Acquisition, Query Forms, and Problem Extensions

An acquisition function is the rule that decides what to ask next (Section 12.1). In preferential Bayesian optimization (PBO) it chooses a pair, or a set, of options to show a person, and Section 19.3 and Section 19.4 introduced the main candidates. This chapter follows those rules through the research record from 2017 to September 2026: where each came from, what is proved about it, under which conditions it did well or badly, and what forms a query can take besides "A or B?".

The design of acquisition functions went through three phases: heuristic and information-theoretic rules from 2017 to 2021; a turn to decision theory in 2022 and 2023, when the expected utility of the best option (EUBO) and its multi-option form qEUBO were proved one-step Bayes optimal for noise-free answers; and, from 2024 to 2026, kernelized dueling algorithms driven by frequentist regret (Chapter 29) alongside criticism and repair of the EUBO default. Two findings frame everything below. The failure modes reported by different groups agree with one another. And the empirical comparisons were run in settings that cannot be compared with one another; at small budgets with realistic noise, random queries sometimes keep up.

28.1 2017 to 2021: heuristics and information #

Two rules from before 2017. The interactive Bayesian optimization of Brochu et al. (2007) chose as the first option the queried point with the largest posterior mean, and as the second the point with the largest expected improvement over it (Section 12.3); Astudillo et al. (2023) later noted that qEUBO with two options, if forced to include the current best point, reduces to exactly this rule. Bayesian active learning by disagreement (BALD), proposed for Gaussian process classifiers by Houlsby et al. (2011) (a preprint), chooses the query whose answer has the largest mutual information with the model's parameters (Section 6.3): the query on which plausible models disagree most. The dueling information gain of Benavoli et al. (2021c) extends BALD to preferences.

The three rules of González et al. González et al. (2017) proposed three acquisition functions on the dueling space (Section 19.2). Pure exploration (PE) picks the pair whose duel outcome has the largest variance. Copeland expected improvement (CEI) computes the one-step lookahead improvement in the soft-Copeland value of the Condorcet winner, the option that beats every other with probability above one half. Dueling Thompson sampling (DTS) picks the first point to maximize the soft-Copeland score computed from one continuous Thompson sample of the preference function (Section 12.5), and the second purely to explore, as the point whose duel against the first is most uncertain. In their experiments on one- and two-dimensional functions (Table 28.3), DTS was consistently the best strategy, and the dueling-bandit baseline Sparring needed about 4000 iterations to approach what DTS reached in 200 duels.

Duels among many, and easy questions. SelfSparring reduced multi-dueling, in which several options are compared in each round, to an ordinary bandit problem solved by Thompson sampling; its kernel version, KernelSelfSparring, adds a Gaussian process prior so that information is shared between options (Sui et al., 2017b). Bıyık et al. (2019) showed (their Theorem 1) that the global optimum of the commonly used volume removal objective, which scores a query by how much of the space of reward parameters an answer is expected to rule out, is a trivial query of $K$ identical options. Replacing it with information gain favors queries a person can answer with confidence, and learned faster in simulation and in a user study. This work belongs to preference-based reward learning with a parametric reward, not to Gaussian process PBO (Section 36.1).

A burst of new rules, 2020 and 2021. Projective PBO (Mikkola et al., 2020) came with five rules for its projective queries, among them projective expected improvement and preferential coordinate descent. Benavoli et al. (2021c) defined three rules relative to the current winner $\vx_r$: a dueling upper confidence bound, the upper end of the 95% credible interval of $f(\vx) - f(\vx_r)$; dueling Thompson sampling; and EIIG, the logarithm of the probability of improvement plus $k$ times the dueling information gain, with $k = 0.1$ or $0.5$. Nguyen et al. (2021) proposed multinomial predictive entropy search (MPES), which they describe as the first information-theoretic acquisition function for Bayesian optimization with preference observations (Section 12.7); it optimizes all inputs of a query jointly, but it must enumerate the possible answers, so it suits only small query sets. Siivola et al. (2021) adapted batch expected improvement (qEI) and batch Thompson sampling to batch-winner feedback, in which the person names the best option of a set.

