Standard Gaussian Process is All You Need for High-Dimensional Bayesian Optimization

Abstract: 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 Mat\'ern 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 Mat\'ern 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.

• The paper demonstrates that standard Gaussian Processes with Matérn kernels achieve competitive performance in high-dimensional Bayesian optimization by outperforming specialized methods on multiple benchmarks.
• The research evidences robust surrogate learning as standard GPs accurately model high-dimensional functions, aligning closely with ground truth on diverse datasets.
• The study highlights computational efficiencies through simplified training schemes such as MAP and MLE, reducing overhead compared to full Bayesian approaches without compromising performance.

An Analysis of Standard Gaussian Processes in High-Dimensional Bayesian Optimization

The longstanding perception within the optimization and machine learning research community is that Bayesian Optimization (BO) using standard Gaussian Processes (GPs)—termed here as standard BO—is ill-suited for high-dimensional optimization problems. This perception has been largely based on theoretical considerations about GPs struggling with high-dimensional covariance modeling without empirical backing. The paper by Xu and Zhe challenges this belief through a systematic empirical investigation, highlighting the potential of standard BO methods equipped with Matérn kernels and Upper Confidence Bound (UCB) acquisition functions to outperform specialized high-dimensional optimization frameworks.

Key Insights and Findings

The investigation by Xu and Zhe revolves around deploying standard GP in BO, especially focusing on high-dimensional optimization tasks that were previously thought to exceed the practical capabilities of such models. The main findings include:

1. Empirical Performance Benchmarking:

• The study evaluated standard BO using ARD Matérn kernels across eleven synthetic and real-world benchmarks, with optimization dimensions ranging from 30 to 388. It demonstrated consistent top-tier performance.
• The methods often surpassed state-of-the-art high-dimensional BO techniques, indicating that standard BO can excel even without enforcing low-dimensional structure assumptions, typically required by other frameworks.

2. Robust Surrogate Learning:

• Analysis revealed that standard GPs equipped with Matérn kernels are not only capable surrogates for learning high-dimensional functions but can also outperform complex models that incorporate low-rank structurization.
• The investigation into the prediction accuracy confirmed that the standard GP with Matérn kernels aligns closely with ground truth across various benchmark datasets, assuring its efficacy as a functional approximation tool.

3. Computational Considerations:

• The paper points out the efficiency of using simplified training schemes such as Maximum A Posteriori (MAP) estimation with diffuse priors or Maximum Likelihood Estimation (MLE). These approaches negate the need for computationally expensive methods like Markov Chain Monte Carlo (MCMC) without compromising the optimization performance of the BO.
• The research demonstrated that while full Bayesian approaches provide incremental benefits, they entail significant computational overheads.

Discussion and Implications

The paper by Xu and Zhe compels a reassessment of the efficacy of standard BO in high-dimensional settings, positioning it as a viable and perhaps preferable approach due to its robustness and simplicity. The insights from the empirical comparison not only challenge prevailing myths about the limitations of standard GPs but also suggest that the broader application of these methods could simplify optimization tasks in various fields.

The findings inspire future work to further investigate the broader adaptability of standard GPs in diverse and complex optimization landscapes and encourage exploration into theoretical justifications for the empirical success observed. Additionally, this research prompts investigations into hybrid methods that might blend the strengths of standard GPs with specialized model structures for enhanced high-dimensional function learning and optimization efficiency.

By motivating deeper inquiry into the latent capabilities of standard BO, this paper provides an invaluable contribution to the continuing evolution of high-dimensional optimization methodologies within artificial intelligence and machine learning disciplines. It emphasizes the importance of empirical validation and challenges researchers to rethink conventional wisdom regarding GPs and BO in high-dimensional spaces.