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arxiv: 2512.14263 · v2 · submitted 2025-12-16 · 💻 cs.LG · cs.AI· math.OC

DT-PBO: an Interpretable Tree-based Surrogate Model for Preferential Bayesian Optimization

Nick Leenders , Thomas Quadt , Boris Cule , Roy Lindelauf , Herman Monsuur , Joost van Oijen , Mark Voskuijl This is my paper

Pith reviewed 2026-05-16 21:56 UTC · model grok-4.3

classification 💻 cs.LG cs.AImath.OC
keywords preferential Bayesian optimizationdecision tree surrogateinterpretable surrogate modelspairwise comparisonsBayesian optimizationLaplace approximation
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The pith

A tree-based surrogate matches Gaussian process performance in preferential Bayesian optimization while adding direct interpretability from pairwise data.

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

Preferential Bayesian optimization identifies a preferred solution through minimal pairwise comparisons. Standard approaches rely on Gaussian process surrogates that deliver strong results but hide the preference structure. DT-PBO replaces the Gaussian process with shallow decision trees built by a new splitting rule on the comparison data and uses Laplace approximation to assign probabilities to each leaf. The resulting model reaches comparable convergence on benchmark functions, especially rugged ones, while remaining transparent and computationally light. Experiments also confirm noise robustness and the ability to surface explicit preference rules that Gaussian process models obscure.

Core claim

DT-PBO constructs interpretable shallow decision trees directly from pairwise comparison data via a novel splitting heuristic and obtains probabilistic estimates at each leaf through Laplace approximation, enabling efficient preference modeling without Gaussian processes.

What carries the argument

The novel splitting heuristic that constructs shallow decision trees directly from pairwise comparison data.

If this is right

  • Competitive convergence to GP-based PBO across eight benchmark functions.
  • Stronger performance on functions with rugged optimization landscapes.
  • Robustness to noise in the comparison data.
  • Faster running time than GP-based alternatives.
  • Direct inspection of preference structure through the learned tree rules.

Where Pith is reading between the lines

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

  • The tree representation could let domain experts inspect and manually correct modeled preferences before further queries.
  • Because trees are built from comparisons alone, the method may extend naturally to settings where only ordinal feedback is available.
  • Leaf probabilities from the Laplace step could be reused to guide active query selection without retraining a full covariance model.

Load-bearing premise

The splitting heuristic produces shallow trees whose leaves capture enough of the preference structure for optimization to succeed without a Gaussian process.

What would settle it

A benchmark function or real dataset where the tree model diverges from the true preference ordering while a Gaussian process model recovers it.

Figures

Figures reproduced from arXiv: 2512.14263 by Boris Cule, Herman Monsuur, Joost van Oijen, Mark Voskuijl, Nick Leenders, Roy Lindelauf, Thomas Quadt.

Figure 1
Figure 1. Figure 1: A schematic overview of the PBO loop with the tree￾based surrogate model being depicted in the red rectangle. At the start of the process, randomly sampled initial comparisons are asked to the decision-maker and the model is fitted. The process ends when the decision-maker is satisfied with the final solution. taste of a slice of cake as 7? Why not 8? On the other hand, expressing preferences through pairw… view at source ↗
Figure 2
Figure 2. Figure 2: Average Regret over 20 runs of the DT-qEUBO, GP-qEUBO and SkewGP with HB-EI models. The methods are tested on, from left to right, top to bottom, eight increasingly spiky benchmark functions. of Takeno et al. (2023) is used and for GP the implemen￾tation in Botorch (Balandat et al., 2020) is used. Since the implementations of qEUBO and SkewGP are incompatible due to package dependencies, the comparison to … view at source ↗
Figure 3
Figure 3. Figure 3: Running time of DT, GP, and SkewGP for the Levy2D and Michalewicz5D optimization functions. on the 5D function. The results show that our models scales much better in dataset size and can therefore be used on much larger datasets than the GP-based alternatives. 5. Application: Sushi preference dataset To assess the interpretability and capability of handling categorical data of our DT-based method, we appl… view at source ↗
Figure 4
Figure 4. Figure 4: ρ-regret for sushi preference without user features for DT and GP. Kendall Tau is also plotted for reference. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An example plot of an item DT for users from regions Skikoku, Tohoku or Kanto. The tree has not been post-pruned. and reduces the dimensionality each item tree must handle, which is something our algorithm struggles with. Essen￾tially this approach builds a user DT that partitions users in clusters of similar users based on their preferences. For each cluster of users, a regular item DT is constructed to r… view at source ↗
Figure 6
Figure 6. Figure 6: Average regret for 900 users as training the user tree progresses. After every 50 users, the user tree is updated. The maximum tree depth is limited to 5 for both trees. ±1 Std. dev. is shown in the error bars for the users within these cohorts. the time to reach zero regret remains about the same. For dataset B, the final regret decreases as the item tree grows. However, it does not reach zero because the… view at source ↗
Figure 7
Figure 7. Figure 7: Surface plots of each optimization function. The functions are ordered from left to right, top to bottom in increasing order of ”spikiness”. ”Spikiness” refers to both the frequency and the strength of sudden changes (i.e. spikes) in the function values. 0 1000 2000 Rosenbrock 5D 0 1000 2000 3000 4000 Hartmann 6D 0 500 1000 1500 2000 Branin 2D 0 500 1000 1500 Levy 2D 0 50 100 150 200 0 1000 2000 3000 Micha… view at source ↗
Figure 8
Figure 8. Figure 8: Running time for each benchmark optimization function for DT, GP and SkewGP. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

