GLOBAL SOLUTIONS TO FOLDED CONCAVE PENALIZED NONCONVEX LEARNING.
GLOBAL SOLUTIONS TO FOLDED CONCAVE PENALIZED NONCONVEX LEARNING.
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DOI：
10.1214/15-aos1380
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发表时间：
2016-04
影响因子：
4.5
通讯作者：
Li R
中科院分区：
文献类型：
--
作者：
Liu H;Yao T;Li R
本文致力于解决带有折叠凹罚分的非凸学习问题。尽管它们的全局解决方案需要理想的统计特性，但缺乏保证一般环境下全局最优性的优化技术。在本文中，我们证明了一类非凸学习问题等价于一般的二次规划。这种等价性有助于我们开发混合整数线性规划重构，它允许有限算法找到可证明的全局最优解。我们将这种基于重构的技术称为基于混合整数规划的全局优化（MIPGO）。据我们所知，这是第一个为具有 SCAD 惩罚和 MCP 惩罚的折叠凹惩罚非凸学习提供理论保证的全局优化方案。数值结果表明，在解质量方面，MIPGO 明显优于文献中最先进的解方案、局部线性近似和其他替代解技术。
This paper is concerned with solving nonconvex learning problems with folded concave penalty. Despite that their global solutions entail desirable statistical properties, there lack optimization techniques that guarantee global optimality in a general setting. In this paper, we show that a class of nonconvex learning problems are equivalent to general quadratic programs. This equivalence facilitates us in developing mixed integer linear programming reformulations, which admit finite algorithms that find a provably global optimal solution. We refer to this reformulation-based technique as the mixed integer programming-based global optimization (MIPGO). To our knowledge, this is the first global optimization scheme with a theoretical guarantee for folded concave penalized nonconvex learning with the SCAD penalty and the MCP penalty. Numerical results indicate a significant outperformance of MIPGO over the state-of-the-art solution scheme, local linear approximation, and other alternative solution techniques in literature in terms of solution quality.