Informed Initialization for Bayesian Optimization and Active Learning
Authors: Carl Hvarfner
David Eriksson
Eytan Bakshy
Maximilian Balandat
Venue: NeurIPS 2025 poster ()
Keywords: Bayesian Optimization
Gaussian Process
Experimental Design
Bayesian Active Learning
Primary area: probabilistic_methods
TL;DR: We propose an information-theoretic initialization strategy for Bayesian optimization that jointly improves predictive accuracy and hyperparameter learning, enabling better surrogate models and optimization efficiency in few-shot settings..

Abstract: Bayesian Optimization (BO) is a widely used method for optimizing expensive black-box functions, relying on probabilistic surrogate models such as Gaussian Processes (GPs). The quality of the surrogate model is crucial for good optimization performance, especially in the few-shot setting where only a small number of batches of points can be evaluated. In this setting, the initialization plays a critical role in shaping the surrogate's predictive quality and guiding subsequent optimization. Despite this, practitioners typically rely on (quasi-)random designs to cover the input space. However, such approaches neglect two key factors: (a) random designs may not be space-filling, and (b) efficient hyperparameter learning during initialization is essential for high-quality prediction, which may conflict with space-filling designs. To address these limitations, we propose Hyperparameter-Informed Predictive Exploration (HIPE), a novel acquisition strategy that balances space-filling exploration with hyperparameter learning using information-theoretic principles. We derive a closed-form expression for HIPE in the GP setting and demonstrate its effectiveness through extensive experiments in active learning and few-shot BO. Our results show that HIPE outperforms standard initialization strategies in terms of predictive accuracy, hyperparameter identification, and optimization performance, particularly in large-batch, few-shot settings relevant to many real-world BO applications.

Informed Initialization for Bayesian Optimization and Active Learning
Carl Hvarfner
David Eriksson
Eytan Bakshy
Meta
Meta
Meta ebakshy@meta.com
hvarfner@meta.com
deriksson@meta.com
Max Balandat
Meta balandatt@meta.com com
Abstract
Bayesian Optimization is widely used method for optimizing expensive black-box functions, relying probabilistic surrogate models such Gaussian Processes. The quality of the surrogate model is crucial for good optimization performance, especially in the few-shot setting where only small number of batches of points can be evaluated. In this setting, the initialization plays a critical role in shaping the surrogate's 's predictive quality and guiding subsequent optimization. Despite this, practitioners typically rely on (quasi-)o -random designs to cover the input space. However, such approaches neglect two key factors: (a) space-filling filling designs may not be desirable to reduce predictive uncertainty, and (b) efficient hyperparam- learning during initialization essential for high -uality prediction. which may conflict with space-filling designs. To address these limitations, we propose Hyperparameter-Inte Informed Predictive Exploration (HIPE), a novel acquisition strat- egy that balances predictive uncertainty reduction with hyperparameter learning using information-theores
principles. We derive closed-form expression for HIPE in the Gaussian Process setting and demonstrate its effectiveness through extensive experiments in active learning and few-shot BO, Our results show that HIPE outperforms standard initialization strategies in terms of predictive accu- racy, hyperparameter identification, and subsequent optimization performance, particularly in large-batch few-shot settings relevant to many real-worls Bayesian Optimization applications.
Introduction
Bayesian Optimization (BO) (Fazzerr 2018 Garnetl 2023 Jones et 1998 Mockus al. 1978)
is principled framework for comple-efficient global optimization of black-box functions with applications across diverse fields such as biological discovery (Grifiths and Hemández--Lobato 2020
Stanton et al. (002),, materials science (Ament et al., 2023b Attia et al. 2020 Frazier and Wang Letham al. 2019), and machine
2016), online A/B testing (Agarwal et al. 2018 Feng et al. 2025
Snoek et al. 2014). BO combines a probabilistic surrogate model- commonly Gaussian Process (GP) with an acquisition function to select where to evaluate the unknown objective function. In many applications, the runtime of single black-box function evaluation may restrict the experimenter to small number of batches the number of sequential rounds of experiments but many real-world experimental setups permit conducting multiple experiment simultancously (e.g., on a parallel compute cluster, in a randomized controlled trial, batch-testing multiple specimens in lab).
learning hyperparameter optimization (HPO) (Feurer et al. 2015
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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The success of Bayesian Optimization in practice is highly sensitive to the quality of the surrogate model (Eriksson and Jankowiak 202T; Foldager et al., 2023] Hvartner et al. 2023). This challenge particularly during the early stages of optimization when few observations are available. In these early stages, a small set of inputs typically selected at random, or via space-filling or quasi-random sampling strategies such as Latin Hypercube Sampling (LHS) or scrambled Sobol' sequences (Bossek et al. 2020] Owen 2023). While such strategies aim to achieve broad coverage of the imput space for initialization of the surrogate model, they may not necessarily result in improving predictive accuracy, or reduce predictive variance, across the entire input space Ren and Sweet (2024). Moreover, they neglect a second crucial aspect of modeling: the need to accurately infer the model's hyperparameters (Zhang et al., 2019), such as the kernel lengthscales accurate hyperparameter estimates, variation in unimportant dimensions will have less influence the selection of points in subsequent iterations of BO, leading to more sample-efficient optimization (Eriksson and Jankowiak 2021 Hvarfner et al. 2023 Müller et al. 2023). Conversely, poor hyperparameter estimation may cause subsequent BO iterations to fail to make meaningful progress (Berkenkamp al., 2019). which may be caused by misidentifying signal for noise, exploring
GP. With
irrelevant dimensions, or returning poor terminal recommendations (Hvarfner et al. 2022
In this work, provide principled approach to addressing this initialization challenge. Our main contributions are as follows:
We propose Hyperparameter-Inoore Predictive Exploration (HIPE), an acquisition function for initialization that optimizes for both predictive uncertainty reduction and hyperparameter learning.
2. We derive a closed-form expression for this objective the case of Gaussian Process models, and implement practical Monte Carlo approximation to make it amenable to batched optimization.
3. We conduct extensive experiments in active learning and Bayesian Optimization on synthetic and real-world BO tasks, demonstrating that HIPE outperforms competing methods in terms of both model accuracy metries and BO performance in few-shot, large-batch settings.
2 Background
2.1 Gaussian Processes
Gaussian Processes (GPs) widely used surrogate model in BO due their flexibility, closed- form and well-calibrated predictive distributions. GPs define distribution over functions, f gP((m()), k(;;)) )), specified by a mean function m(-) and a covariance (kernel) function k(,,-). For a given location æ, the function value f(x) is normally distributed, with closed-for form expressions for the predictive mean p(x) and variance o² axx). In practice, the mean function often kept constant, leaving the covariance function to capture the structural properties of the objective.
To model differences in variable importance in GPs with stationary kernels, each input dimension is commonly scaled by a lengthscale hyperparameter é, a practice known as Automatic Relevance Determination (ARD) (Williams and Rasmussen 1995). Additional, optional hyperparameters include a learnable noise variance of and signal variance of. The full set of hy perparaaaetssss 0 = (L.o2. o
may be learned either by maximizing the marginal likelihood p(D 0) (Maximum Likelihood Estimation, MLE) or by incorporating hyperpriors p(0) to perform Maximum A Posteriori (MAP) estimation. Alternatively, fully Bayesian treatment (Lalchand and Rasmussen) 2020: Osborne 2010, integrates over approximate the full Bayesian posterior distribution using Markov Chain Monte Carlo (MCMC) methods, thereby explicitly accounting for hyperparameter uncertainty. For additional background on GPs, see (Rasmussen and Williams 2005).
2.2 Bayesian Optimization
Bayesian Optimization (BO) (Frazier 2018) sample-efficient framework for finding the maximizer
2 arg max x (() of black-box function f X' - R over D-dimensional input space X' [0,1] P. The function f assumed to expensive to evaluate and observable only through noisy point- wise measurements, y(ι) f(æ) + E, where (000

