Order-Preserving Bayesian Optimization (OPBO)

• Order-Preserving Bayesian Optimization (OPBO) is a framework that preserves the ordinal structure of objective values, reducing sample complexity and improving noise robustness compared to traditional methods.
• It leverages various surrogate models, including neural networks and Gaussian processes, optimized through ranking losses or ordinal warping to capture relative performance accurately.
• OPBO employs novel acquisition strategies, such as order-based Expected Improvement and Lower Confidence Bound, to efficiently guide parallel and batch evaluations in complex settings.

Order-Preserving Bayesian Optimization (OPBO) is a class of Bayesian optimization (BO) frameworks that focus on preserving and exploiting the ordinal structure (ordering) of objective values or preferences, rather than modeling their absolute values directly. OPBO is motivated by the observation that in many high-dimensional, noisy, or practically constrained environments, only the relative ordering—rather than precise function values—can be reliably obtained or is ultimately required. Methods in this family encompass both ordered (global ranking) and pairwise preference (dueling) feedback, are typically more robust to noise, and can scale more efficiently in high-dimensional settings than traditional GP-based BO.

1. Core Principles and Motivation

Order-preserving Bayesian Optimization fundamentally contrasts with classical value-based BO, which fit surrogates to the true black-box objective $f: \mathcal X \to \mathbb{R}$ via regression losses like mean squared error. Value-based surrogates incur high sample and computational complexity in high-dimensional spaces or when function values are dominated by noise, making precise numerical interpolation infeasible. OPBO circumvents these issues by requiring only that the surrogate $h_\theta$ (parametric, e.g., neural, or nonparametric, e.g., GP) preserve the ranking of the observed data:

$$f(x_i) \geq f(x_j) \iff h_\theta(x_i) \geq h_\theta(x_j), \quad \forall\,i,j.$$

This ordinal perspective enables two key shifts:

• Sample efficiency: Learning to rank requires fewer points than fitting values, as the goal is to model the order, not fine-grained differences (Peng et al., 22 Dec 2025).
• Noise and outlier robustness: Rankings are far less sensitive to observation variance or heavy tails—a claim supported by empirical studies (Wang et al., 5 Feb 2024).

In practical terms, many settings (A/B testing, recommender systems, human-in-the-loop optimization) present only comparisons or rankings, rather than cardinal values (Benavoli et al., 2020, Xu et al., 8 Feb 2024).

2. Mathematical Formalizations and Surrogate Models

OPBO implementations span several modeling paradigms:

2.1 Parametric Neural Surrogates

In high-dimensional regimes, a feedforward neural network $h_\theta$ is trained to preserve observed orders by minimizing a ranking loss $L(\theta)$ over a set of pairs $P$:

$$L(\theta) = \sum_{(i,j) \in P} \ell\big(\mathrm{sign}(y_i - y_j),\, \mathrm{sign}(h_\theta(x_i) - h_\theta(x_j))\big),$$

with $\ell$ typically the logistic or hinge loss. This reduces to maximizing the probability of observed permutations under the Plackett–Luce model, optimized via stochastic gradient descent (Peng et al., 22 Dec 2025).

2.2 Latent GP Surrogates and Ordinal Warping

The "Ordinal Bayesian Optimisation" approach warps both input and output spaces into latent rank spaces, then fits a zero-mean GP to these coordinates. Ordinal regression likelihoods with learnable thresholds handle the non-conjugacy induced by ordering constraints, and variational inference is used for tractable posterior computation (Picheny et al., 2019).

