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arxiv: 2502.06178 · v5 · pith:N7DAL225new · submitted 2025-02-10 · 🧮 math.OC · cs.LG· stat.ML

Bayesian Optimization by Kernel Regression and Density-based Exploration

Tansheng Zhu , Hongyu Zhou , Ke Jin , Xusheng Xu , Qiufan Yuan , Lijie Ji This is my paper

Pith reviewed 2026-05-23 04:12 UTC · model grok-4.3

classification 🧮 math.OC cs.LGstat.ML
keywords Bayesian optimizationkernel regressionkernel density estimationglobal convergencequadratic complexityblack-box optimizationnoisy evaluationsconfidence bound
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The pith

BOKE replaces Gaussian processes with kernel regression and density estimation to reduce Bayesian optimization to quadratic cost while proving global convergence under noisy evaluations.

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

The paper presents BOKE as a method that approximates the black-box function via kernel regression and directs exploration via kernel density estimation, both folded into a confidence-bound acquisition rule. This design targets the cubic per-step cost of Gaussian processes that produces quartic total time as iterations grow. The authors supply a convergence proof that holds when observations contain noise. Experiments indicate that the resulting algorithm matches the solution quality of standard Bayesian optimization on synthetic and engineering tasks while running substantially faster. A reader would care because the change opens Bayesian optimization to problems whose scale previously made Gaussian-process methods impractical.

Core claim

BOKE integrates kernel regression for efficient function approximation and kernel density estimation for exploration into the confidence-bound acquisition function, thereby lowering per-iteration complexity from cubic to quadratic while establishing rigorous global convergence guarantees under noisy evaluations.

What carries the argument

Kernel regression paired with kernel density estimation inside the upper-confidence-bound acquisition criterion.

If this is right

  • BOKE performs competitively with Gaussian-process Bayesian optimization and other baselines on both synthetic and real-world tasks.
  • The method exhibits markedly lower wall-clock time, making it suitable for resource-constrained engineering settings.
  • Global convergence holds even when function evaluations are corrupted by noise.
  • Overall runtime scales quadratically rather than quartically with the number of iterations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same substitution technique could be tested on acquisition functions other than upper confidence bound.
  • Quadratic scaling may allow Bayesian optimization on data sets an order of magnitude larger than current Gaussian-process limits permit.
  • Density-based exploration might automatically adapt step sizes on problems with widely separated local optima.

Load-bearing premise

Replacing the Gaussian-process posterior with kernel regression and kernel density estimates inside the acquisition function still yields the same convergence guarantees that standard analyses require.

What would settle it

A single-dimensional noisy test function on which BOKE repeatedly converges to a suboptimal point while a Gaussian-process Bayesian optimizer reaches the known global optimum.

Figures

Figures reproduced from arXiv: 2502.06178 by Hongyu Zhou, Ke Jin, Lijie Ji, Qiufan Yuan, Tansheng Zhu, Xusheng Xu.

Figure 1
Figure 1. Figure 1: Kernel regression with Gaussian kernels at various bandwidths. The objective function Eq. (5) is sampled 10 times, with each sample evaluated incorporating additive noise ε ∼ N (0, 0.01). 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Kernel density estimation with Gaussian kernels at various bandwidths, where ˆp(x) = (2π) −1/2 (ℓt) −1Wt(x). The 20 samples are generated from p(x) = 0.7 Beta(3, 8) + 0.3 Beta(10, 2). Here, we propose the density-based exploration (DE) strategy for space-filling design, that is, xt+1 ∈ arg min x∈X Wt(x). (10) This means that a query sequence XT of size T ∈ Z+ is generated by sequentially minimizing the KDE… view at source ↗
Figure 3
Figure 3. Figure 3: Fill distances of different space-filling design methods across various dimensions. Each curve represents the mean result from 100 random seeds. Both axes are displayed using a logarith￾mic scale. function, which is defined as follows: at(x) = a(x; Dt) := m(x; Dt) + βtσˆ(x; Xt), (13) where t denotes the number of queried points, βt > 0 is a tuning parameter that varies with t, and σˆ is the density-based e… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the IKR-UCB acquisition function. The gray dashed line represents the analytical function, while the blue curve shows the KR approximation of the objective function. The upper light blue shaded area indicates the uncertainty quantification based on kernel density estimation. The lower shaded plot visualizes the acquisition function. Algorithm 2: BOKE algorithm Input: initial dataset DT0 , b… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of BOKE (top two rows) and BOKE+ (bottom two rows) on the one￾dimensional problem defined in Eq. (5). The gray dashed line represents the analytical function, while the blue curve shows the KR approximation of the objective function. The shaded plot visualizes the acquisition function. can be preprocessed during dataset fitting, whereas the inference phase involves the computations necessary for… view at source ↗
Figure 6
Figure 6. Figure 6: Simple regret with respect to noise-free evaluations on benchmark optimization func￾tions. Each curve is the mean of results from 30 random seeds. The left area of the dashed gray line represents the sampling points initialized by LHS. 5.1 Synthetic benchmark functions We empirically evaluate our method on a range of synthetic benchmark optimization tasks, including Forrester function (1D), Goldstein-Price… view at source ↗
Figure 7
Figure 7. Figure 7: Simple regret with respect to noisy evaluations on benchmark optimization functions. Each curve is the mean of results from 30 random seeds. The left area of the dashed gray line represents the sampling points initialized by LHS. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simple regret with respect to both iteration step and average CPU time on the sprinkler computer problem (8D). Each curve is the mean of results from 30 random seeds. 5.3 Hyperparameter tuning We empirically evaluate our method on hyperparameter optimization for random forest (RF) and XGBoost (XGB) regression tasks using datasets including California Housing [48] and Wine Quality [16]. For both problems, w… view at source ↗
Figure 9
Figure 9. Figure 9: Goodness-of-fit measure on the random forest tuning (5D) and the XGBoost tuning problems (7D). Each curve is the mean of results from 30 random seeds. the best. Conversely, PseudoBO, KR-UCB, and GP-UCB perform worst in this task ( [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

