We provide the proof (Section A) of our main result presented in Section 3. Section B is about an additional numerical illustration in the context of kernel ridge regression on the importance of hard shape constraints in case of increasing level of noise. 'above' the affine hyperplane defined by normal vector Our results are summarized in Figure 1(b). To illustrate the tightening property of Theorem 3, i.e. that (n 1) (n 1) By construction both measures are zero for SOC. ( d 2)

Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard way (for example at all points of an interval) for rich function classes is a notoriously challenging problem. We propose a unified and modular convex optimization framework, relying on second-order cone (SOC) tightening, to encode hard affine SDP constraints on function derivatives, for models belonging to vector-valued reproducing kernel Hilbert spaces (vRKHSs). The modular nature of the proposed approach allows to simultaneously handle multiple shape constraints, and to tighten an infinite number of constraints into finitely many. We prove the consistency of the proposed scheme and that of its adaptive variant, leveraging geometric properties of vRKHSs. The efficiency of the approach is illustrated in the context of shape optimization, safety-critical control and econometrics.

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.