论文标题
各向异性近端梯度
Anisotropic Proximal Gradient
论文作者
论文摘要
本文研究了一种非凸复合材料最小化的新型算法，可以用双空间非线性预处理来解释经典近端梯度方法。所提出的方案可以应用于加性复合最小化问题，其平滑部分相对于参考函数表现出各向异性下降不平等。事实证明，如果共轭参考函数产生的布雷格曼距离是联合凸的，各向异性下降特性在平均值下关闭。更具体地说，对于指数参考函数，我们证明了其在圆锥形组合下的闭合度。我们分析了该方法的渐近收敛性，并在各向异性近端梯度优势条件下证明了其线性收敛。讨论了应用程序，包括指数正规化的LP和逻辑回归，并以非平滑的正则化。在数值实验中，我们显示了所提出的方法比其欧几里得对应物的显着改善。
This paper studies a novel algorithm for nonconvex composite minimization which can be interpreted in terms of dual space nonlinear preconditioning for the classical proximal gradient method. The proposed scheme can be applied to additive composite minimization problems whose smooth part exhibits an anisotropic descent inequality relative to a reference function. It is proved that the anisotropic descent property is closed under pointwise average if the Bregman distance generated by the conjugate reference function is jointly convex. More specifically, for the exponential reference function we prove its closedness under pointwise conic combinations. We analyze the method's asymptotic convergence and prove its linear convergence under an anisotropic proximal gradient dominance condition. Applications are discussed including exponentially regularized LPs and logistic regression with nonsmooth regularization. In numerical experiments we show significant improvements of the proposed method over its Euclidean counterparts.
