Hard Shape-Constrained Kernel Machines
This addresses the need for interpretable and reliable shape-constrained models in fields like economics and trajectory analysis, offering a flexible solution with proven guarantees, though it builds incrementally on existing kernel methods.
The paper tackled the problem of enforcing hard shape constraints (e.g., non-negativity, monotonicity, convexity) in kernel machines, which is challenging due to lack of out-of-sample guarantees or reliance on specialized methods. It proved that hard affine shape constraints on derivatives can be encoded in kernel machines, presenting a reformulation with performance guarantees and demonstrating efficiency in applications like joint quantile regression.
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.