上非凸和非平滑复合问题的最著名的迭代复杂性结果。对于Riemannian平滑随机梯度方法，可以实现$ \ MATHCAL {O}（ε^{ - 5}）$的迭代复杂性，用于驱动广义$ε$ - 稳定点。进行数值实验以验证我们算法的优越性。
In this paper, we introduce the notion of generalized $ε$-stationarity for a class of nonconvex and nonsmooth composite minimization problems on compact Riemannian submanifold embedded in Euclidean space. To find a generalized $ε$-stationarity point, we develop a family of Riemannian gradient-type methods based on the Moreau envelope technique with a decreasing sequence of smoothing parameters, namely Riemannian smoothing gradient and Riemannian smoothing stochastic gradient methods. We prove that the Riemannian smoothing gradient method has the iteration complexity of $\mathcal{O}(ε^{-3})$ for driving a generalized $ε$-stationary point. To our knowledge, this is the best-known iteration complexity result for the nonconvex and nonsmooth composite problem on manifolds. For the Riemannian smoothing stochastic gradient method, one can achieve the iteration complexity of $\mathcal{O}(ε^{-5})$ for driving a generalized $ε$-stationary point. Numerical experiments are conducted to validate the superiority of our algorithms.