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At the core of BO is an funncionnonnuuncionnn which uses a surrogate model to quantify the (expected) utility of candidate points æ. Acquisition functions balance exploration and exploitation typically through greedy heuristics. Popular examples include Expected Improvement (EI) (Bull 2011 Joness et al., 1998) and its numerically stable variant, LogEI (Ament al. (023a). as well as the Upper Confidence Bound (UCB) (Srinivas et al., [012 2010). In batch BO, multiple points are selected in parallel to accelerate data collection. This is often achieved by computing and jointly optimizing (quasi-)MC estimate of the utility associated with acquisition function over the full
batch X
{2i, 02 ..., F of size q (Balandal et al. 2020
Wilson al. 2017
Wilson et al. 2020).
2.3 Bayesian Active Learning
Bayesian Active Learning (BAL) and Bayesian Experimental Design (BED) (Chaloner and Verdinelli 1995) aim to improve predictive models by selecting data points that are most informative, either with regard to the model or to future predictions. A key quantity is the Expected Information Gain (EIG):
((G())((Z)) = H[{] - En)))) [H[{lv(=)]].
(1)
where H is the Shannon (differential) entropy and & is parameter of interest. Importantly, EIG is symmetric, and can be equivalently formulated as an entropy reduction over y(z), instead of g.
Bayesian Active Learning by Disagreement Bayesian Active Learning by Disagreement (BALD) (Houlsby et al.) 2011 Kirsch et al., 2019) selects query points that maximize the mu- tual information between model predictions and hyperparameters 0:
BALD(x)) œ) EIG(y( = H[y(=)|D) - ED(
BALD identifies locations where models within an ensemble exhibit the greatest disagreement in predictive uncertainty. In the GP setting. this often leads axis-aligned queries when there is high uncertainty in the lengthscales, and to repeated queries when observation noise is highly uncertain. Notably, BALD is model-agnostic and can be applied wide range of surrogate models and hyper- parameters, including subspace models (Garnett et al. 2014) and additive decompositions (Gardner)
al. 2017 Hvarfner et al. 2023).
Negative Integrated Posterior Variance The Negative Integrated Posterior Variance (NIPV) (Seo)
al. 2000) criterion selects queries that minimize the expected posterior variance over test distribution
p* p
NIPV(a;;P)) - -E [x²(x)) |x, D..
Expected Predictive Information Gain The Expected Predictive Information Gain (EPIG) (Bick ford Smith et al.) 2023) selects a candidate point æ by maximizing the mutual information between its label y(æ) and the label y(x*) at randomly drawn test input 2s (x))..
EPIG( p+) = Emz-pe [ [T(())):((z))
[D]]
4
which can equivalently be expressed in terms of an expectation over difference of entropies. In its general, model-agnosin form, this constitutes of nested expectation over both test inputs xx and outcomes and selected locations y(æ):
EPIG((;; p&) = Egg Ps H[)((x)) D] E (H]((x))(([(()))) D, x, y(z)]]].
(5)
EPIG for Gaussian Processes Standard GP regression with Gaussian likelihoods constitutes a special case for EPIG, as the predictive variance (and therefore the predictive entropy) after conditioning on (æ, v())) is independent of the realized value y(æ). Thus,
H[y(x,) DD ær, y(æ))] = H[y(æ.) (D, æ),
and the inner expectation over y(x)) in (5) disappears. EPIG therefore simplifies to an expected reduction in predictive entropy:
EPIGGP(#; pe) = Exy <p [H()(;)) [D] - -Hy((xx)) (D, =]]-
(6)
We will use this Gp-specific form of EPIG throughout the remainder of this paper.
3

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Both NIPV and EPIG promote the selection of data that reduces uncertainty over the test distribution, but without considering the effect on hyperparameter learning. The test distribution encodes how much emphasis is put on different parts of the domain. It can be specified by subject matter experts; in the case of no prior knowledge is typically the uniform distribution. Throughout the remainder of the paper, we will exclusively consider NIPV with uniform P*
3 Related Work
Initialization has received surprisingly little attention in the context of BO. Bossek al. (2020) conducts a study on the effect of various random initial designs on BO performance. Alternatively, minimax or maximin criteria (Johnson et al.] 1990) may be used to accomplish evenly distributed designs. Mayhe closest to our work is (Zhang et al. 2019)) which proposes LHS-Beta, an initial design criterion which alters samples drawn by LHS to achieve pairwise distances between points which matches a Beta distribution. LHS-Beta pursues diverse pairwise distances in the data, in order to best learn the lengthscale of a GP with an isotropic kernel. Mülllrr and Zimmerman (1999); Zimmerman (2006) address the problem of learning parameters of Kriging estimators using a empirical estimates of optimal experimental design criteria, limiting candidates to fixed grid of points.
1995) or pariance-miniminn et al. 2024) designs have been explored in active learning for optimal sensor placement (Krause
Entropy-maximizing (Guestrin et al. 2005 MacKay 1992
(ark
et al.
2006 2008) and other applications involving GPs, such as geostatistics (Sauer et al.) 2023)
and contour finding (Cole et al. (202)). Moreover, parameter-related EIG criteria
bedrock
of the broader topic of BED [Bickford Smith et al. 2023; Chaloner and Verdinelli 1995 Kirsch
ctt al. 2021 Rainnooth et al. 2024), which focuses on selecting data that is most informative about
model parameters or future predictions. These prediction or model-oriente
criteria have yet to see
widespread use in BO, particularly for initialization. However, information-theorell acquisition functions (Hennig and Schuler
et al. 2014 Hvarfner et al. 2022; Moss
ct al. 2021 Neiswanger et al.) 2021 2024] Tu et al., 2022] Wang and Jegelka [017 address the
optimization problem from an information theoretic perspective, albeit not with a primary focus on initialization or model predictive performance.
The problem of actively learning model hyperparameters during BO (post initialization) has previously been investigated by HHarrnnr al. (2023), who propose a combined BO-BAL framework actively learn the hyperparameters of the GP along with the optimum, and demonstrate that better-calibrated surrogates significantly enhance BO performance. [Houlsyy et al. (2011) proposes BALD, an active learning acquisition functions for hyperparameters in preference learning in GPs. Ris et al. (2022) proposes a a Query-By Buuery-By-Conge Coomimteee-riiietteeee et al.] (2019); Ziomek et al. (2024) address BO performance under hyperparameter uncertainty from a theoretical perspective, proving regret bounds when hyperparameters of the objective are unknown.
acquisition function for BAL in GPs. Lastly, Berkenkamp
4 Method
We consider the case of large-batch, few-shot BO, where the batch size is large (g > 8), and only small number of sequential batches B can be evaluated (often only one for initialization and one for BO, so B = 2). With only a handful of batches, there is little opportunity to correct for poor initialization. If the initial design fails to reduce uncertainty in key hyperparameters or leaves large regions of the input space unexplored, both hyperparameter inference and optimization become challenging. This setting is common if evaluations are lengthy but can be effectively parallelized, which is the case for instance in online A/B tests or lab experiments for biology González-Duquue al. 20 Stanton al. (002) or material design Yik al. 2025).
In practice, the objective(s) often exhibit complex structure such as moderate or high dimensionality D (D > 5) and significant uncertainty in the lengthscales l, noise and signal variance of and oj. respectively. and the importance of each input dimension. Standard space-filling designs, such as scrambled Sobol sequences, are widely used for initialization due to their theoretical uniformity properties. However, these methods are agnostic to the underlying model and its uncertainties. As result, they often fail to uncover critical hyperparameter dependencies (Zhang et al., 2019), leading to poorly calibrated surrogate models and suboptimal acquisition decisions in the subsequent BO phase.

