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arxiv: 2512.00170 · v2 · submitted 2025-11-28 · 💻 cs.LG · stat.ML

We Still Don't Understand High-Dimensional Bayesian Optimization

Colin Doumont , Donney Fan , Natalie Maus , Jacob R. Gardner , Henry Moss , Geoff Pleiss This is my paper

Pith reviewed 2026-05-17 03:21 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords optimizationbayesianlinearhigh-dimensionalmethodstasksadvantagesafford
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The pith

Simple linear models match state-of-the-art performance in high-dimensional Bayesian optimization after a geometric fix.

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

The paper establishes that methods for high-dimensional Bayesian optimization that encode assumptions such as locality, sparsity, or smoothness are outperformed by Bayesian linear regression. After a geometric transformation to prevent boundary-seeking behavior, Gaussian processes with linear kernels achieve comparable results on tasks spanning 60 to 6,000 dimensions. This challenges prior intuitions about the necessity of complex structural encodings and highlights practical advantages of linear models, including closed-form sampling and linear scaling with data size, which enable handling of molecular optimization problems with over 20,000 observations.

Core claim

Existing high-dimensional Bayesian optimization methods aim to overcome the curse of dimensionality by carefully encoding structural assumptions, from locality to sparsity to smoothness, into the optimization procedure. Surprisingly, these approaches are outperformed by arguably the simplest method imaginable: Bayesian linear regression. After applying a geometric transformation to avoid boundary-seeking behavior, Gaussian processes with linear kernels match state-of-the-art performance on tasks with 60- to 6,000-dimensional search spaces. Linear models offer numerous advantages over their non-parametric counterparts: they afford closed-form sampling and their computation scales linearly, a

What carries the argument

Geometric transformation applied to Gaussian processes with linear kernels to avoid boundary-seeking behavior

Load-bearing premise

The selected benchmark tasks and the molecular optimization problem represent the high-dimensional regimes where complex structural assumptions were previously considered necessary, and the transformation does not systematically favor linear models.

What would settle it

A new high-dimensional optimization benchmark in which linear-kernel Gaussian processes with the geometric transformation fall short of methods that encode locality or sparsity would falsify the performance-matching claim.

Figures

Figures reproduced from arXiv: 2512.00170 by Colin Doumont, Donney Fan, Geoff Pleiss, Henry Moss, Jacob R. Gardner, Natalie Maus.

Figure 1
Figure 1. Figure 1: Our linear kernel on spherically-mapped inputs matches state-of-the-art high-dimensional BO performance. (Left): benchmarks with N ≈ D evaluation budgets. While standard linear kernels can fail to make any optimization progress, our modified kernel matches or exceeds competitive methods. (Right): benchmarks with N ≫ D. The natural scalability of linear kernels, coupled with the improved optimization perfor… view at source ↗
Figure 3
Figure 3. Figure 3: Ablation over the spherical mapping func [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Differences in the acquired inputs of standard versus modified linear models. (Left): the “boundary %” depicts, for each acquisition xt, how many dimensions of the vector lie on the boundary of X (i.e. ±1). Standard linear models lead to acquisitions with points nearly in the corners of the hypercube (i.e. 100% of dimensions on the boundary). Our linear model acquires non-corner points (≈ 75% boundary on M… view at source ↗
Figure 5
Figure 5. Figure 5: Spherical mappings affect BO perfor￾mance, but not supervised regression perfor￾mance. (Left): predictive RMSE of GPs trained and tested on quasi-random data. The standard and mod￾ified linear models make worse predictions than the RBF model. (Right): predictive RMSE on adaptively￾chosen BO acquisitions. Unlike with random data, linear models match the ability of RBF models for pre￾dictions at future BO ac… view at source ↗
Figure 6
Figure 6. Figure 6: When applied to our spherical mapping with global lengthscale, both linear and RBF kernels are [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The spherical projection and centering of the hypercube search-space play a key role in our proposed [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Our proposed linear kernel from Equation (2) significantly outperforms Vanilla BO on molecular tasks [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Extension of Figure 2 (high-order polynomials do not improve upon our linear model). [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Extension of Figure 3 (the choice of spherical projection can have a drastic impact on performance). [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Extension of Figure 4 (our linear kernel exhibits “locality”, similar to Vanilla BO and unlike the [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Extension of Figure 4 (standard linear kernels lead to acquisitions in the corners of the hypercube, [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Extension of Figure 5 (our spherical projection does not lead to better predictive performance for the [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
read the original abstract

Existing high-dimensional Bayesian optimization (BO) methods aim to overcome the curse of dimensionality by carefully encoding structural assumptions, from locality to sparsity to smoothness, into the optimization procedure. Surprisingly, we demonstrate that these approaches are outperformed by arguably the simplest method imaginable: Bayesian linear regression. After applying a geometric transformation to avoid boundary-seeking behavior, Gaussian processes with linear kernels match state-of-the-art performance on tasks with 60- to 6,000-dimensional search spaces. Linear models offer numerous advantages over their non-parametric counterparts: they afford closed-form sampling and their computation scales linearly with data, a fact we exploit on molecular optimization tasks with >20,000 observations. Coupled with empirical analyses, our results suggest the need to depart from past intuitions about BO methods in high-dimensions.

