Hard Shape-Constrained Kernel Machines

arXiv2005.12636MaRDI QIDQ6341433

Pierre-Cyril Aubin-Frankowski, Zoltan Szabo

Publication date: 26 May 2020

Abstract: 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.

Has companion code repository: https://github.com/PCAubin/Hard-Shape-Constraints-for-Kernels

Mathematics Subject Classification ID

Nonparametric regression and quantile regression (62G08) Convex programming (90C25) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22)

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