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arxiv: 2605.20249 · v1 · pith:ZYR3OH67new · submitted 2026-05-18 · 💻 cs.LG · cs.AI

Automated Kernel Discovery Towards Understanding High-dimensional Bayesian Optimization

Taeyoung Yun , Woocheol Shin , Inhyuck Song , Jaewoo Lee , Jinkyoo Park This is my paper

Pith reviewed 2026-05-21 08:22 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords kernel discoverybayesian optimizationgaussian processeshigh-dimensional optimizationlarge language modelsevolutionary algorithmsautomated machine learning
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The pith

An LLM-driven evolutionary framework discovers novel Gaussian process kernels for high-dimensional Bayesian optimization without conditioning on raw observations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an automated method called Kernel Discovery to design kernels for Gaussian processes used in Bayesian optimization when inputs have many dimensions. Manual kernel design is difficult in these settings, and prior automated approaches are restricted to simple combinations of base kernels or must feed raw data into language models, which hits practical limits. The new approach prompts large language models in two stages to first suggest new mathematical kernel expressions and then convert those expressions into working code. Kernels are then chosen using a leave-one-out continuous ranked probability score that discourages overfitting. Tests on five standard high-dimensional benchmarks show the method ranks first on average among seventeen competing approaches.

Core claim

We introduce Kernel Discovery, a LLM-driven evolutionary framework for high-dimensional BO that searches a broader kernel space beyond predefined composition rules and does not require conditioning on observations. 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 selection criterion that penalizes overfitted kernels. On five high-dimensional BO benchmarks, our

What carries the argument

The two-stage LLM process in which one model proposes novel mathematical kernel forms and a second model turns them into executable code, together with LOO-CRPS selection that penalizes overfitting.

Load-bearing premise

Large language models can reliably propose mathematically novel and functionally distinct kernel forms without access to raw observations, and LOO-CRPS will select kernels that generalize rather than overfit in high-dimensional regimes.

What would settle it

Running Kernel Discovery on a new high-dimensional Bayesian optimization benchmark and checking whether the average rank across methods stays near 1.2 or whether the discovered kernels fail to beat standard automatic relevance determination kernels.

Figures

Figures reproduced from arXiv: 2605.20249 by Inhyuck Song, Jaewoo Lee, Jinkyoo Park, Taeyoung Yun, Woocheol Shin.

Figure 1
Figure 1. Figure 1: Motivation Figure. (Left): In high-dimensional BO, a conventional LLM-based BO does not work due to the out-of-context limits and the difficulty of extracting meaningful patterns. (Right): Our pipeline enables us to discover effective kernels for high-dimensional BO beyond compositions. these breakthroughs are promising, identifying such effective priors or transformations still requires extensive manual e… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the method. Given an initial population, we instruct an LLM to generate novel [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Prompts for Kernel Discovery. To discover novel kernels for high-dimensional BO, we [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Learning curve of various methods across standard benchmarks. We visualize the top-2 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ablation studies on the two-stage approach for kernel discovery. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Ablation on each compo￾nent of Kernel Discovery [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Post-analysis on kernel discovery. Both experiments are conducted in SVM benchmark. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mathematical form discovery prompt template [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Code conversion prompt template. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Composition prompt template. Kernel Validation Failure Rates. The failure probability in Tables 2 and 3 denotes the empirical fraction of LLM-generated kernels rejected by our structural validator. A generated kernel knew is accepted only when V(knew) = Vagn(knew) ∧ Vpsd(knew) = 1. Here, Vagn checks whether the generated kernel implementation is dimension- and shape-agnostic, as required by the BO pipelin… view at source ↗
Figure 11
Figure 11. Figure 11: Full learning curves for all methods across five benchmarks. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Functional redundancy example from direct code generation. Both snippets compute [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Functional redundancy example from direct code generation. Both snippets compute [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Functional redundancy example from direct code generation. Both snippets compute [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Ablation studies on each component of our method across other benchmarks. [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Ablation studies on the base population for Compositional Search and CAKE baselines. [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Ablation studies on the evaluation metric for Compositional Search and CAKE baselines. [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Representative cross-covariance failures caught by [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Representative PSD failures caught by Vpsd. Both include terms that do not generally preserve positive semi-definiteness and can yield indefinite Gram matrices. Problematic terms are highlighted in red. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Forward code for the discovered kernel k1. 1 def forward ( self , x1 , x2 , diag = False , ** params ): 2 if x1 . dim () == 1: x1 = x1 . unsqueeze ( -1) 3 if x2 . dim () == 1: x2 = x2 . unsqueeze ( -1) 4 arc_weight = self . raw_arc_weight_constraint . transform ( self . raw_arc_weight ) 5 global_scale = self . raw_global_scale_constraint . transform ( self . raw_global_scale ) 6 angular_weights = self . r… view at source ↗
Figure 21
Figure 21. Figure 21: Forward code for the discovered kernel k2. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Forward code for the discovered kernel k3. Evolving Population on Other Benchmarks [PITH_FULL_IMAGE:figures/full_fig_p031_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Post-analysis of kernel discovery on four additional benchmarks. For each benchmark, we [PITH_FULL_IMAGE:figures/full_fig_p031_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Analysis on boundary-seeking behavior of several kernels. [PITH_FULL_IMAGE:figures/full_fig_p033_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Code snippet of the discovered kernel from SVM benchmark. [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Code snippet of an additional discovered kernel with strong transferability. This kernel [PITH_FULL_IMAGE:figures/full_fig_p035_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Code snippet of another additional discovered kernel with strong transferability. This [PITH_FULL_IMAGE:figures/full_fig_p036_27.png] view at source ↗
read the original abstract