Bayesian optimization is highly effective for optimizing expensive-to-evaluate black-box functions, but it faces significant computational challenges due to the cubic per-iteration cost of Gaussian processes, which results in a total time complexity that is quartic with respect to the number of iterations. To address this limitation, we propose a novel algorithm, Bayesian optimization by kernel regression and density-based exploration (BOKE). BOKE uses kernel regression for efficient function approximation, kernel density for exploration, and integrates them into the confidence bound criteria to guide the optimization process, thus reducing computational costs to quadratic. Our theoretical analysis rigorously establishes the global convergence of BOKE under noisy evaluations. Through extensive numerical experiments on both synthetic and real-world optimization tasks, we demonstrate that BOKE not only performs competitively compared to Gaussian process-based methods and several other baseline methods but also exhibits superior computational efficiency. These results highlight BOKE's effectiveness in resource-constrained environments, providing a practical approach for optimization problems in engineering applications.

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

2 major / 1 minor

Summary. The manuscript proposes BOKE, an algorithm replacing Gaussian process posteriors in Bayesian optimization with kernel regression for mean/variance approximation and kernel density estimation for an exploration term, both inserted into a confidence-bound acquisition function. It claims this reduces per-iteration cost such that total complexity is quadratic in the number of evaluations, rigorously proves global convergence under noisy black-box evaluations, and reports competitive empirical performance versus GP-based BO and baselines on synthetic and real tasks.

Significance. A correctly proved result would be significant: it would supply a scalable, non-GP Bayesian optimization method whose acquisition function still guarantees global convergence, directly addressing the O(n^4) bottleneck while retaining theoretical backing. The claimed quadratic complexity and noise-robust convergence, if established with explicit error control, would be a useful contribution to the optimization literature.

major comments (2)
  1. [§4 (Convergence Analysis)] §4 (Convergence Analysis), Theorem 1 (or equivalent statement of global convergence): the proof does not derive explicit bounds on the deviation between the kernel-regression estimates and the true posterior mean/variance, nor on the bias introduced by the KDE exploration term inside the acquisition function. Standard UCB regret analyses rely on controlling these quantities to ensure the modified acquisition still forces sufficient exploration; without such rates showing that the total approximation error vanishes fast enough relative to the information gain, the claimed extension of existing theory does not hold.
  2. [§5 (Numerical Experiments)] §5 (Numerical Experiments), Tables 1–3 and associated text: the experimental protocol omits the number of independent runs, the precise noise model and variance schedule, the cross-validation or selection procedure for kernel bandwidths in both regression and KDE, and any statistical significance testing. These omissions make it impossible to verify the competitiveness and efficiency claims against the GP baselines.
minor comments (1)
  1. [§2–3] Notation for the kernel regression estimator and the density-based bonus term is introduced without a dedicated preliminary section; a short subsection defining all symbols before the algorithm would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [§4 (Convergence Analysis)] §4 (Convergence Analysis), Theorem 1 (or equivalent statement of global convergence): the proof does not derive explicit bounds on the deviation between the kernel-regression estimates and the true posterior mean/variance, nor on the bias introduced by the KDE exploration term inside the acquisition function. Standard UCB regret analyses rely on controlling these quantities to ensure the modified acquisition still forces sufficient exploration; without such rates showing that the total approximation error vanishes fast enough relative to the information gain, the claimed extension of existing theory does not hold.

    Authors: We agree that explicit error bounds are necessary to rigorously extend the UCB convergence analysis to our approximated acquisition function. The current Section 4 provides a high-level argument but lacks the detailed deviation rates. In the revised version, we will add a new lemma deriving bounds on the kernel regression approximation error to the GP posterior (using properties of kernel ridge regression) and on the KDE bias term. We will show that with bandwidth h_t scaling appropriately (e.g., h_t ~ t^{-1/(d+4)}), the total error is sufficiently small relative to the information gain term, preserving the sublinear regret and thus global convergence. This will be incorporated into an updated proof of Theorem 1. revision: yes

  2. Referee: [§5 (Numerical Experiments)] §5 (Numerical Experiments), Tables 1–3 and associated text: the experimental protocol omits the number of independent runs, the precise noise model and variance schedule, the cross-validation or selection procedure for kernel bandwidths in both regression and KDE, and any statistical significance testing. These omissions make it impossible to verify the competitiveness and efficiency claims against the GP baselines.

    Authors: We acknowledge these omissions in the experimental description. In the revision, we will update Section 5 to include: the number of independent runs (20 runs per task), the noise model (additive Gaussian noise with fixed variance 0.01 for synthetic functions, and task-specific for real-world), the bandwidth selection via 5-fold cross-validation on a held-out validation set for both kernel regression and KDE, and statistical tests (reporting mean and standard deviation, with p-values from paired t-tests comparing to baselines). The tables will be expanded to reflect these details and ensure reproducibility. revision: yes

Circularity Check

0 steps flagged

No circularity; convergence established by analysis rather than by construction or self-citation.

full rationale

The provided abstract and context present BOKE as using kernel regression and KDE to modify the acquisition function, with a separate theoretical analysis claimed to prove global convergence under noise. No equations, self-citations, or fitted quantities are quoted that reduce the convergence result to an input by definition. The derivation chain is described as independent analysis controlling approximation errors, which is the standard non-circular structure for such algorithmic papers. Absence of explicit reduction steps in the visible text supports a finding of no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities can be extracted from the given text.

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