Mixed Variable Bayesian Optimization with Frequency Modulated Kernels

TL;DR

This paper introduces a frequency modulated kernel for Gaussian Process-based Bayesian optimization that effectively models dependencies in mixed variable spaces, leading to improved sample efficiency and superior performance in complex optimization tasks.

Contribution

The paper proposes a novel frequency modulated kernel for Gaussian Processes, enabling better modeling of mixed variable dependencies in Bayesian optimization.

Findings

BO-FM outperforms competitors on synthetic and hyperparameter optimization problems.

Frequency modulation principle significantly improves sample efficiency.

BO-FM surpasses RE and BOHB even with fewer evaluations.

Abstract

The sample efficiency of Bayesian optimization(BO) is often boosted by Gaussian Process(GP) surrogate models. However, on mixed variable spaces, surrogate models other than GPs are prevalent, mainly due to the lack of kernels which can model complex dependencies across different types of variables. In this paper, we propose the frequency modulated (FM) kernel flexibly modeling dependencies among different types of variables, so that BO can enjoy the further improved sample efficiency. The FM kernel uses distances on continuous variables to modulate the graph Fourier spectrum derived from discrete variables. However, the frequency modulation does not always define a kernel with the similarity measure behavior which returns higher values for pairs of more similar points. Therefore, we specify and prove conditions for FM kernels to be positive definite and to exhibit the similarity measure behavior. In experiments, we demonstrate the improved sample efficiency of GP BO using FM kernels (BO-FM).On synthetic problems and hyperparameter optimization problems, BO-FM outperforms competitors consistently. Also, the importance of the frequency modulation principle is empirically demonstrated on the same problems. On joint optimization of neural architectures and SGD hyperparameters, BO-FM outperforms competitors including Regularized evolution(RE) and BOHB. Remarkably, BO-FM performs better even than RE and BOHB using three times as many evaluations.