Editorial analysis

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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

2 major / 2 minor

Summary. The paper claims that high-dimensional Bayesian optimization methods relying on structural assumptions (locality, sparsity, smoothness) are outperformed by the simplest alternative: Bayesian linear regression. After a geometric transformation to mitigate boundary-seeking behavior, Gaussian processes with linear kernels match state-of-the-art performance on benchmark tasks with 60–6,000-dimensional search spaces. The authors further demonstrate computational advantages of linear models, including closed-form sampling and linear scaling with data size, on a molecular optimization task involving >20,000 observations, and argue that these results challenge prevailing intuitions about the necessity of complex models in high-dimensional BO.

Significance. If the central empirical claim holds under fair and reproducible comparisons, the work would be significant for the Bayesian optimization community. It provides a direct counterexample to the assumption that high-dimensional problems inherently require sophisticated structural encodings, potentially simplifying algorithm design and enabling better scalability. The emphasis on linear scaling and closed-form properties is a concrete practical strength for large-scale applications such as molecular design. The results could prompt re-examination of why simple linear models succeed in regimes previously thought to demand non-parametric complexity.

major comments (2)
  1. [Abstract and Section 4] Abstract and Section 4 (Experimental Setup): The central claim that linear-kernel GPs match SOTA after the geometric transformation is load-bearing on the transformation being a neutral preprocessing step available to every baseline. The manuscript does not explicitly state whether the same geometric mapping was applied uniformly to locality-, sparsity-, and smoothness-based competitors. If the transformation interacts favorably only with the closed-form properties of the linear model, the results demonstrate the value of the preprocessing rather than the sufficiency of linear kernels without complex assumptions.
  2. [Section 5] Section 5 (Results on Molecular Optimization): The reported linear scaling with >20,000 observations is a key advantage, but the manuscript should include an ablation or timing comparison against at least one non-linear high-dimensional baseline under identical data regimes to substantiate that the scaling benefit is not offset by poorer sample efficiency.
minor comments (2)
  1. [Figure 2] Figure 2 and associated caption: axis labels and legend entries should explicitly indicate whether the geometric transformation was used for each plotted method.
  2. [Abstract] The abstract states 'match state-of-the-art performance' but the main text should report exact quantitative gaps (e.g., mean and standard deviation over repeated runs) rather than qualitative statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments. We address each major point below with clarifications and indicate where revisions will be made to improve the manuscript.

read point-by-point responses

  1. Referee: [Abstract and Section 4] Abstract and Section 4 (Experimental Setup): The central claim that linear-kernel GPs match SOTA after the geometric transformation is load-bearing on the transformation being a neutral preprocessing step available to every baseline. The manuscript does not explicitly state whether the same geometric mapping was applied uniformly to locality-, sparsity-, and smoothness-based competitors. If the transformation interacts favorably only with the closed-form properties of the linear model, the results demonstrate the value of the preprocessing rather than the sufficiency of linear kernels without complex assumptions.

    Authors: We appreciate this observation on ensuring fair comparisons. The geometric transformation is a general preprocessing step intended to mitigate boundary-seeking behavior and is applicable to any method. In the reported experiments, it was applied uniformly across all baselines, including those based on locality, sparsity, and smoothness assumptions. We will revise Section 4 to explicitly state this uniform application and describe the implementation details for non-linear methods to remove any ambiguity. revision: yes

  2. Referee: [Section 5] Section 5 (Results on Molecular Optimization): The reported linear scaling with >20,000 observations is a key advantage, but the manuscript should include an ablation or timing comparison against at least one non-linear high-dimensional baseline under identical data regimes to substantiate that the scaling benefit is not offset by poorer sample efficiency.

    Authors: We agree that an explicit timing comparison would further substantiate the practical advantages. While sample efficiency is already compared on the benchmark tasks in Sections 4 and earlier, Section 5 emphasizes computational scaling on large datasets where cubic-scaling methods become infeasible. We will add a timing ablation against one representative non-linear high-dimensional baseline on a subsampled version of the molecular dataset to directly address this concern. revision: yes

Circularity Check

0 steps flagged

Empirical demonstration with no load-bearing derivation

full rationale

The paper advances an empirical claim that Bayesian linear regression (Gaussian processes with linear kernels) plus a geometric transformation matches state-of-the-art high-dimensional BO performance on 60- to 6000-dimensional tasks. This rests on direct benchmark comparisons and scaling observations rather than any first-principles derivation, fitted-parameter prediction, or self-citation chain that reduces the result to its own inputs by construction. No equations or uniqueness theorems are invoked that would create self-definitional or fitted-input circularity; the central result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper's central claim is supported by empirical benchmarking rather than new theoretical derivations, so it introduces no free parameters beyond standard model fitting, relies on conventional domain assumptions of Gaussian processes, and postulates no new entities.

axioms (1)
  • domain assumption Standard assumptions of Gaussian process regression and Bayesian optimization remain valid when the kernel is restricted to linear form in high dimensions.
    The comparison implicitly treats these modeling assumptions as holding for the linear case without additional justification.

pith-pipeline@v0.9.0 · 5433 in / 1322 out tokens · 32672 ms · 2026-05-17T03:21:56.671621+00:00 · methodology

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