Separating two kinds of uncertainty. Fauvel and Chalk (2021) (a preprint) split the uncertainty about a duel's outcome into an epistemic part, which more data would remove, and an aleatoric part, the person's own randomness, which it would not. Their maximally uncertain challenge (MUC) takes as champion the point of largest posterior mean and as challenger the point that maximizes the epistemic variance of $\Phi(f(\vx_1) - f(\vx))$, in closed form and with a batch version. In the same year came the first kernelized dueling algorithm with a cumulative regret guarantee (Kirschner and Krause, 2021), whose feedback model Section 29.2 examines.

28.2 2022 to 2023: EUBO and the decision-theoretic turn #

EUBO in preference exploration. EUBO first appeared in Bayesian optimization with preference exploration (BOPE), where experiments produce several outcomes and a decision maker's utility over those outcomes is learned from comparisons. Lin et al. (2022) used EUBO to choose the two outcome vectors to show the decision maker, and proved that it is the one-step Bayes optimal preference-exploration policy (Section 19.4.1). Two variants address the fact that some outcomes are not achievable: EUBO-ζ generates outcomes from one posterior sample of the outcome model, and EUBO-f̃ compares only outcomes that are likely achievable. On an unrestricted outcome set, EUBO tended to over-explore outcomes that could not be achieved. The paper also concluded that a Monte Carlo version of BALD, BALD-f̃, is a strong and fast baseline.

qEUBO. Astudillo et al. (2023) generalized EUBO to

where $q$ is the number of options shown to the person in one query, from which the person picks the best, and $f$ is the latent utility (written $g$ in Section 19.4). It is not a batch of separate queries. Section 19.4.1 summarized their four theorems; their conditions matter for how far they reach:

1. Theorem 1. With noise-free answers, qEUBO is one-step Bayes optimal and equivalent to the knowledge gradient (Section 12.6).
2. Theorem 2. Under logistic (multinomial logit) noise of scale $\tau$ (Equation (16.4); the paper writes $\lambda$), the one-step value of qEUBO's maximizer is at least the noise-free one-step optimal value minus $\tau\, W\!\left((q - 1)/e\right)$, where $W$ is the Lambert W function, the inverse of $w \mapsto w e^w$ (Exercise 28.1 computes how fast the guarantee loosens with $q$).
3. Theorem 3. On a finite domain, with $q = 2$ and further technical conditions, the Bayesian simple regret of qEUBO, the expected gap between the best utility and that of the recommended option averaged over the prior (Section 13.1), is $o(1/n)$. Examples of sufficient conditions are a logistic likelihood with a prior under which, almost surely, $\delta \le \lvert f(\vx) - f(\vy)\rvert \le \Delta$ for all $\vx \neq \vy$; or a nondegenerate Gaussian process prior with a likelihood equal to a constant $a > 1/2$ whenever $f(\vx_1) \neq f(\vx_2)$.
4. Theorem 4. On some instances satisfying the same assumptions, the Bayesian simple regret of qEI stays above a constant $R > 0$ for every $n$: qEI is not asymptotically consistent.

Two readings follow (inference). EUBO's one-step optimality was first proved in BOPE; qEUBO's contribution is the extension to noise and to $q > 2$, and consistency on finite domains. And the conditions behind the $o(1/n)$ rate, a finite domain with utility gaps bounded away from zero, turn the problem into one of identifying the best of finitely many options, so the rate cannot be compared with the rates for continuous domains in Section 29.4. In software, BoTorch 0.10.0 (February 2024) added qEUBO (Meta Platforms, Inc., 2026e), and its preference tutorial runs the loop with PairwiseGP and the analytic EUBO (Meta Platforms, Inc., 2026c).