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While Sobol and other space-filling sequences are designed to uniformly cover the input space, this property alone does not imply that the resulting model will be desirably helpful for subsequent BO. In fact, space-filling is not synonymous with informativeness: model trained on a space-filling design may still exhibit high predietive uncertainty in regions most relevant for optimization. For example, random designs increasingly sample near the boundaries of the search space dimensionality grows howssKKppen|//e (2000); Swersky (2017), whereas a point located at the center of the search space may be far more informative for optimization, as it can yield greater reduction in predictive variance throughout the search space.
To this point, a strategy that explicitly reduces predictive uncertainty within the region of inter- est est [Bickford Smith al. (2023) namely, the search space- not only seems better suited for our goals, but is also better aligned with both theoretical (Berkenkamp et al.) [2019] (Sriniaas et al.) 2010) and information gain-based criteria, as well as the practical objective of achieving a more informed. in an informal sense, model gost-initialization Garnett (2023). Methods like NIPV (Zhang et al., 2019) which minimize predictive variance over the input space achieve this goal better than Sobol, and yields designs that are spread out but not necessarily informative for model calibration and hyperparameter learning.
On the other hand, methods that focus exclusively on hyperparameter informativeness, such as BALD (Houlsby et al., 201T), tend to cluster queries along specific axes in order to resolve length- scales, thereby sacrificing coverage and potentially overlooking important regions of the search space. When additional hyperparameters are present, such as a learnable noise variance, BALD may even select duplicate queries at the same location to better estimate said noise variance. While such behavior is not inherently problematic, it can be undesirable in the few-shot setting, where initialization must address hyperparameter learning and space-filling simultaneously.
To overcome these limitations, we introduce HIPE, a method that explicitly balances predictive By jointly considering coverage and informa-
uncertainty reduction with syperparameter-awarreen tiveness, HIPE produces initial designs that effectively reduce both predictive and hyperparameter uncertainty, ensuring robust model calibration and improved downstream optimization.
The interplay between all these initialization strategies is illustrated in Figure [ which visualizes the acquisition surfaces for a GP model in two dimensions under lengthscale uncertainty. Each subplot highlights the distinct behavior of different method. Sobol emphasizes uniform coverage of the input space, but its designs may not achieve the desired reduction in predictive uncertainty, especially given the inherent randomness and lack of model awareness. BALD, in contrast, focuses on reducing lengthscale uncertainty by selecting axis-aligned queries, which can lead to clustering along specific directions and a loss of broader coverage. NIPV aims minimize average predictive variance across the input space, resulting in well-spread points, but it can neglect hyperparameter informativeness since its queries are not tailored to resolve model parameter uncertainty. Our proposed method, HIPE, strikes balance between predictive uncertainty reduction and hyperparameter-an awareness::. it selects points that are both well-distributed and aligned with the axes of uncertainty, thereby achieving low predictive and hyperparameter uncertainty simultaneously.
Sobol
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NIPV
HIPE
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Figure 1: Visual comparison of initialization strategies for BO with GP in two dimensions under lengthscale uncertainty. Each plot shows the acquisition surface used guide batch selection. Sobol emphasizes space- filling, but may not accomplish this to the desired degree. BALD focuses on reducing lengthseale uncertainty via axis- aligned queries. NIPV spreads points to minimize average predictive variance over the input space. Lastly, HIPE balances space- filling and hyperparameter awareness, choosing spread-on outt points while preserving axis-alignan alignment between queries.
5

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4.1 Ixperparammter--nnoree Predictive Exploration
We approach the problem of initialization through the lens of optimization, by simultaneously
maximizing the coverage of region of interest and the information acquired about model-level
uncertainty. A natural way to achieve this is to optimize a criterion that combines EPIG and BALD:
HIPEg(X)) -Eg v(x)
[Eugg
v XII]] +B (Hl [ Eg [H[ 01) (7)
EPIG objective
BALD objective
where > 0 is scalar weighting parameter. For large B, this objective favors hyperparameter learning, while for small / it favors space-filling designs.
It turns out that for the choice of B = 1, the maximizer of Eq. 1 is exactly the maximizer of the joint information gain over test function values and model hyperparameters (for proof see Appendix B
Proposition (Equivalence HIPE,=1 to Joint Information Gain). The HIPE, acquisition function with 3 =
1 is equivalent to maximizing the expected joint information gain over test function values yxx*) and model hyperparameters O acquired by a candidate batch X. Formally,
arg max HIPE8=1(X;p))) X
arg max Ew-ps BIG (y( (v( 0: X)]. XEROXD
(8)
While Proposition]] provides quantities often have vastly different scales. Therefore the choice of 3 1 will generally not result in optimal performance since the optimization will inadvertently focus on the larger of the two terms.
provvide an intuitive connection between HIPE, EPIG, and BALD, the constitunt
A key observation is that not all hyperparameter information gain amounts to information gained on the test set this depends on multiple aspects, including downstream test distribution and hyper- parameterization. To what extent the reduction in hyperparameter entropy manifests in reduction in test set entropy can be quantified through the mutual information between the hyperparameters O and the test points y(x,), Fx Pa (xx):
B EIG(y( ODD) E.,, [H(v(x,) ) [P] - E, [H[p(=>))) DJ]]
(9)
Intuitively, Eq. quantifies how well the knowledge of the hyperparameters, expectation, informs us about the values y(xx).. Importantly, EIG(y(x*+) D does not depend on the candidate set X and can thus be pre-computed.
Setting 3 (IG(y((*)) ); OJD) balances the two competing objectives in Eq. according their
effect on downstream predictive uncertainty, without introducing any additional hyperparameters. We refer to the resulting acquisition function Hyperparameter-Innoon Predictive Exploration (HIPE):
HIPE(X) : -Ex BEag (x()(()))))))))))))))))))))))[) (0, y X)]] EIG(y(x*);
H[y(X)]
(10)
Notably, optimizing HIPE does not require having observed any data for this problem to be well-posed only requires test distribution, model structure, and model parameter hyperpriors.
4.2 A Parallel Monte Carlo Implementation of HIPE
Following the Monte Carlo approach used in modern acquisition functions (Balandat et al. 2020 Wilson et al. 2020), we implement a parallel version of HIPE that enables joint optimization. For a candidate batch X {....... Fyy, we estimate the acquisition objective using M MC samples over the hyperparameters, T test locations of the EPIG objective, and N samples from the predictive posterior for the BALD objective:
OBALD(X;
log p (Eo [vr X ()))
Eo Mogp(s(X) [0]]
(11a)
1
log
Σ p (y(m) 10(m)) m=1
Σ M m=1
log (y(m) (
N
M
(11b)
where (mm) p(0) D) are i.i.d samples from the belief over hyperparameters, and (()) y( y(X) are ii.d, g-dimensional (joint) samples from the predictive posterior. Thus, the BALD objective amounts

