A warped kernel improving robustness in Bayesian optimization via random embeddings

Mickaël Binois (1, 2) , David Ginsbourger (3, 4) , Olivier Roustant (3, 2)

1 Technocentre Renault [Guyancourt]
2 DEMO-ENSMSE - Département Décision en Entreprise : Modélisation, Optimisation
3 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
4 IMSV - Institute of Mathematical Statistics and Actuarial Science [Bern]

The scope of Bayesian Optimization methods is usually limited to moderate-dimensional problems. Wang et al. recently proposed to extend the applicability of these methods to up to billions of variables, when only few of them are actually influential, through the so-called Random EMbedding Bayesian Optimization (REMBO) approach. In REMBO, optimization is conducted in a low-dimensional domain Y, randomly embedded in the high-dimensional source space X. New points are chosen by maximizing the Expected Improvement (EI) criterion with Gaussian process (GP) models incorporating the considered embeddings via two kinds of covariance kernels. A first one relies on Euclidean distances in X. It delivers good performance in moderate dimension, albeit its main drawback is to remain high-dimensional so that the benefits of the method are limited. A second one is defined directly over Y and is therefore independent from the dimension of X. However, it has been shown to possess artifacts that may lead EI algorithms to spend many iterations exploring equivalent points. Here we propose a new kernel with a warping inspired by simple geometrical ideas, that retains key advantages of the first kernel while remaining of low dimension.

Mots clés: High-dimensional Bayesian optimization; Gaussian Processes; Warping; Kernel methods; Expected Improvement

Domaines: Optimisation et contrôle [math.OC]; Machine Learning [stat.ML]

Mickaël Binois, David Ginsbourger, Olivier Roustant. A warped kernel improving robustness in Bayesian optimization via random embeddings. 2014. ⟨hal-01078003v1⟩