The hallucination believer. In the same year, Takeno et al. (2023) proposed the hallucination believer (HB) of Section 19.3: take the current winner as the first point of the duel, and apply expected improvement or an upper confidence bound to a Gaussian process fitted to one sample of the latent comparison values from the truncated posterior (a hallucination). They had observed that standard acquisition functions applied to a preference Gaussian process keep choosing similar duels, because a duel carries little information and the variance of the preference model hardly decreases. Ignatenko et al. (2025) later proposed remaining system uncertainty, from a minimax view of data collection, as a performance measure that needs no ground truth.

28.3 2024 to 2026: noise, exact knowledge gradient, amortization #

2024 and 2025. Ozaki et al. (2024) used a mutual-information active-learning acquisition function to choose preference queries in a multi-objective setting (see Section 28.6). Sinaga et al. (2026) (first posted in 2024; its 2026 version appeared in the proceedings track of ProbML 2026) proposed a risk-averse acquisition function that trades utility against how hard a comparison is to answer, and showed that their risk-adjusted EUBO stays one-step Bayes optimal up to an additive constant. POP-BO (Xu et al., 2024b) and MaxMinLCB (Pásztor et al., 2024) are optimistic algorithms with regret bounds (Section 29.3), and in 2025 PABBO (Zhang et al., 2025a) used a pretrained transformer policy that outputs query pairs directly.

2026: the default under examination. Most of the 2026 method work examined the default pipeline, in two preprints reported with the failure modes below (Wu and Gardner, 2026; Shao et al., 2026). Other work adapted classical ideas. Erarslan et al. (2025) (a preprint) adapted max-value entropy search to a production-cost constraint under which every comparison must involve a candidate that has already been produced. Haltia et al. (2026) (a preprint) used a cost-aware value of information to choose between a direct evaluation and a pairwise query; they report performance close to the convex hull of the two single-source trajectories, and the method falls back to standard Bayesian optimization when queries are expensive or noisy. Local PBO (Menn et al., 2026a) (a preprint) ported trust-region search (TuRPBO) and derivative-guided local search (GIPBO, PrefSQP) to pairwise feedback (Section 30.4). And PF-TS (Lazzaro et al., 2026) chooses a pair by drawing two independent posterior samples and maximizing each against a common anchor point.

28.4 Documented failure modes #

Section 19.6 previewed these failures. Here they are with the mechanism each paper gives and the conditions under which each was seen.

Adapted expected improvement stalls, again and again. Four groups reported the same phenomenon independently. González et al. found that CEI over-exploits and that the interactive Bayesian optimization of Brochu et al. performs poorly (González et al., 2017). Fauvel and Chalk found Brochu et al.'s expected improvement only slightly better than random, and attributed it to a frequent pathology in which the function samples the same duel members (Fauvel and Chalk, 2021). Takeno et al. found that expectation propagation with expected improvement often stalls through over-exploitation (Takeno et al., 2023). And Astudillo et al. found that qEI tended to stall late in a run, on the 7-dimensional Alpine1 function with initial data that included comparisons against a known good point, and proved the mechanism in their Theorem 4: when the value of the incumbent, the point of largest posterior mean, is already known fairly precisely, qEI is reluctant to include it in a query, so it learns only how the other options compare with one another (Astudillo et al., 2023). Rules of this family stop testing a well-known incumbent (inference; Section 19.3).

EUBO over-exploits, and its pipeline becomes ill-conditioned. On BOPE's unrestricted outcome set, EUBO over-explored unachievable outcomes (Lin et al., 2022). In single-objective PBO, EUBO's queries collapse toward the estimated maximum, as Section 19.6 reported and Section 19.5.1 reproduces in simulation. The source, Wu and Gardner (2026) (a preprint), adds the mechanism: under a Gaussian process prior and a probit likelihood, the one-step lookahead posterior is an extended skew-normal distribution, a skewed relative of the Gaussian whose mean has a closed form, and EUBO is only a lower bound on the exact knowledge gradient, approximately equal to it only when the probit noise goes to zero. Their test case is the two-dimensional Levy function, and their abstract also acknowledges a case showing that the knowledge gradient has limits in some situations. KappaSharp, also a preprint, reports that EUBO's queries create isolated comparison pairs and a rank-deficient Hessian (Shao et al., 2026) (Section 27.6). And POP-BO's authors report that, on instances sampled from a Gaussian process, qEUBO's reported solution was slightly better than theirs, but its cumulative regret was more than 2.5 times higher (Xu et al., 2024b).