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 automated approaches struggle in high dimensions for two bottlenecks: their kernel search space is limited to additions and multiplications of base kernels, and LLM-based approaches require conditioning on raw observations, which becomes infeasible due to context-length limits and the difficulty of extracting meaningful patterns. We introduce \textbf{Kernel Discovery}, a LLM-driven evolutionary framework for high-dimensional BO that searches a broader kernel space beyond predefined composition rules and does not require conditioning on observations. 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 selection criterion that penalizes overfitted kernels. On five high-dimensional BO benchmarks, our method achieves an average rank of \textbf{1.2 out of 17}, outperforming competitive baselines. We further analyze the discovered kernels to identify which kernels lead to improvements in high-dimensional BO.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces Kernel Discovery, an LLM-driven evolutionary framework for automated kernel design in high-dimensional Bayesian optimization. It uses a two-stage process in which one LLM proposes novel mathematical kernel forms and a second converts them to executable code, avoiding direct observation conditioning to sidestep context-length limits. A leave-one-out continuous ranked probability score (LOO-CRPS) serves as the selection criterion to penalize overfitting. The method is evaluated on five high-dimensional BO benchmarks, where it reports an average rank of 1.2 out of 17 competing approaches, and the discovered kernels are further analyzed to identify properties that improve high-dimensional BO performance.

Significance. If the performance claims are substantiated, the work offers a practical advance in automated kernel engineering for GPs in regimes where manual design is intractable. The two-stage LLM proposal and LOO-CRPS selection address documented bottlenecks in prior search-based and LLM-based kernel methods. Credit is due for the broader kernel space explored beyond fixed composition rules and for the explicit analysis of which kernel characteristics benefit high-dimensional BO. The empirical nature of the contribution makes rigorous experimental reporting essential to its significance.

major comments (3)
  1. [§4] §4 (Experimental results): the headline claim of average rank 1.2 out of 17 is presented without the number of independent runs, standard deviations across runs, or any statistical significance tests (e.g., paired Wilcoxon or Friedman tests) against the 16 baselines. This information is load-bearing for the central performance assertion.
  2. [§3.3] §3.3 (LOO-CRPS selection): the criterion is motivated as penalizing overfit kernels, yet the manuscript provides no ablation or held-out generalization test demonstrating that LOO-CRPS prefers kernels that extrapolate rather than fit noise in the sparse high-dimensional regime. Given the skeptic concern that leave-one-out on limited data may still reward excessive flexibility, this validation is required to support the selection mechanism.
  3. [§3.2] §3.2 (two-stage proposal): the claim that the two-stage LLM process yields functionally distinct kernels is central to the broader search space argument, but the paper does not quantify functional diversity (e.g., via kernel Gram-matrix distances or predictive behavior on synthetic functions) or describe explicit diversity maintenance in the evolutionary loop.
minor comments (2)
  1. [Abstract / §4] The abstract and §4 refer to “five high-dimensional BO benchmarks” without naming them or providing a summary table of their dimensions and characteristics; explicit identification would aid reproducibility.
  2. [§3.3] Notation for the LOO-CRPS formula in §3.3 could be clarified by explicitly stating the predictive distribution used for each left-out point.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. The comments highlight important aspects for strengthening the experimental rigor and supporting arguments. We address each major comment point by point below, indicating the revisions we plan to make.

