论文标题
弱凸复合函数的可变平滑
Variable Smoothing for Weakly Convex Composite Functions
论文作者
论文摘要
我们研究结构化目标函数的最小化，是平滑函数的总和和与线性算子的弱凸功能的组成。应用程序包括与标准凸正则化器相比，引入偏差少的正规化器的图像重建问题。我们基于莫罗包络线以减少平滑参数的序列开发可变的平滑算法，并证明$ \ MATHCAL {O}（ε^{ - 3}）的复杂性以实现$ε$ -AppRoximate解决方案。此绑定在$ \ mathcal {o}（ε^{ - 2}）$之间绑定了为平滑案例绑定的$，而$ \ Mathcal {o}（ε^{ - 4}）$绑定了子级别方法。我们的复杂性绑定与其他涉及弱凸功能的结构化非平滑度的作品一致。
We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with a decreasing sequence of smoothing parameters, and prove a complexity of $\mathcal{O}(ε^{-3})$ to achieve an $ε$-approximate solution. This bound interpolates between the $\mathcal{O}(ε^{-2})$ bound for the smooth case and the $\mathcal{O}(ε^{-4})$ bound for the subgradient method. Our complexity bound is in line with other works that deal with structured nonsmoothness of weakly convex functions.
