Automated Kernel Discovery Towards Understanding High-dimensional Bayesian Optimization

Gaussian Process (GP) kernels are central to Bayesian optimization (BO), yet designing effective kernels for high-dimensional problems still relies on extensive manual engineering. Existing kernel design methods restrict the search to additive and multiplicative compositions of base kernels, while LLM-based BO approaches condition on raw observations, which are infeasible in high dimensions due to context-length constraints and the difficulty of extracting meaningful patterns from high-dimensional numerical observations. We introduce **Kernel Discovery**, an LLM-driven population-based evolutionary framework that overcomes both limitations. Motivated by the observation that directly prompting an LLM to generate kernel code yields syntactically varied but functionally identical kernels, we adopt a two-stage approach: an LLM first proposes novel mathematical forms, then a second LLM call converts each form into validated, executable code. We also propose a leave-one-out continuous ranked probability score (LOO-CRPS) as a held-out predictive scoring criterion that penalizes overconfident fits more directly than marginal log-likelihood. On five standard high-dimensional BO benchmarks, our method achieves an average rank of **1.2 out of 17**, consistently outperforming competitive baselines. We further analyze the discovered kernels to examine which kernel characteristics lead to superior performance in high-dimensional BO.