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to repeated evaluation the g-dimensional multivariate normal posterior x(X) for each candidate X, and estimating the posterior entropy from it. Secondly, we estimate the EPIG objective analogously to Balandat et al. (2020) as
OEPIG (aPiG(X:P0 Pip::
MT Σ
11
M
T Σ
C C
H[y(a g(m) DI
(12)
where C H[y(x (2) [D] is constant w.r.t. X and does not need to be computed. Since v(x))) (T) X ((T²), D in Eq. (12) is the GP posterior predictive at it is a Gaussian random variable and entropy can be computed in closed form without observing or simulating the outcome at x+. Thus, Eq. (12) can be evaluated in closed form for each of the M GPs' T test points after conditioning on the set X.
With these two MC estimators, the subsequent optimization can be carried out jointly for the entire q-batch in a gDD-dimensional space, as the entropy reduction is computed with regard to the entire batch of candidates as opposed to singular one. Using Sample Average Approximation (Balandat al. 2020), the HIPE objective is deterministic and auto-differentiable.
One downside of this formulation is that the nested MC estimator imposes substantial computational runtime, making HIPE less suited for nigh-throughput applications (Daulton et al. 2022 Eriksson
ct al. 2019 Maus et al. 2022). However, in those applications the quality of the initialization batch generally much less crucial, this is not limitation in practice.
5 Results
We evaluate HIPE across two main types of tasks: Active Learning (AL) and Bayesian Optimization (BO). We consider various synthetic and real-world problems and number of different baselines. In both settings, we maintain large batch sizes and few batches.
Setup For AL,, we simply run a number of batches with HIPE and the other baselines with the goal of achieving the best model fit. For BO, we consider the "two-shot" setting which we first have to select an initialization batch and then can perform a single iteration of (batch) BO using qLogNEI (Ament et al., 2023a) as the acquisition function. We benchmark against the conventional initializations Sobol and Random Search, as well as BALD, NIPV and
(Zhang et al.
2019). On all tasks, we utilize fully Bayesian GP (Eriksson and Jankowiak 2021; Snoek et al. 2012) using MCMC with NUTS (Hoffman and Gelman 2014) in Pyro (Bingham et al. 2018). We implement HIPE and all baselines in Bo'Torch (Balandat et al. 2020p. For all experiments, the hyperparameter set f consists of lengthscales l for each dimension with D-scaled prior (Hvarfner) al. 2024), a constant mean c, and an inferred noise variance of unless otherwise specified. All baselines, including random algorithms, are given the center of the search space as part of their initial design, as both HIPE and NIPV select the center of the search space by design under a uniform Pe. Complete details on the experimental setup, including code pointers and benchmarks can be found in Appendix
Evaluation Criteria We measure model fit quality with Root Mean Square Error (RMSE) of the mean prediction and Negative Log-Likelihood (NLL) against large number of (ground truth) test point sampled uniformly from the domain. In the "two-shot" optimization setting, we are interested in how the initialization affects the quality of the GP surrogate after both the first (initialization) and second (BO) batch. We compute relative rankings, the performance of each algorithm compared to its competitors, for each seed of each function, and average across the task type. As such, the relative rankings aggregate anter-algorithm performance across rows for Figs
We also study how these model quality improvements translate to better optimization performance. To this end, we consider the out-ol-sample inference performance (Hernández-Lobar) et al.] [014] Hvarfner et al. 2022), that is, the performance of the point x'' = arg max μ(æ a D) selected as the maximizer of the posterior mean of the surrogate model fit on the data available in each batch. We choose this metric since using observed points directly performs very poorly in noisy settings, and only considering in-sample points is rather limiting in the shw-shot setting.
7

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5.1 Batch Active Learning on GP Surrogates and Synthetic F Functions
We first evaluate the ability of HIPE to learn accurate surrogate models through batch active learning on noisy synthetic test functions and surrogate LCBench (Zimmer et al.) 202T) tasks. The LCBench tasks derived from complete neural network training 2014) datasets, with 7D GP surrogate models fitted as described in Appendix AA] Additionally. we evaluate performance variant, where dummy input dimensions are added following standard practice in high-dimensional BO (Eriksson and Jankowiak 2021). These dummy dimensions introduce additional challenge, as effectively identifying and ignoring irrelevant features is critical for accurate predictions. All evaluations subject substantial observation noise, detailed in Appendix A
various OpenML (Vanschoren et al.)
the Hartmann 6D function and nigh-dimenssonal Hartmann 6D (12D)
We rum each algorithm for 4 batches of size = 16 and measure the NLL and RMSE after each batch, displaying mean and one standard eπror on all tasks. In Fig. 2 we see that, across all tasks, HIPE is the only method that consistently ranks in the top two for both RMSE and NLL., demonstrating that the models it produces
both accurate and well-calibrated. On NLL, HIPE performs comparably to BALD, which targets hyperparameter learning and thus excels at model calibration. Similarly, HIPE competitive with NIPV on RMSE, metric for which NIPV is particularly well-suited (Gramacy and Lee 2009).
Australian
Car
Har Harimann 6D
Hartmann 6D, (12D)
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Random
Figure 2: Model accuracy results in the batch active learning setting. We report RMSE across various synthetic and LCBench surrogate tasks over 4 batches of 16 evaluations across 100 seeds. HIPE consistently ranks the top in relative rankings both metrics. achieving strong balance between hyperparameter learning and predictive accuracy. BALD performs competitively on marginal aoglliilidood (MLL) but underperforms on RMSE due to poor space-filling, while NIPV excels at reducing RMSE but struggles with calibration. On aggregate, random initialization methods lag significantly behind across all benchmarks.
5.2 Noisy Synthetic Test Functions
Next, we evaluate HIPE on synthetic benchmark functions in the two-shot Bayesian optimiza- tion setting, using B batches and batch size of = 24. We consider three standard test functions functions-Ackles -Acssssccyy (4D), Hartmann (4D), and Hartmann (6D) aa well as two higher-dimensional variants of the Hartmann function. Observation noise is added to all tasks (a. 2 for Ackley and o, = 0.5 for Hartmann), further increasing task difficulty. In Fig. shows that across all benchmarks, HIPE consistently achieves the best or second-best performance, followed by NIPV. Random and space-filling initialization methods (LHS-Bete Random, and Sobol) perform noticeably worse across all settings. In App. CTT
we show the runtime of HIPE and various other methods on Hartmann (6D). The extra cost of HIPE acquisition is minority of iteration runtime for fully Bayesian model.

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Ackley 4D
Hartmann 4D
Hartmann 4Dg (BD)
Hartmann 6D
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Value of infered maximum
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-9
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Beta
Random
Figure 3: Out-of-sample inference optimization performance noisy synthetic benchmark functions under the two-shot setting (B 2, 4 24) across 100 seeds per benchmark. HIPE oulperforms matches the best-performnng method across all benchmarks, including on the high-dimensional Hartmann variants with added dummy variables. NIPV performs well on most tasks, but its performance degrades on tasks where hyperparameter identification is critical. Random initialization strategics perform the worst throughout.
5.3 LCBench HPO Tasks
We evaluate five additional tasks from LCBench the two-shot optimization setting: Fashion-MNIST. MiniBooNE, Car, Higgs, and Segment, using fixed, minimal observation noise. Fig. Leg.-Aissaaay that HIPE substantially outperforms the competition on Fashion-MNIS" Mini-BooNE and Higgs, and performs competitively on the remaining Segment and Car. Overall, HIPE consistently delivers the highest relative rank, outperforning competing algorithms by a substantial margin. In App. we demonstrate the performance of the same initialization schemes on 8, B 5. In this setting, HIPE remains the top- cop-performing method, and the overall ranking of algorithms remains intact.
Fashion-MNIST
MiniBooNE
Car
Higgs
Segment
Relative Rank
Value of infered moximum
65.0
2.6
$3.5
64.5
78
79
2.8
83.0
85.0
64.0
76
3.0
$2.5
H/
74
63.5
77
12.0
Batch
Batch
Batch
Batch
Batch
Batch
HIPE
NIPV
LHS-Beta
Sobol
Random
Figure 4: Two-shot optimization results five hyperparameter optimization tasks from LCBench seeds each task. HIPE achieves the highest final performance on four competitive on the remaining one (Segment). Relative rankings across the five problems show that HIPE consistently outperforms other methods. This confirms the practical relevance of our approach for real-world HPO scenarios, where both accurate predictive modeling and effective exploration erucial.
five tasks and remains
5.4 High- Dighenninnnin SVM Tasks
Finally, we evaluate HIPE and baseline methods on challenging high-dimensional SVM hyperparam- eter optimization tasks with D 20 and D 40 input dimensions, considered in similar variants by Ament et al. (2023a); Eriksson and Jankowiak (2021): Hvarfner et al. (2024): Papeameier et al. (2023). For both tasks, only the last two dimensions corresponding parameters- parameters--sinificcan
the SVM's global regularization influence the objective, while the remaining dimensions, corresponding to of lesser importance. Effectively identifying and prioritizing these
feature-specific lengthscales, relevant dimensions critical for successful optimization. We again consider the two-shot setting, using larger batch size of 32 due the higher dimensionality of the problems.
The left panel of Fig. reports the out-of sample inference performance after each batch. 20D task, HIPE achieves competitive, mid-range performance relative to the evaluated methods. the more challenging 40D task, HIPE obtains the highest performance, albeit by narrow margin over the next-best alternatives. The limited budget relative to the dimensionality presents

