Bayesian Optimization over Bounded Domains with the Beta Product Kernel

TL;DR

This paper introduces the Beta kernel for Bayesian optimization, which naturally models functions on bounded domains and outperforms traditional kernels like Matérn and RBF in various optimization tasks.

Contribution

The paper proposes a novel Beta kernel for Gaussian processes that effectively models bounded domains and demonstrates its superior performance over existing kernels.

Findings

Beta kernel exhibits exponential eigendecay rate.

Outperforms Matérn and RBF kernels in synthetic and real-world tasks.

Robustly models functions with optima near domain boundaries.

Abstract

Bayesian optimization with Gaussian processes (GP) is commonly used to optimize black-box functions. The Mat\'ern and the Radial Basis Function (RBF) covariance functions are used frequently, but they do not make any assumptions about the domain of the function, which may limit their applicability in bounded domains. To address the limitation, we introduce the Beta kernel, a non-stationary kernel induced by a product of Beta distribution density functions. Such a formulation allows our kernel to naturally model functions on bounded domains. We present statistical evidence supporting the hypothesis that the kernel exhibits an exponential eigendecay rate, based on empirical analyses of its spectral properties across different settings. Our experimental results demonstrate the robustness of the Beta kernel in modeling functions with optima located near the faces or vertices of the unit hypercube. The experiments show that our kernel consistently outperforms a wide range of kernels, including the well-known Mat\'ern and RBF, in different problems, including synthetic function optimization and the compression of vision and language models.