read point-by-point responses

  1. Referee: [§4] §4 (Experimental results): the headline claim of average rank 1.2 out of 17 is presented without the number of independent runs, standard deviations across runs, or any statistical significance tests (e.g., paired Wilcoxon or Friedman tests) against the 16 baselines. This information is load-bearing for the central performance assertion.

    Authors: We agree that the experimental reporting in §4 requires these details to fully substantiate the performance claims. In the revised manuscript, we will explicitly state the number of independent runs, report standard deviations across runs, and include statistical significance tests such as the Friedman test with post-hoc analysis to compare our method against the baselines. revision: yes

  2. Referee: [§3.3] §3.3 (LOO-CRPS selection): the criterion is motivated as penalizing overfit kernels, yet the manuscript provides no ablation or held-out generalization test demonstrating that LOO-CRPS prefers kernels that extrapolate rather than fit noise in the sparse high-dimensional regime. Given the skeptic concern that leave-one-out on limited data may still reward excessive flexibility, this validation is required to support the selection mechanism.

    Authors: We appreciate the referee's concern regarding the validation of LOO-CRPS. While the criterion is intended to penalize overfitting through its leave-one-out predictive evaluation, we agree that additional evidence would strengthen the case. In the revised manuscript, we will add an ablation study in §3.3 comparing LOO-CRPS to other selection criteria, along with a held-out generalization test on synthetic high-dimensional data to show that it favors kernels with better extrapolation properties. revision: yes

  3. Referee: [§3.2] §3.2 (two-stage proposal): the claim that the two-stage LLM process yields functionally distinct kernels is central to the broader search space argument, but the paper does not quantify functional diversity (e.g., via kernel Gram-matrix distances or predictive behavior on synthetic functions) or describe explicit diversity maintenance in the evolutionary loop.

    Authors: We thank the referee for this observation. To better support the claim that the two-stage process enables a functionally broader kernel space, we will revise §3.2 to include quantitative analysis of functional diversity, for example by computing distances between kernel Gram matrices evaluated on synthetic functions and by examining predictive performance differences. We will also describe the diversity-promoting aspects of the evolutionary loop, such as the mutation and selection procedures. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical search procedure with external benchmark evaluation

full rationale

The paper introduces an LLM-driven evolutionary framework for automated kernel discovery in high-dimensional BO. It relies on a two-stage proposal process and LOO-CRPS selection, then reports comparative performance (avg rank 1.2/17) on five external benchmarks. No mathematical derivations, first-principles predictions, or fitted parameters are presented that could reduce to self-definitions or self-citations. The central claims rest on algorithmic description plus independent experimental results rather than any internal consistency loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the unverified capability of current LLMs to generate useful novel kernel expressions and on the effectiveness of LOO-CRPS as a generalizable selection metric; no explicit free parameters or invented physical entities are described.

axioms (2)
  • domain assumption LLMs prompted directly produce syntactically varied but functionally identical kernels, motivating the two-stage separation of form proposal and code generation.
    Stated as motivation in the abstract for adopting the two-stage approach.
  • domain assumption LOO-CRPS penalizes overfitting kernels more effectively than standard likelihood in high-dimensional BO settings.
    Introduced as the selection criterion without further justification in the abstract.

pith-pipeline@v0.9.0 · 5760 in / 1408 out tokens · 44341 ms · 2026-05-21T08:22:27.705539+00:00 · methodology

discussion (0)

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