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significant challenge, as the ability to accurately learn the model hyperparameters diminishes with increasing dimensionality. Despite this, we observe in the right panel of Fig. that HIPE identifies the important hyperparameters remarkably well after initialization on the 40l task aas--nnign the last two dimensions lengthscales that are, on average, nearly half an order of magnitude smaller than those inferred under a Sobol initialization. In [ we display the inferred pyperprrmmeter values for NIPV and Random as well. While these methods infer lower lengthscales for the last two dimensions than Sobol, they do not manage to infer as low values as HIPE.
Finally, we note that the best solutions to the SVM problem were almost exclusively located near the boundaries of the search space particularly in the last two dimensions, which neither NIPV nor HIPE naturally explore during initialization.
SVM 20D
SVM 40D
SVM 40D Lengthscale Distribution
Valuo of infered maximuum
0.348
-0.20
-0330
-022
-0.352
Lengthscale
-0.356
-0.23
10
-0.358
Batch
Batch
HIPE
LHS-Beta
Random
NIPV
Sobol
5 10 15
20
25
30
35
40
Figure 5: Results on 20D and 40D SVM hyperparameter optimization tasks. Left: Objective value of inferred maximum after each batch. On 20D, HIPE achieves average performance, climbing to a mid tier result in the second batch after focusing on hyperparameter learning in the first batch. HIPE obtains a large standard error in the first batch, two repetitions poorly infers the maximizer and suggests ill-performing points result. On 40D, outperforms all baselines, demonstrating strong robustness in higher dimensions and the ability to recover after a less successful first batch. Notably, the performance of Random decreases between batches on the 20D task, demonstrating the difficulty of accurately inferring the optimum. Right: Log-mean estimated lengthscale hyperparameters after initialization on the 40D task. HIPE identifies the last the SVM's global regularization parameters much more effectively than Sobol, assigning substantially smaller lengthscales to the relevant inputs.
two dimeension--correspoodiii
Discussion
Contributions We introduced HIPE, a principled, hyperparameter-ffee information-theoree method for initializing Bayesian Optimization and Bayesian Active Learning algorithms. its HIPE yields initial designs that balance coverage of (relevant) areas of the domain with the ability to effectively learn model hyperparameters. HIPE is especially useful in the few-shot, large-batch setting, where achieves superior surrogate model quality compared to various initialization baselines.
Limitations HIPE can become computationally expensive, especially for large batch sizes in higher dimensions D, though in many applications this cost is still insignificant compared to the time and resources required to evaluate the underlying black-box function. Finally, the current paper focuses on GP surrogates, and while our main insights and the general approach apply also to other surrogate types, our implementation does not translate directly. Thus, the computation of the acquisition function would have to be re-derived for other types Bayesian models.
Future work Here we studied the "cold start" problem of initializing Bayesian Optimization and Bayesian Active Learning from scratch. In practice, we may have access to data from related (but not necessarily identical) problems. This motivates an extension of HIPE to the transfer learning setting. e.g., by means using a multi-task GP surrogate. Additionally, incorporating prior knowledge of domain experts in principled fashion is of high practical relevance, and can readily be utilized in the form of a non-uniform ps. Finally, we are also interested in studying the multi-objective setting in which different surrogates of potentially different form with different hyperparameters and priors model different objectives but share observations at the same input locations.
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A Experimental Setup
We describe the full experimental setup used in the paper: the design of the Bayesian optimization and active learning loops, the benchmarks used, the compute as well the licenses of all software and datasets. The code to run the experiments in the paper, including all the benchmarks and plotting to reproduce our results, available at htttp::///// / tuub. hwtp://eeeeeeruiaee/iii.....
An easy-to-run, tested version of the algorithm and accompanying notehook is available open-source in BoTorrc., The HIPE acquisition is found at plob/main/botowe wwwmunity/
botocch/ http://caaiiiitttoaaaaaaaaaaaaaa actieeeeiiiaaaiieeeeeeeeeeeeeeee Seaiiveeeeeannnnneeeeee py The note-
book, which demonstrates how to run the algorithm should be run its indented setting and the expected results, can be found at http:///tt.t......aaatattttttttr
A.1 Bayesian Optimization Loop
All experiments were conducted using standardized pipeline based on BoTorrh (Balandat et al. 2020) and GPy (Gardner et al. 2018). We use fully Bayesian Gaussian Process (GP) models. with hyperparameter inference performed via the No-U-Turn Sampler (NUTS (HHffman and Gelman 2014) as implemented in Pyro (Bingham et al.) 2018). Unless otherwise specified, we draw 192 burn-in samples followed by 288 hyperparameter samples, retaining every 24th sample for evaluation.
Our GP prior is adapted from Hvarfner et al. (2024) to better suit a fully Bayesian setting. Specifically, we set (40- = -0 -0.75 and = 0.75, resulting in La ~ (0.00.5 + log( D)/2, 0.75). The noise standard deviation is modeled as og (-55, 0.75). and the constant mean parameter follows ~ ~0, 0.25). These modifications ensure that the means of the priors for la and o² approximately match the modes of the corresponding parameters in the prior proposed by Hvarfner et al. (2024). producing similar hyperparameter values in practice.
Acquisition functions are optimized jointly over the batch using multi-start L-BFGS-B optimization with 4 random restarts and 384 initial samples drawn from a scrambled Sobol sequence. For the optimization of LogEI, we additionally sample 384 points from Gaussian distribution centered at the current incumbent to improve local search performance.
For our proposed HIPE method and relevant baselines (BALD and NIPV), Monte Carlo estimators use M- = 12 hyperparameter samples, T 1024 test points drawn uniformly from the search space, and N 128 predictive posterior samples.
A.2 Benchmarks
We use three types of benchmarks: synthetic optimization test functions, surrogate-based based hyper- parameter optimization tasks from LCBench (Zimmer et al., 2021), and high-dimensional SVM hyperparameter optimization problems. Synthetic functions are standard benchmarks for evaluating active learning and Bayesian optimization under controlled noise and dimensionality. LCBench tasks are GP surrogate models trained on 2,000 evaluations of multi-layer perceptrons (MLPs) on real-world datasets, using Matern 3/2 kernel; all surrogates are included as part of our code release. The SVM benchmarks follow the setup in Ament et al. (2023a), based on the problem originally introduced in Eriksson and Jankowiak (2021). ). For these tasks, a Support Vector Regressor (Drucker et al.. 1996) is fit to the CTSlice dataset, using fixed subset of 5,000 data points, with 20% reserved for validation. To vary the dimensionality, an XGBoost model is fit to the original 988-dimensiionl dataset, and the most important features are retained according to XGBoost's 's default feature selection criterion. The objective is to minimize the validation RMSE. synthetic functions, additive Gaussian noise is introduced directly to the function evaluations (see Table [ for details). The LCBench benchmarks rely on pre-trained surrogate models, where the posterior mean is evaluated, and in cases where o, is non-zero, additional noise is added to the evaluation of the posterior mean.
A.3 Licenses
The following software packages, libraries and datasets were used in our experiments and for presenting the results in the paper:
23

