论文标题
通过近端梯度方法确定的流形的牛顿加速度
Newton acceleration on manifolds identified by proximal-gradient methods
论文作者
论文摘要
已知近端方法可以识别非平滑优化问题的基本子结构。更重要的是，在许多有趣的情况下，接近操作员的输出无需额外的成本即可带有其结构，并且一旦与最小化器的结构匹配，收敛就会得到改善。但是，通常不可能知道当前的结构是否最终结构。这样的高价信息必须自适应地利用。为此，我们将自己置于近端梯度方法可以识别非平滑目标可不同性的歧管的情况下。利用这种歧管识别，我们表明类似牛顿的方法可以与近端梯度步骤交织在一起，以极大地增强收敛性。我们证明算法在求解某些非平滑非平滑非凸优化问题时，算法的超值收敛性。我们提供有关$ \ ell_1 $ -norm或Trace-norm正规的优化问题的数值插图。
Proximal methods are known to identify the underlying substructure of nonsmooth optimization problems. Even more, in many interesting situations, the output of a proximity operator comes with its structure at no additional cost, and convergence is improved once it matches the structure of a minimizer. However, it is impossible in general to know whether the current structure is final or not; such highly valuable information has to be exploited adaptively. To do so, we place ourselves in the case where a proximal gradient method can identify manifolds of differentiability of the nonsmooth objective. Leveraging this manifold identification, we show that Riemannian Newton-like methods can be intertwined with the proximal gradient steps to drastically boost the convergence. We prove the superlinear convergence of the algorithm when solving some nondegenerated nonsmooth nonconvex optimization problems. We provide numerical illustrations on optimization problems regularized by $\ell_1$-norm or trace-norm.
