硬形状约束核机器
Hard Shape-Constrained Kernel Machines
摘要：
形状约束（如非负性，单调性，凸性）在大量的应用中发挥着核心作用，因为它们通常可以提高小样本量的性能，并有助于解释。然而，以硬性方式强制执行这些形状要求是一个极具挑战性的问题。传统上，这个任务是解决（i）在一个软的方式（没有样本外的保证），（ii）在个案的基础上，通过专门的变量变换，或（iii）通过使用高度限制的函数类，如多项式或多项式样条。在本文中，我们证明了硬仿射形状约束的函数导数可以编码在内核机器，这是机器学习和统计中最灵活和强大的工具之一。特别是，我们提出了一个收紧的二阶锥约束的重新制定，可以很容易地实现凸解算器。我们证明了性能保证的解决方案，并证明了联合分位数回归的方法的效率与经济学的应用和飞机轨迹的分析，等等。
英文摘要：
Shape constraints (such as non-negativity, monotonicity, convexity) play a central role in a large number of applications, as they usually improve performance for small sample size and help interpretability. However enforcing these shape requirements in a hard fashion is an extremely challenging problem. Classically, this task is tackled (i) in a soft way (without out-of-sample guarantees), (ii) by specialized transformation of the variables on a case-by-case basis, or (iii) by using highly restricted function classes, such as polynomials or polynomial splines. In this paper, we prove that hard affine shape constraints on function derivatives can be encoded in kernel machines which represent one of the most flexible and powerful tools in machine learning and statistics. Particularly, we present a tightened second-order cone constrained reformulation, that can be readily implemented in convex solvers. We prove performance guarantees on the solution, and demonstrate the efficiency of the approach in joint quantile regression with applications to economics and to the analysis of aircraft trajectories, among others.
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