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Table 1: Noise levels for all Active Learning (AL) and Bayesian Optimization (BO) benchmarks.
Category
Benchmark
Task Type
Dimensionality
d²
Ackley (4D)
BO
4
2.0
Hartmann (6D)
BO / AL
6
0.5
Synthetic
Hartmann (6D)
BO/ / AL
12
0.5
Hartmann (4D)
BO
4
0.5
Hartmann (4D)
BO
8
0.5
Car
AL
7
2.5
Australian
AL
7
2.5
Fashion-MN
BO
7
0.0
LCBench
MiniBooNE
BO
7
0.0
Car
BO
7
0.0
Higgs
BO
7
0.0
Segment
BO
7
0.0
Feature-reduced SVM
BO
20
0.0
SVM
Feature-reduced SVM
BO
40
0.0
GPyTorch, BoTorch, Hydra: MIT License
Py Torch, NumPy, SciPy, Pandas: BSD Licenses
Matplotlib, Seaborn: PSF/BSD Licenses
LCBench: Apache License
A.4 Compute Resources
All experiments were condueted using NVIDIA A40 GPU cluster. The compute usage to all experiments the main paper amounts to approximately 1000 GPU hours, and an additional 500 GPU hours to produce all the results provided Appendix C
B Derivation of HIPE Joint Information Gain
Recall Proposition D from section
Proposition (Equivalence of HIPE3-1 to Joint Information Gain). The HIPE, acquisition,function with 3 1 equivalent maximizing the expected joint information gain over test function values
y(x)) and model hyperparameters 0 acquired by candidate batch X. Formally,
arg max
HIPE (X:;)) = arg max Eg-pa EIG (y (e),,) 0; X)].
(8)
XEROXD
X
Proof of Proposition For 3 = 1, we have
HIPE, Ee,v(X) Ery [y((xx) 0, (X)]]] (H[v(X)) E. (X) O]))
Therefore, with X = arg max X HIPE, (X),
X arg max Ex E (
0, X)]]| (H[ (X)) Ep [H[b(X)
0]]) (13a)
X
arg max X
- (x))(x)
Exg [H
| X)11
(13b)
arg max En,, [H[y(**), 0)] - [[(]]]))))))))))),,,.._)_)))))pa
(by def)
(13c)
X D
arg max Eyg [EIG(y(Z4), X)]
(13d)
X
where the equalities follow from the Bayes' rule of conditional entropy and the fact that the arg max independent of quantities which do not involve X.

---

C Additional Experiments
We present supplementary experiments to further analyze the behavior of the evaluated methods under varying experimental conditions. These results provide additional insights into the robustness of the methods with respect to batch size during initialization and active learning, as well as their performance trade-offs across different evaluation metrics.
C.1 Runtime Analysis
We compare acquisition function optimization runtime of HIPE and NIPV, and put that in relation to the runtime of the other components of a BO loop that employs fully Bayesian GP modeling. In 2 we demonstrate that while HIPE is substantially slower to optimize than NIPV, its total runtime still amounts to only a minor share of the total runtime of the overall BO run.
Runtime Component (sec)
HIPE
NIPV
Random N/A
Model fit (initial)
21.91 + 0.15
21.91 - 0.15
Acquisition Opt.
19.61 + 1.12
8,62 + 0.37
Model fit (BO loop)
181.79 +: 7.60
189.90 5.17
177.33 + 7.69
qLogNEI Optimization
4.97 + 0.18
4.44 + 0.05
4.35 0.06
Table 2: Total runtime comparison (in seconds) for HIPE, NIPV, and Random across different components over HIPE is time-consuming to optimize compared to NIPV.
the course of 4-batch, q optimization While NIPV takes approxinately half as long to optimize, the runtime to optimize HIPE is small relative to the time required to fit the fully Bayesian GP with NUTS.
C.2 Impact Batch Size on Predictive Performance
We investigate how the initialization batch size g affects predictive quality and inference performance on the LCBench tasks introduced Section [.3 We vary q from 6 to in increments of 2 up to q 16, then in steps of 4. For each batch size, evaluate models trained the initialization batch only before any active learning iterations under a fixed-noise setting. This isolates the impact of the initial design and avoids the confounding effects of subsequent data acquisition.
Fig. summarizes the results across three key metrics: test-set NLL., test-set RMSE, and out-of- sample inference performance. HIPE consistently achieves the lowest NLL across most benchmarks and batch sizes, particularly in the g E [12, 16] range, demonstrating strong calibration and hyperpa- rameter learning. In contrast, RMSE results show that HIPE often underperforms relative to simpler baselines like Sobol and Random, suggesting a trade-off between predictive calibration and pointwise accuracy. Inference performance shows HIPE leading for smaller batch sizes (q 16), although performance plateaus or declines at higher q. indicating that additional samples are not necessarily helpful for inference when uncertainty in relevant parameters remains high. This occurs because, as batch size increases for our experiments, the inferred arg max has an increased propensity to be located boundary. As these boundary points may not be well-performing. inference performance occasionally stagnates or decreases with increasing batch size. Notably, as HIPE's 'ss initialization is generally more centered than competing methods", it obtains higher in-sample values across most batch sizes.
Unlike the active learning setup, these tasks are noiseless and assume fixed, known noise levels during training. This removes uncertainty in the noise model and creates a distinct evaluation setting focused solely input selection and hyperparameter inference.
C.3 Bayesian Optimization with Different Batch Sizes
We further investigate the impact of batch size on Bayesian optimization performance by evaluating all methods on the LCBench tasks with a reduced batch size of q = 8. As shown in Fig. 6) HIPE maintains its position the best-performing method overall, achieving the lowest inference regret across the majority of tasks. Notably, HIPE benefits more than competing methods from identifying high-values points during the initialization phase, which can be attributed to HIPE biasing towards
25

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Fashion-MNIST
Higgs
Segment
MiniBooNE
Negatvve Log Libelibood
12
10
20
10
20
10
20
Roat Mean Squared Breor
5.5
2.4
1.7
5.0
1.6
14
4.5-
2.0
12
1.5
1.4
1.0
10
20
10
Valuu at Inferred Maxtmizor
78
03.5
64.0
85.0
63.5
84.8
76
83.0
79
84.6
82.5
78
84.4
62.5
84.2
72-
82.0
76
84.0
62.0
81.5
75
83.0
10
20
10
10
20
Batch
Batch Size
Batch Size
Botch
Batch Size
HIPE
LHS-Bett --S-Beta
NIPV
Random
Sobol
Figure 6: Effect of initialization batch size 4 on predictive quality and inference across LCBench tasks. Each row shows performance on one of three metrics after initialization batch: (top) NLL., (middle) RMSE, and (bottom) out-of-sample inference. HIPE consistently leads in NLL and small-q inference, while other methods achieve lower RMSE, indicating trade-off between model calibration and pointwise prediction accuracy.
the center of the search space. However, HIPE maintains its advantage throughout, demonstrating its impact on subsequent BO iterations.
Rashion-
NiniBooNE
Higgs
Segment
Value of infered maximum
65.0
1.75
84.0
3.25
82
82
2.00
33.00
2.25
03.5
80
2.50
70
03.0
2.75
02.5
78
3.00
74
3.25
82.0
62.0
76
3.50
33.75
2 3 Batch
3 Batch
2
3
5
Batch
Batch
Batch
Batch
Sobol
Random
NTPV
I.HS-
Figure 7: Inference regret per method using batch size of 8 across LCBench tasks. HIPE is once again the pest-performing method overall, but benefits from finding high valued points during initialization greater extent than other methods.
C.4 Active Learning with Diffferent Batch Sizes
analyze the effect of batch size the active learning setting by comparing model performance under small batches (g 8) and large batches (q 24). This evaluation assesses each method's
26