Preferential Bayesian Optimization (PBO) aims to find a decision-maker's most preferred solution in as few pairwise comparisons as possible. Existing approaches rely on Gaussian Process (GP) surrogates, which provide strong performance but limited interpretability. This limits real-world usability in high-stakes domains, such as healthcare, where interpretability and trust are essential. We propose DT-PBO, a novel tree-based surrogate model for PBO that is inherently interpretable while capturing preference uncertainty. Specifically, we introduce a novel splitting heuristic that constructs interpretable shallow decision trees directly from pairwise comparison data, and use Laplace approximation to obtain probabilistic estimates for each leaf. This enables efficient preference modeling without sacrificing interpretability. Across eight benchmark functions, our method achieves competitive convergence to GP-based PBO, particularly on functions with rugged optimization landscapes. Additional experiments show robustness against noise and a fast computational running time. Experiments on real-world datasets further demonstrate that our model provides interpretable insights into decision-maker preferences that would remain opaque under GP-based approaches.

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. DT-PBO proposes an interpretable tree-based surrogate for Preferential Bayesian Optimization. It introduces a novel splitting heuristic to construct shallow decision trees directly from pairwise comparison data and applies Laplace approximation to obtain probabilistic estimates at each leaf. The paper claims this achieves competitive convergence to GP-based PBO across eight benchmark functions (especially rugged landscapes), plus robustness to noise, fast runtime, and interpretable preference insights on real-world datasets.

Significance. If the performance claims hold, the work supplies a practical, inherently interpretable alternative to Gaussian-process surrogates in PBO. This is valuable for high-stakes domains such as healthcare, where decision-makers need to understand and trust the recovered preference structure. The emphasis on shallow trees and computational speed could also broaden the applicability of preference-based optimization.

major comments (2)
  1. [Section 3.2] Section 3.2: The novel splitting heuristic is load-bearing for the central claim that shallow trees plus leaf-wise Laplace posteriors suffice to reach near-GP performance. No formal analysis or recovery guarantee is supplied showing that the heuristic captures non-axis-aligned or high-order interactions; competitive results on rugged benchmarks could therefore be benchmark-specific rather than evidence of general modeling power.
  2. [Experiments] Experiments section: The abstract asserts competitive convergence and noise robustness on eight benchmarks, yet the provided text supplies no quantitative tables, error bars, or statistical tests. Without these, the data-to-claim link for the splitting heuristic cannot be verified and the cross-benchmark claim remains ungrounded.
minor comments (1)
  1. [Abstract] Abstract: The statement that the method is 'particularly' strong on rugged landscapes would be clearer if the specific benchmarks and quantitative advantage (e.g., regret curves or iteration counts) were identified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive feedback. We have carefully considered the comments and provide point-by-point responses below. We believe the revisions will strengthen the manuscript.

read point-by-point responses

  1. Referee: [Section 3.2] Section 3.2: The novel splitting heuristic is load-bearing for the central claim that shallow trees plus leaf-wise Laplace posteriors suffice to reach near-GP performance. No formal analysis or recovery guarantee is supplied showing that the heuristic captures non-axis-aligned or high-order interactions; competitive results on rugged benchmarks could therefore be benchmark-specific rather than evidence of general modeling power.

    Authors: We acknowledge that the paper does not provide formal recovery guarantees for the splitting heuristic. The heuristic is designed to prioritize splits that maximize the separation of preferences based on pairwise comparisons, allowing the tree to capture interactions through recursive partitioning. While formal analysis would be valuable, our focus is on practical performance, and the competitive results across eight diverse benchmarks, including rugged landscapes, provide empirical support for its effectiveness. We have added a discussion in Section 3.2 on the heuristic's ability to model interactions via multiple splits. revision: partial

  2. Referee: [Experiments] Experiments section: The abstract asserts competitive convergence and noise robustness on eight benchmarks, yet the provided text supplies no quantitative tables, error bars, or statistical tests. Without these, the data-to-claim link for the splitting heuristic cannot be verified and the cross-benchmark claim remains ungrounded.

    Authors: We apologize for the omission in the initial submission. The full manuscript includes convergence plots with error bars in Figure 3 and additional results in the appendix. To address this, we have added Table 1 summarizing the average number of queries needed to reach the optimum with standard deviations across 20 runs, along with statistical significance tests comparing to GP-PBO. revision: yes

Circularity Check

0 steps flagged

No circularity: novel splitting heuristic and Laplace leaf posteriors are independently constructed from pairwise data

full rationale

The paper defines a new splitting heuristic directly on pairwise comparisons to build shallow trees, followed by per-leaf Laplace approximation for uncertainty; neither step is shown to be a re-expression of fitted parameters from the same model nor dependent on self-citations that carry the central claim. Benchmark comparisons are external evaluations rather than internal predictions forced by construction. The derivation chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unstated assumption that pairwise comparisons contain enough signal for a shallow tree plus Laplace approximation to recover a usable preference model; no free parameters, axioms, or invented entities are enumerated in the abstract.

axioms (1)
  • domain assumption Pairwise comparisons are sufficient to model latent preferences
    Standard premise of all PBO methods; invoked by the problem statement.

pith-pipeline@v0.9.0 · 5503 in / 1205 out tokens · 35761 ms · 2026-05-16T21:56:39.158465+00:00 · methodology

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