论文标题
通过MM算法对双重规范的数值评估
Numerical evaluation of dual norms via the MM algorithm
论文作者
论文摘要
我们处理数值计算双重规范的问题，这对于研究稀疏性诱导正规化非常重要（Jenatton等，2011； Bach等，2012）。双重规范在优化和统计学习中找到了应用程序，例如在设计工作集策略的设计中，以表征双重梯度方法，双重分解以及增强拉格朗日功能的定义。然而，在分析上尚未获得一些众所周知的稀疏性重新极化方法的双重规范。示例是重叠组$ \ ell_2 $ -norm（Jenatton等，2011）和Zhou and Hastie（2005）的弹性净规范。因此，我们求助于Lange（2016）的大型最小化原理，以提供有效的算法，该算法利用双重约束优化问题的重新绘制为无限制的优化障碍。为了验证操作的正确性并评估所提出的方法的性能，已经进行了广泛的仿真实验。我们的结果证明了该算法在检索双重规范方面的有效性，即使对于大尺寸。
We deal with the problem of numerically computing the dual norm, which is important to study sparsity-inducing regularizations (Jenatton et al. 2011,Bach et al. 2012). The dual norms find application in optimization and statistical learning, for example, in the design of working-set strategies, for characterizing dual gradient methods, for dual decompositions and in the definition of augmented Lagrangian functions. Nevertheless, the dual norm of some well-known sparsity-inducing regolarization methods are not analytically available. Examples are the overlap group $\ell_2$-norm of (Jenatton et al. 2011) and the elastic net norm of Zhou and Hastie (2005). Therefore we resort to the Majorization-Minimization principle of Lange (2016) to provide an efficient algorithm that leverages a reparametrization of the dual constrained optimization problem as unconstrained optimization with barrier. Extensive simulation experiments have been performed in order to verify the correctness of operation, and evaluate the performance of the proposed method. Our results demonstrate the effectiveness of the algorithm in retrieving the dual norm even for large dimensions.