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robustness to changes in batch size and its capability to maintain predictive accuracy and effective hyperparameter learning under varying evaluation budgets.
As shown in Fig. 8 for smaller batches of = 8, HIPE achieves the best performance on the NLL metric and significantly outperforms BALD second-best method on NI..-in terms of MSE. In this setting, NIPV and BALD exhibit inconsistent behavior, alternating between the worst performance on MLL and RMSE, respectively.
When the batch size increases to y = 24, as visualized in Fig. g HIPE continues to perform strongly, consistently ranking among the top methods across both evaluation metries. Notably, its relative performance improves with larger batch sizes, particularly in terms of RMSE, indicating that HIPE scales more effectively with increased parallelism in query selection. This suggests that HIPE is better suited for scenarios requiring efficient learning under larger batch evaluations.
Australian
Car
Hartmann 6D
Hartmann 6D₃ (12D)
Ishigami
Relative Rank
14
0.65
4.0
Roet Moan Squared
0.50
10
0.50
3.0
0.45
3.0
e-
0.40
0.45
3.5
0.35
0.30
0.35
2.0
4.0
2
1
10
Negative Lon Likejthood
2.0
1.75
14
2.0
12
7-
6-
1.25
10
2.5
8-
3.0
4
0.75
0.5
0.50
3
1 3 7.8
1
2
1
2
4 Batch
2 4
Batch
Batch
Batch
Batch
Batch
HIPE
BALD
NIPV
Sobol
Random
Figure 8: Model accuracy results in the batch active learning setting with smaller batches (7= 8). RMSE is reported across various synthetic and LCBench surrogate tasks over 8 batches, using 100 random seeds per benchmark. HIPE achieves the best performance NLL and outperforms BALD, the second-best method on NLL, terms of RMSE. NIPV and BALD alternate exhibiting worst performance on MLL and RMSE, respectively.
C.5 Complementary Hyperparameter Visualization on SVM
We display the inferred hyperparameter values of the left-out methods (Random, NIPV) the 40D-SVM. Both methods infer more accurate hyperparameter values than Sobol, but less accurate than HIPE. Compared Sobol, both Random and NIPV assign smaller lengthscales the relevant dimensions, indicating improved identification of important features. However, their estimates remain less precise than those obtained by HIPE, which more effectively distinguishes the relevant dimensions, namely the last two.
27

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Australian
Car
Hartmann 6D
Hartmann (12D)
Ishigami
Relative
Root Mean Squared Ervor
7.5-1
0.55
4.0
2.50
0.50
2.0-
6.5-
0.50
3.5
3.00
0.45
8.0
3.25
5.5
0.40
3.0
0.40
2.5
3.75
4.5
4.00
Negative: Log. lakehhhond
1.0
2.0
1.2
2.5
1.0
3.0
0.8
0.6
0.6
3.5
3
0.4
0.5
4.0
2 Batch
2
2
Batch
Batch
Batch
Batch
Balch
HIPE
BALD
INPV
Sobol
Random
Figure 9: Model accuracy results in the batch active learning setting with larger batches (g = 24). RMSE reported across various synthetic and LCBench surrogate tasks over 3 batches, using 100 random seeds per benchmark. HIPE consistently ranks among the top two methods across both metrics, achieving strong balance between hyperparameter learning and predictive accuracy. With larger batch sizes, HIPE exhibits improved relative performance, particularly RMSE, demonstrating effective scalability under increased parallel evaluations.
SVM 40D Lengthscale Distribution
Lengthscale
10
10
15
20
25
30
35
40
Figure 10: Estimated lengthscale hyperparameters Sobol, Random, NIPV HIPE on the 40D-SVM task. Random and NIPV infer low lengthscales for the last two dimensions, which are known to be the most important. However, HIPE does so with greater effectiveness.
28