2.3 Preferential (Pairwise) Models

Here, the latent function $f$ is paired with a likelihood over observed pairwise preferences $y_{ij}$, modeled via a probit or logit (Bradley–Terry–Luce) response:

$$p(y_{ij}=1\,|\,f(x_i), f(x_j)) = \Phi\big(f(x_i) - f(x_j)\big) \quad\text{(probit)}$$

or

$$\sigma(f(x_i)-f(x_j)) = \frac{1}{1 + \exp(-[f(x_i) - f(x_j)])} \quad\text{(logit)}.$$

The true posterior is a Skew Gaussian Process (SkewGP) (Benavoli et al., 2020); the surrogate can also be learned via maximum likelihood in an RKHS ball, enabling nonparametric uncertainty quantification (Xu et al., 8 Feb 2024).

2.4 Poisson Process Surrogate (Global Rank Modeling)

PoPBO models the number of points in the design space beating candidate $x$ as a (truncated) non-homogeneous Poisson process. The rate $\lambda(x; \theta)$ is fitted by maximum likelihood to the observed ranking data, efficiently parameterized by a small neural network. Predictive order probabilities and tailored acquisition functions follow analytically (Wang et al., 5 Feb 2024).

3. Acquisition Strategies and Order-Preserving Search

The core difference in OPBO is that acquisition functions are defined and used on the surrogate's order-statistics, not its best-fit values.

• Classical Acquisitions (with Order Surrogate): Expected Improvement, UCB, and Thompson sampling can be applied to the order-preserving surrogate, over a large candidate pool $X_\text{cand}$ (Peng et al., 22 Dec 2025).
• Good-enough Ordinal Set: Rather than selecting only the argmax of the acquisition, OPBO often selects the top-$g$ "good-enough" candidates by rank for parallel or batch evaluation. This increases robustness and data efficiency (Peng et al., 22 Dec 2025).
• Order-based Acquisitions:
  • OP-LCB (Order-Preserving Lower Confidence Bound): For PoPBO, this takes the predicted mean rank minus a multiple of the standard deviation, with exploration rectification (Wang et al., 5 Feb 2024).
  • OP-EI (Expected Ranking Improvement): Directly quantifies expected improvement in rank under the predictive order distribution (Wang et al., 5 Feb 2024).
  • Domain Decomposition/Cell Partition: In GP-based ordinal frameworks, the input rank lattice partitions the search domain into "cells"; acquisition optimizes over these cells, not over metric distances (Picheny et al., 2019).
  • Optimistic Dueling/Preference-Based Acquisition: Selects candidate duels to maximize the best-case improvement over a reference, solved via convex optimization in the RKHS (Xu et al., 8 Feb 2024).

4. Theoretical Guarantees and Regret Analysis

Theoretical properties of OPBO are established under oracle and model assumptions:

• The regret is measured naturally in the rank-latent space, not on absolute values.
• For GP-based ordinal BO, the main LCB regret bound is:

$$R(N) \leq \sqrt{4 N \beta_N (C_1 \gamma_N + C_2 \log (N+1))} + C_3,$$

where $\gamma_N$ is the maximum information gain of the kernel, and $\beta_N = O(\log N + d \log N)$. This establishes sublinear regret in the latent order space under mild GP assumptions (Picheny et al., 2019).

• Preferential (pairwise) BO under a BTL model yields information-theoretic cumulative and simple regret bounds, e.g.,

$R_T = O(\sqrt{\beta_T T \gamma_T^{ff'}})$

for appropriate kernel classes and covering numbers (Xu et al., 8 Feb 2024).

• Classical regret bounds derived for value-based GP-UCB can be extended to the rank structure, under control of the step-size drift in the latent mappings (Picheny et al., 2019).
• No formal theoretical guarantees for neural-network-based OP surrogates have yet been established; establishing such bounds remains an open problem (Peng et al., 22 Dec 2025).

