Sparse optimization problems are ubiquitous in many fields such as statistics, signal/image processing and machine learning. This has led to the birth of many iterative algorithms to solve them. A powerful strategy to boost the performance of these algorithms is known as safe screening: it allows the early identification of zero coordinates in the solution, which can then be eliminated to reduce the problem's size and accelerate convergence. In this work, we extend the existing Gap Safe screening framework by relaxing the global strong-concavity assumption on the dual cost function. Instead, we exploit local regularity properties, that is, strong concavity on well-chosen subsets of the domain. The non-negativity constraint is also integrated to the existing framework. Besides making safe screening possible to a broader class of functions that includes beta-divergences (e.g., the Kullback-Leibler divergence), the proposed approach also improves upon the existing Gap Safe screening rules on previously applicable cases (e.g., logistic regression). The proposed general framework is exemplified by some notable particular cases: logistic function, beta = 1.5 and Kullback-Leibler divergences. Finally, we showcase the effectiveness of the proposed screening rules with different solvers (coordinate descent, multiplicative-update and proximal gradient algorithms) and different data sets (binary classification, hyperspectral and count data).
翻译:在统计、信号/图像处理和机器学习等许多领域,偏差优化问题无处不在。这导致产生了许多迭代算法来解决这些问题。提高这些算法绩效的强大战略被称为安全筛选:它允许早期确定解决方案中的零坐标,然后可以消除这些坐标,以减少问题的规模和加速趋同。在这项工作中,我们通过放松全球对双重成本功能的强力兼容性假设,扩展现有的差距安全筛选框架。相反,我们利用地方常规性特性,即域内选取的精密子组的强烈混杂性。非强化性制约也被纳入现有框架。除了使安全筛选成为更广泛的功能类别外,包括乙型差异(例如库尔贝克-利伯尔差异),拟议办法还改进了以前适用案例(例如物流回归)上的现有差距安全筛选规则。拟议的一般框架以一些显著的特殊情况为例证:物流功能、乙型=1.5和库尔克斯-后期数据级化(不同版本、不同版本数据级、不同版本的升级和跨级)的拟议数据结构。