Statistics with set-valued functions: applications to inverse approximate optimization
Statistics with set-valued functions: applications to inverse approximate optimization
复制标题
集值函数统计：逆近似优化的应用
DOI：
10.1007/s10107-018-1257-5
复制
发表时间：
2017
影响因子：
2.7
通讯作者：
A. Aswani
中科院分区：
文献类型：
--
作者：
A. Aswani
大部分统计学依赖于四个关键要素：大数定律，操作随机收敛的微积分，中心极限定理和构造局部近似的框架。对于向量空间中的对象（例如，点或函数），这些元素是很容易理解的；然而，许多统计理论不能直接转化为集合，因为它们不能形成向量空间。本文以随机集的概率论为基础，利用变分分析开发集值函数统计的操作工具。这些工具首先应用于非参数估计（集值函数的核回归）。第二个应用是逆近似优化问题，其中观察优化问题的近似解（被噪声破坏），然后用于估计解决方案的次优性数量和生成解决方案的优化问题的参数。我们表明，当数据被噪声破坏时，以前处理该问题的方法在统计上不一致，而我们的方法在温和条件下是一致的。
Much of statistics relies upon four key elements: a law of large numbers, a calculus to operationalize stochastic convergence, a central limit theorem, and a framework for constructing local approximations. These elements are well-understood for objects in a vector space (e.g., points or functions); however, much statistical theory does not directly translate to sets because they do not form a vector space. Building on probability theory for random sets, this paper uses variational analysis to develop operational tools for statistics with set-valued functions. These tools are first applied to nonparametric estimation (kernel regression of set-valued functions). The second application is to the problem of inverse approximate optimization, in which approximate solutions (corrupted by noise) to an optimization problem are observed and then used to estimate the amount of suboptimality of the solutions and the parameters of the optimization problem that generated the solutions. We show that previous approaches to this problem are statistically inconsistent when the data is corrupted by noise, whereas our approach is consistent under mild conditions.