Thompson sampling over-explores as the dimension grows. DTS was the best rule in one and two dimensions (González et al., 2017), but rules based on Thompson sampling performed only modestly over 34 functions, where KernelSelfSparring's weaker batch performance was attributed to its choosing the members of a batch independently (Fauvel and Chalk, 2021), and Thompson sampling over-explored on the 4- and 6-dimensional Hartmann functions (Takeno et al., 2023). Every study that tested a Thompson-sampling rule at four or more dimensions found it behind another rule (Figure 28.1). A plausible mechanism, which none of these papers isolates, is that the maximum of one posterior sample tends to fall where the posterior is most uncertain, and the share of the domain that is far from every observation grows quickly with dimension (inference; Exercise 28.2).

The hallucination believer depends on the noise. Takeno et al.'s results were obtained at noise variance $10^{-4}$, exactly the regime in which they showed the Laplace approximation fails worst. Their Appendix G.4 shows that the hallucination believer did relatively poorly when combined with the maximally uncertain challenge or with binary expected improvement, which they suspect is due to over-exploration (Takeno et al., 2023). Under logistic noise, the POP-BO authors report that the hallucination believer got stuck in local optima because it trusts random preference feedback too much, treating it as a hard constraint when it draws its Thompson sample (Xu et al., 2024b). Its advantage, in other words, depends on the noise level (inference).

Other costs. MPES must enumerate the possible answers: in qEUBO's 4- to 7-dimensional experiments it took 12.7 to 24.8 seconds per iteration against about 7 to 12 seconds for qEUBO, and it lost to qEUBO (Astudillo et al., 2023).

28.5 The acquisition functions compared #

Table 28.1 collects the main rules. Every entry under "Favorable evidence" and "Documented failures" carries the dimensions and noise under which it was observed, taken from the study that reported it (Table 28.3 gives each study's full setting). The frequentist regret bounds are detailed in Chapter 29.

The conditions in the table are easier to compare as a picture. Figure 28.1 places every comparison study by its noise model and the input dimensions it tested, and lets you choose a rule to see which studies report on it and with what result.

Some things to look for:

• Thompson sampling (the default view): both favorable findings sit at one to three dimensions, and even there it trailed MPES.
• The hallucination believer: favorable in the top row, near noise-free answers; unfavorable in the logistic-noise row.
• EUBO and qEUBO: favorable at 4 to 7 dimensions with logistic noise; mixed or unfavorable in the 2024 to 2026 studies that measured cumulative regret or looked for collapse.
• The empty regions: above 20 dimensions, only the local-PBO preprint compares rules.

28.6 Query forms #

A query need not be a pair. Chapter 20 teaches the forms and reports the evidence for the main ones. For larger sets (Section 20.1.2), qEUBO found $q = 4$ clearly better than $q = 2$, while Siivola et al. found only marginal gains from larger batches, with different acquisition functions and noise (Astudillo et al., 2023; Siivola et al., 2021). For galleries and projections (Section 20.3), every projective variant beat every pairwise variant in 2 to 20 dimensions (Mikkola et al., 2020), and the Sequential Gallery's plane search beat line search in 5 to 20 dimensions, with 6 participants satisfied after 5.36 iterations on average (Koyama et al., 2020). With pairs alone, Figure 19.4 shows how quickly the share of the gap that forty comparisons close shrinks as parameters are added. Benavoli et al. (2023) let a person pick several mutually incomparable options from a set. This section adds the remaining forms and what the evidence on forms, taken together, says.

