Standard Gaussian Process is All You Need for High-Dimensional Bayesian Optimization
This work addresses the problem of high-dimensional optimization for researchers and practitioners, offering a simple and effective solution that re-evaluates standard methods, though it is incremental in improving existing techniques.
The study challenges the belief that standard Gaussian Process-based Bayesian Optimization underperforms in high dimensions, showing that using Matérn kernels or a robust initialization for the Square Exponential kernel enables it to achieve top-tier results, often surpassing specialized methods, across twelve benchmarks.
A long-standing belief holds that Bayesian Optimization (BO) with standard Gaussian processes (GP) -- referred to as standard BO -- underperforms in high-dimensional optimization problems. While this belief seems plausible, it lacks both robust empirical evidence and theoretical justification. To address this gap, we present a systematic investigation. First, through a comprehensive evaluation across twelve benchmarks, we found that while the popular Square Exponential (SE) kernel often leads to poor performance, using Matérn kernels enables standard BO to consistently achieve top-tier results, frequently surpassing methods specifically designed for high-dimensional optimization. Second, our theoretical analysis reveals that the SE kernel's failure primarily stems from improper initialization of the length-scale parameters, which are commonly used in practice but can cause gradient vanishing in training. We provide a probabilistic bound to characterize this issue, showing that Matérn kernels are less susceptible and can robustly handle much higher dimensions. Third, we propose a simple robust initialization strategy that dramatically improves the performance of the SE kernel, bringing it close to state-of-the-art methods, without requiring additional priors or regularization. We prove another probabilistic bound that demonstrates how the gradient vanishing issue can be effectively mitigated with our method. Our findings advocate for a re-evaluation of standard BO's potential in high-dimensional settings.