5. Algorithmic Implementations and Complexity

Implementations are dictated by the surrogate choice:

Method	Surrogate	Key Complexity	Notes
GP-OPBO	GP + Ordinal mapping	$O(n^3)$	Variational inference, cell search for acquisition (Picheny et al., 2019)
NN-OPBO	Neural Rank MLP	$O(r\,d)$	Linear in dim and sample (Peng et al., 22 Dec 2025)
PoPBO	Poisson process	$O(N^2)$	Analytic likelihood, fast updates (Wang et al., 5 Feb 2024)
SkewGP BO	GP + Skew likelihood	$O(n^3)$	Exact SUN posteriors, outperforms Laplace (Benavoli et al., 2020)

In practice, OPBO with neural surrogates scales efficiently to $d \geq 1000$, since surrogate training and acquisition computations are linear or near-linear in sample and dimension, and batch "good-enough" acquisition further amortizes costs (Peng et al., 22 Dec 2025). Pairwise GP and SkewGP methods are bottlenecked by cubic scaling in data size due to Cholesky decompositions.

6. Empirical Performance and Applications

OPBO methods have demonstrated superior or competitive performance relative to value-based BO in the following contexts:

• High-dimensional synthetic test functions (Ackley, Lévy, Rosenbrock, Dixon-Price) up to $d=1000$, where OP surrogates integrated into TuRBO or HEBO frameworks significantly outperformed both GP and regression NN surrogates (Peng et al., 22 Dec 2025).
• Hyperparameter and Neural Architecture Search: PoPBO achieved lower regret and higher final test accuracy than GP-EI, TPE, SMAC, HEBO, and random search in HPO-Bench and NAS-Bench-201, with particularly notable stability under noise (Wang et al., 5 Feb 2024).
• Noisy, discontinuous, or ill-posed objectives: OPBO is robust to settings where stationary metric assumptions fail, including discrete variables and settings where only ranks or preferences can be observed (Picheny et al., 2019, Benavoli et al., 2020).
• Human-in-the-loop, subjective evaluation tasks: Preferential (dueling) BO/OPBO is naturally suited for optimization under A/B testing paradigms, recommender systems, and applications where only comparative feedback is available (Xu et al., 8 Feb 2024, Benavoli et al., 2020).

Empirical results consistently demonstrate faster convergence, lower wall-clock times, and lower final regret for OPBO surrogates compared to classical GP-BO in the above settings.

7. Current Limitations and Future Directions

Notable open questions and potential extensions for OPBO include:

• Theoretical Analysis for NN Surrogates: No regret or convergence analysis yet exists for OPBO using neural ranking models; developing such theory analogous to GP-UCB remains unresolved (Peng et al., 22 Dec 2025).
• Design of Ordinal-specific Acquisitions: Most current frameworks adapt classical BO acquisitions (EI, UCB, TS) to the ordinal setting, but new acquisition rules that exploit uncertainty in ranks, especially exploiting ordinal surrogates’ uncertainty structure, are an active area for exploration (Wang et al., 5 Feb 2024).
• Scalable Partitioning/Cell Search: For GP-based ordinal BO, exact cell partitioning becomes impractical in very high dimensions; scalable alternatives and sparse approximations are under development (Picheny et al., 2019).
• Multi-objective and Constrained Extensions: Preserving Pareto orderings and handling black-box constraints via ordinal surrogates are active areas for generalization (Peng et al., 22 Dec 2025).
• Adaptive Good-enough Sets: Dynamic tuning of candidate and selection pool sizes could further enhance the efficiency and robustness of OPBO in high dimensions.
• Preferential BO with Theoretical Guarantees: The construction of high-probability confidence sets and acquisition via convex reduction has enabled the first regret-bound guarantees for preferential BO models (Xu et al., 8 Feb 2024).

In summary, Order-Preserving Bayesian Optimization constitutes a principled, scalable, and robust paradigm shift for BO in settings where metric-based modeling is impractical or unnecessary. OPBO leverages the stability and tractability of ordinal information, delivering substantial empirical and theoretical advantages in modern large-scale, noisy, or human-centric optimization problems (Peng et al., 22 Dec 2025, Wang et al., 5 Feb 2024, Benavoli et al., 2020, Picheny et al., 2019, Xu et al., 8 Feb 2024).