Coactive feedback, ordinal labels, and robot-specific forms. CoSpar (Tucker et al., 2020b) adds coactive feedback to the posterior sampling of SelfSparring: the user both compares trials and suggests improvements. LineCoSpar (Tucker et al., 2020a) restricts the computation to a random line (Section 20.3.3) and tuned 6 gait parameters with 6 able-bodied participants. ROIAL (Li et al., 2021) combines ordinal labels with preferences inside a region of interest meant to guarantee safety and comfort. Chapter 24 follows this line through a case study.

Preferences over hypothetical outcomes, and requests for improvement. Astudillo and Frazier (2020) let a decision maker compare attribute vectors, and BOPE alternates a stage in which the person compares outcome vectors that may be hypothetical, sampled from the outcome model, with an experimentation stage (Meta Platforms, Inc., 2026d). Ozaki et al. (2024) add an improvement request: the decision maker indicates which objective of a shown result they want improved, and the utility is a Chebyshev scalarization (a weighted worst-case combination of the objectives) with uncertain weights.

Weak preferences, validity labels, numbers, and language. The "About Equal" answer of Bıyık et al. (2019), C-GLISp's better, worse, or similar, validity labels, and crash reports are extensions of the observation model (Section 27.2). Xu et al. (2020b) allowed both direct queries and duels (COMP-GP-UCB). For finitely many options, Wang et al. (2025b) proved that an efficient algorithm pays, for each option, only the smaller of the two regrets it would incur from reward feedback or from dueling feedback. Social BO (Adachi et al., 2025), a preprint, proved that under mild rationality axioms noisy group feedback alone cannot reach a consensus free of social influence, and mixed cheap public votes with expensive private ones. Natural language enters as a source of labels: PEBOL (Austin et al., 2024a) uses natural-language inference as its likelihood over independent items, and LILO (Kobalczyk et al., 2026) has a language model translate free-text feedback into pairwise labels for PairwiseGP with qEUBO, with a simulated decision maker and no study with people (Section 35.2). Labels generated by a language model are correlated and biased rather than independent probit noise (inference).

What the evidence on forms says. It points two ways. Forms that let each human action carry more information (projections, planes, $q = 4$) beat pairs in the papers that introduced them, while the only direct comparison of batch winners and full rankings found little difference. Forms that ask for a continuous answer (sliders, projections, coactive suggestions) bring other kinds of noise, motor and perceptual precision and cognitive load, which every cited paper handles with a single generic noise term (inference). As of September 2026 we found no controlled study with people that compares pairs, batch winners, full rankings, and sliders under the same acquisition function and budget, and no acquisition function derived for forms other than pairs and multi-option sets: slider, plane, and projective queries use adapted expected improvement or random subspaces. Until such a study exists, the choice of form is best made on the human side, by what people can answer reliably and quickly (Section 20.3.4, Section 32.4), and a richer form should be tested against pairs within the same system (inference).

28.7 Problem extensions #

PBO has been extended in the same directions as ordinary Bayesian optimization: several objectives, constraints, context, many dimensions, mixed inputs, several fidelities, and stopping. Table 28.2 summarizes each direction; the paragraphs after it give the points that matter.

Several objectives. Astudillo et al. (2025) let every objective be observed only through preferences and proposed dueling scalarized Thompson sampling (DSTS): sample from the posterior, apply a random Chebyshev scalarization, then run dueling Thompson sampling. They proved it asymptotically consistent, which they call the first convergence guarantee for dueling Thompson sampling in PBO, while noting that even for single-objective PBO the regret bound of DTS remains unknown; DSTS did best on four synthetic functions and on simulated exoskeleton and autonomous-driving tasks. The "preferences" of Abdolshah et al. (2019) (NeurIPS 2019) are an importance order over objectives, not pairwise feedback on designs; the other works in the table continue the line of Section 14.5.

