Sequential construction and dimension reduction of Gaussian processes under inequality constraints

Abstract

Accounting for inequality constraints, such as boundedness, monotonicity or convexity, is challenging when modeling costly-to-evaluate black box functions. In this regard, finite-dimensional Gaussian process (GP) models bring a valuable solution, as they guarantee that the inequality constraints are satisfied everywhere. Nevertheless, these models are currently restricted to small dimensional situations (up to dimension 5). Addressing this issue, we introduce the MaxMod algorithm that sequentially inserts one-dimensional knots or adds active variables, thereby performing at the same time dimension reduction and efficient knot allocation. We prove the convergence of this algorithm. In intermediary steps of the proof, we propose the notion of multi-affine extension and study its properties. We also prove the convergence of finite-dimensional GPs, when the knots are not dense in the input space, extending the recent literature. With simulated and real data, we demonstrate that the MaxMod algorithm remains efficient in higher dimension (at least in dimension 20), and has a smaller computational complexity than other constrained GP models from the state-of-the-art, to reach a given approximation error.

Notes

• François Bachoc is an Assistant professor in statistics at the Toulouse Mathematics Institute and at the University Paul Sabatier.. Personal website can be found here.