Constraints and safety. StageOpt (Sui et al., 2018b) optimizes a utility under unknown safety constraints and separates expanding the safe region from maximizing utility. Its guarantees of constraint satisfaction and convergence are stated for numerical observations; a variant in an appendix learns the utility from preference feedback while the safety functions still receive numerical measurements, comes without a convergence theorem of its own, and was used clinically for spinal cord stimulation. Constrained PBO (Iwai et al., 2025) proposed EUBOC, which weights EUBO by the probability, modeled with a Gaussian process, that the constraints are satisfied, in the manner of constrained expected improvement (Gardner et al., 2014), and evaluated it in a banner-ad study with 11 professional designers, with predicted click-through rate as the constraint. Constraints have been handled in two ways: by learning feasibility from labels a person provides (C-GLISp, Benavoli et al.'s validity labels, crash feedback), or by measuring the constraint separately (the click-through rate in constrained PBO, StageOpt's safety signal). Which route fits depends on whether the constraint can be observed without the person (inference).

Context and transfer. Context enters through offline utilities learned from expert knowledge (Khan et al., 2025) or through contextual variables such as outdoor temperature in controller tuning (Wang et al., 2025d), a preprint. Transfer has been implemented through amortization (PABBO), stored models of earlier users (Meta-PO), and encoders trained on a population (Granley et al.), not through a multi-task Gaussian process across users (inference); Section 32.5 reports what population priors buy.

High dimension, mixed inputs, fidelities, and stopping. The high-dimensional methods of 2017 to 2020 all restrict each query to a low-dimensional subspace through the current best point, which turns a $d$-dimensional acquisition optimization into a one- or two-dimensional one and lets each human action carry more than one bit (inference); Section 30.3.1 lists how far each reached, and Section 30.4 examines the local methods of 2026 and the lengthscale bound that confounds their comparison with qEUBO. The piecewise affine surrogate of Zhu and Bemporad (2025) uses mixed-integer linear programming to handle known linear constraints and mixed numerical and categorical variables; it is the only preference method for mixed inputs we found, and the Sequential Gallery lists not handling discrete parameters, such as layouts, fonts, or filter types, among its limitations. The only multi-fidelity preference methods are those of Theiner et al. (2026) and elicitation-augmented Bayesian optimization (Haltia et al., 2026), both from 2026. For stopping, the only explicit optimal rule assumes a parametric reward model (Bıyık et al., 2019), and the Sequential Gallery stops when the user presses a "satisfied" button; Section 30.7 reports what scalar Bayesian optimization has learned about stopping and what a preferential rule would need.

28.8 Competing 'first' claims #

PBO sits between dueling bandits, preference-based reinforcement learning, control (the GLISp line), and human-computer interaction, and claims of being "first" in one community often overlook earlier work in another. Each claim below holds only within its exact setting (inference).

• Constrained PBO (Iwai et al., 2025) claims to be the first to introduce inequality constraints, after C-GLISp (Zhu et al., 2022) handled unknown constraints in 2022 and a variant of StageOpt (Sui et al., 2018b) learned a utility from preference feedback under numerically measured safety constraints in 2018; it does not cite the validity labels of Benavoli et al. 2021 (Benavoli et al., 2021c). The claim holds for inequality constraints on a Gaussian process surrogate.
• LineSpar (Cheng et al., 2020), a workshop paper, claims to be the first high-dimensional preference-based Bayesian optimization, after sequential line search (2017) and alongside projective PBO (ICML 2020).
• DSTS (Astudillo et al., 2025) claims the first multi-objective PBO framework, which overlaps with a preprint on choice functions (Benavoli et al., 2021b); the claim holds for latent objectives observed only through preferences.
• POP-BO (Xu et al., 2024b) says existing methods lack cumulative regret or global convergence guarantees, with the qualifier "continuous input space", and does not cite Kirschner and Krause (2021).
• MPES (Nguyen et al., 2021) calls itself the first information-theoretic acquisition function with preference observations (first arXiv version December 2020), while the EIIG rule (Benavoli et al., 2021c) (August 2020) also uses the dueling information gain; both descend from BALD (Houlsby et al., 2011) (inference).
• EUBO's one-step Bayes optimality was first shown in BOPE (Lin et al., 2022); the qEUBO paper's claim holds for its extension to logistic noise and $q > 2$.

28.9 The state of empirical comparison #

Table 28.3 lists the main comparison studies and their settings. In it, qTS is batch Thompson sampling, qNEI is noisy batch expected improvement, HPO-B is a benchmark built from hyperparameter optimization tasks, and a Sobol sequence is a quasi-random space-filling design.

The noise calibration of the qEUBO experiments comes from their Appendix C.2 and code (Astudillo, 2023b).

The comparisons do not add up. The studies differ in metric (Section 31.4.3 lists five definitions of regret), in noise model, from a nearly noise-free probit to 10% flipped answers and moderate Gumbel noise (Section 31.4.2), and in posterior inference: Laplace, expectation propagation, variational inference, or skew Gaussian process sampling. Rankings flip with these choices, as POP-BO's comparison with qEUBO shows (inference). Fauvel and Chalk's Borda analysis over 34 functions is the broadest single comparison, but it predates qEUBO and the hallucination believer, uses expectation propagation throughout, and runs only 80 iterations. Takeno et al.'s comparison comes next, but its duels are almost noise-free and it does not include qEUBO. No comparison has breadth, realistic noise, and the acquisition functions of 2023 and later at the same time.

Random queries sometimes keep up. The PABBO authors write that the random strategy often beat some of the Gaussian process baselines (Zhang et al., 2025a). On real data with a low signal-to-noise ratio, Siivola et al. found that all methods only barely beat the baseline, and wrote that preference observations are inherently less informative than direct ones, that larger batches do not alleviate this, and that for very noisy data, random search with large batches may be a good choice (Siivola et al., 2021). In a single replication, BoTorch's BOPE tutorial prints candidate utilities of $-0.473$ for EUBO-ζ, $-0.216$ for random preference exploration, $-0.101$ for the true utility, and $-1.380$ for random experimentation, and notes that EUBO-ζ's win is not guaranteed in a single replication (Meta Platforms, Inc., 2026d). The three pieces of evidence point the same way: at small budgets and realistic noise, the gap between an elaborate acquisition function and a random design may be small, and a single run says little (inference).

Evidence from people. The human studies in this chapter are all small, and none was designed to compare acquisition functions: projective PBO's materials-science user, the Sequential Gallery's 6 participants, LineCoSpar's 6, constrained PBO's 11 designers, and Meta-PO's 36. As of September 2026, we found no study that randomizes people to different acquisition functions, such as qEUBO, DTS, and MUC, under the same interface and budget. Section 47.4 describes the experiment that would settle it. Until then, a practitioner should treat the ranking of acquisition functions as unknown for people and keep a share of random queries in every session as a control, which costs little given the evidence above (inference). Nor is there a benchmark for PBO with agreed metrics, noise models, and budgets: each paper reuses Forrester, six-hump camel, Hartmann, Ackley, Sushi, and Candy as it sees fit (Section 31.5).

28.10 Settled, contested, missing #

28.11 Exercises #

Further reading #

• Astudillo et al. (2023) is the reference for qEUBO; read its four theorems with their conditions, and its Appendix C.2 on how the noise was calibrated.
• Fauvel and Chalk (2021) is the broadest comparison of rules (34 functions, nine rules) and the clearest treatment of epistemic against aleatoric uncertainty in acquisition.
• Takeno et al. (2023) and Xu et al. (2024b) disagree about the hallucination believer; together they show how much the noise level decides.
• Wu and Gardner (2026) derives the exact knowledge gradient under a probit likelihood and is the best place to see why EUBO's equivalence with it breaks under noise.
• Mikkola et al. (2020) and Koyama et al. (2020) are the two reference papers on queries richer than a pair.