Reviews:
[{"review_id": "zULg0LaXKu", "reviewer": "Reviewer_MUHR", "summary": "This paper introduces Hyperparameter-Informed Predictive Exploration (HIPE), a principled initialization strategy for Bayesian Optimization and active learning that jointly balances space-filling exploration with efficient hyperparameter learning using information-theoretic objectives. The method is proposed for the large-batch and few-shot setting.", "questions": "1. Could the authors justify why setting $\\beta = $EIG$(y(x_*); \\theta)$ achieves a balance between the two competing objectives?\n\n2. What do the red and blue boxes represent in the right subplot of Figure 5? A legend or explanation would improve clarity.", "limitations": "Yes.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**:\n\n1. The paper explores an interesting direction direction about the initialization for Bayesian Optimization and Active Learning in few-shot, large-batch scenarios.\n\n2. The experimental evaluation spans multiple settings, deterministic and noisy test functions and hyperparameter tuning, including LCBench and SVM.\n\n**Weaknesses**:\n\n1. The core idea behind HIPE offers limited novelty, essentially combining two well-established acquisition strategies, and introduces a hyperparameter $\\beta$ that may be difficult to tune. Moreover, the fast implementation of the objective builds heavily on existing techniques.\n\n2. While the authors introduce NIPV, it is not empirically evaluated in combination with BALD, leaving its practical benefits underexplored. The authors are encouraged to also test this combination.\n\n3. Aside from Proposition 1, which establishes an information-gain equivalence, the paper lacks theoretical guarantees (e.g., regret analysis) to support the proposed method’s performance.\n\n4. The current experiment presentation lacks sufficient details to enable reproducibility, particularly regarding the choice and tuning of the hyperparameter $\\beta$, which is not explicitly specified.", "was_revised": true, "final_justification": "The authors have addressed most of my concerns. However, I still think that the proposed method lacks novelty, as the new acquisition is essentially a linear combination of BALD and EPIG. While a combination approach is not necessarily a problem, the paper does not convincingly demonstrate that such a simple combination yields a significant improvement in most of the experiments. Moreover, even though the authors explain why $\\beta=EIG$ is suggested, I still feel this choice to be rather heuristic and not theoretically grounded.", "rebuttal": "We would like to thank the reviewer for their feedback on our work. Moreover, we are appreciative that the reviewer acknowledges the relevance of our problem setting. Regarding the design of the hyperparameter $\\beta$, we would like to bring to the reviewer’s attention that we provided a method for setting it in the paper, in Section 4.1, L192. All our responses are outlined below.\n\n\n1. **The current experiment presentation lacks sufficient details to enable reproducibility, particularly regarding the choice and tuning of the hyperparameter beta, which is not explicitly specified.**\n\n   We use a principled approach to set the hyperparameter beta, as described at line 192 in the paper, and apply this consistently throughout all experiments. Appendix A1 provides detailed information on the experimental setup, including hyperpriors and software, to support reproducibility. Our full experimental code is available on GitHub, the link to which is in Appendix A1 as well. We are happy to provide further clarifications if needed.\n\n&nbsp;\n\n2. **Could the authors justify why setting β = EIG achieves a balance between the two competing objectives?**\n\n\n   The EPIG and BALD objectives are generally on very different scales, making weighting necessary. Setting $β = \\text{EIG(}y(x_*); \\theta)$ provides a principled weighting that reflects how much hyperparameter information actually translates to predictive information on the test set. This choice ensures that the hyperparameter learning component (BALD) is weighted according to its actual relevance for prediction, rather than using an arbitrary constant. When hyperparameter uncertainty has little impact on test predictions, $\\beta$ will be small, emphasizing space-filling. When hyperparameter uncertainty significantly affects predictions, $\\beta$ will be larger, emphasizing hyperparameter learning. This adaptive weighting achieves the desired balance automatically without manual tuning.\n\n&nbsp;\n\n3. **While the authors introduce NIPV, it is not empirically evaluated in combination with BALD, leaving its practical benefits underexplored.**\n\n   We acknowledge that NIPV + BALD can have practical value on its own. However, NIPV is not combined with BALD because it does not adhere to an information-theoretic framework like HIPE. Consequently, the same derivation as in Prop. 1 cannot be made, nor can beta be derived in a similar manner to achieve the desired trade-off between hyperparameter learning and space-filling. Given this, combining the two would be more heuristic than principled. \n\nWith that said, We have included the NIPV + BALD combination in the table below. While it wins on 1/5 LCBench problems, the BALD term generally dominates the NIPV term, leading to under-exploration.\n\n   | Function       | HIPE            | NIPV + BALD     | LHS-Beta        | NIPV            | Random          | Sobol           |\n   | -------------- | --------------- | --------------- | --------------- | --------------- | --------------- | --------------- |\n   | Fashion-MNIST  | 84.06 ± 0.09    | **84.15 ± 0.07**| 83.77 ± 0.08    | 83.45 ± 0.09    | 83.75 ± 0.09    | 83.62 ± 0.08    |\n   | MiniBooNE      | **85.37 ± 0.02**| 85.31 ± 0.03    | 85.25 ± 0.03    | 85.27 ± 0.03    | 85.20 ± 0.04    | 85.21 ± 0.03    |\n   | car            | **83.34 ± 0.24**| 82.99 ± 0.21    | 82.91 ± 0.26    | 82.31 ± 0.33    | 82.36 ± 0.27    | 80.95 ± 0.56    |\n   | higgs          | **65.17 ± 0.04**| 64.91 ± 0.04    | 64.77 ± 0.06    | 64.99 ± 0.05    | 64.61 ± 0.08    | 64.54 ± 0.08    |\n   | segment        | **81.55 ± 0.28**| 81.09 ± 0.29    | 81.31 ± 0.24    | 80.81 ± 0.31    | 81.01 ± 0.30    | 80.92 ± 0.29    |\n\n&nbsp;\n\n4. **Aside from Proposition 1, which establishes an information-gain equivalence, the paper lacks theoretical guarantees (e.g., regret analysis) to support the proposed method's performance.**\n\n   Our acquisition function is designed to produce a well-informed initialization to improve subsequent Bayesian Optimization performance rather than to minimize regret immediately. As such, regret bounds are not currently available. While theoretical guarantees on model predictive performance might be possible, such results are rare in the broader Bayesian active learning literature. We acknowledge this as a limitation and an interesting direction for future work.\n\n&nbsp;\n\n5. **What do the red and blue boxes represent in the right subplot of Figure 5?**\n\n   The red and blue boxes in Figure 5 (right) represent the interquartile ranges of the estimated lengthscale distributions for different initialization methods. The red boxes show lengthscales estimated after HIPE initialization, while the blue boxes show lengthscales after Sobol initialization. The plot demonstrates that HIPE correctly identifies the last two dimensions (the SVM's global regularization parameters) as most important by assigning them substantially smaller lengthscales compared to Sobol. We will add a legend to clarify this in the camera-ready version.\n\n&nbsp;\n\n---\nWe once again thank the reviewer for their service, and hope our rebuttal addresses the reviewer’s questions and concern regarding how the hyperparameter $\\beta$ is automatically set. Should the reviewer have any further questions or require additional details, we would be more than happy to provide them."}, {"review_id": "GZhgLQju1Q", "reviewer": "Reviewer_g6zV", "summary": "In this work, the authors propose a new initialisation scheme for sequential experimental design methods, specifically for active learning and Bayesian Optimisation. The authors argue that previously proposed initialisation schemes may not be sufficiently space-filling and that they often do not sufficiently consider how to learn hyperparameters of the surrogate model. Based on this the authors propose hyperparameter-informed predictive exploration (HIPE) which essential combines initialisation/sampling strategies that either optimise for predictive information gain or hyperparameter learning. The initial formulation of HIPE contains a hyperparameter that should balance either optimising predictive information gain or hyperparameter learning which can be cumbersome to tune at run time. However, the authors show that this hyperparameter can be found optimally and precomputed without traditional hyperparameter tuning. The authors then illustrate the efficacy of the proposed method via a number of experiments.", "questions": "I firstly list larger questions here, and below more optional questions.\n\nMy main questions in relation to aforementioned weaknesses and clarifications, which if most of were addressed/discussed would cause me to raise my score:\n1. The authors discuss how the presented method may be too expensive for some applications - would the authors consider including compute times in the work to make it clear how expensive the method is?\n2. Am I understanding correct that using the proposed method requires a candidate set of initial data points?\n3. I think some ablation studies and some more insights on experimental results would be useful. I elaborate here:\n    - Do the authors have any insights on why the init scheme seems to be relatively problem sensitive? In Figure 2, 3 and 4 there are some problem types where HIPE perform a lot worse than in other problems. Some discussion on why this might be the case would be very useful for practitioners and further development.\n    - Is there any reasons the authors only show the Sobol results in Figure 5 (right) and not all the other methods?\n    - Did the authors look at performance of their proposed method in the BO setting for iterations > 2?\n4. Do the authors have any insights on how the proposed init scheme would work for other surrogate models such as BNNs or Random Forests?\n5. I think the title is a little confusing (although this is an opinion) whilst I think the abstract is overclaiming. To me the title indicates something where the init scheme is prior knowledge dependant, but HIPE is actually \"fully automized\" requiring no prior from the user - perhaps including Hypeparameter in the title as in the method would be helpful. In the abstract, the authors write that HIPE outperforms other standard init schemes, which is not always the case - I believe this should be amended.\n\nMinor comments or questions:\n1. There is a citation missing in relation to BO surrogate quality and calibration in lines 34 and lines 136-139, see [1].\n2. The authors write \"at the core of BO is an acquisition function\", but I would argue that the surrogate is equally important.\n3. In line 162 the authors write that Scrambled Sobol sequences fail to achieve the desired coverage, but I do not see any citation providing evidence for this.\n4. In lines 163-166 the authors repeat much of what was already said in the paragraph running from lines 149-158 if I am not mistaken.\n5. Based on Figure 1, I have a hard time seeing qualitative differences between NIPV and HIPE. Maybe the authors could discuss this more in detail.\n6. In line 306 there appears to be a grammatical error in the sentence beginning with \"its HIPE\".\n\n[1]: Jonathan Foldager, Mikkel Jordahn, Lars Kai Hansen, and Michael Riis Andersen. On the role of\nmodel uncertainties in Bayesian optimization. In Uncertainty in Artificial Intelligence (UAI), 2023.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The paper reads clearly and is motivated well.\n- There are essentially _no_ hyperparameters in the method which is a big plus for the sequential experimental design setting.\n- Good initialisation schemes for sequential experimental design setting is an often overlooked problem that in practice can translate to big differences in performance - this paper contributes significantly to this issue.\n\nWeaknesses\n- Missing run time costs of experiments which can be important for these types of applications.\n- The method is presented only for GPs, which may be a shortcoming as BNNs and NNs are becoming more used surrogates in the sequential experimental design setting.\n- The experimental setup and evaluation is somewhat strict and specific in some senses. I elaborate below.", "was_revised": true, "final_justification": "Increased score from borderline accept to accept based on informative replies in rebuttal. See comment to authors for details.", "rebuttal": "We thank the reviewer for their detailed feedback and positive evaluation of our work. We appreciate that the reviewer finds our paper to read well, and that they recognize the potential significance and historical lack of attention for our problem setting. Below, we address each of the weaknesses and questions raised.\n\n*&nbsp;*\n\n1. **The authors discuss how the presented method may be too expensive for some applications - would the authors consider including compute times…**\n\nCertainly. Below is a runtime table for $q=8$ and $N=32$, namely 4 batches of 8 evaluations each with the first batch being initialization. We show runtime across 20 seeds for HIPE, NIPV and random on Hartmann (6D). The \"BO loop\" components are summed across all batches, meaning that \"Model fit during BO loop\" involves 3 unique model fittings for the three BO batches. All experiments are run on one A40 GPU. In our experience, this makes acquisition function optimization substantially faster, but accelerates model fitting less. HIPE is slower than NIPV and Random, but the runtime of initialization is still less than 20% of the overall runtime of BO in this setup. Naturally, the proportion of the total runtime allocated to initialization increases with fewer total batches.\n\n| Runtime Component (sec)      | HIPE          | NIPV          | Random        |\n|------------------------------|---------------|---------------|---------------|\n| Initial model fit / sampling | 21.91 ± 0.15  | 21.73 ± 0.18  | 0.02 ± 0.00   |\n| AcqOpt during init           | 19.61 ± 1.12  | 8.62 ± 0.37   | —             |\n| Model fit during BO loop     | 181.79 ± 7.60 | 189.90 ± 5.17 | 177.33 ± 7.69 |\n| AcqOpt during BO loop        | 4.97 ± 0.18   | 4.44 ± 0.05   | 4.35 ± 0.06   |\n| Objective evals (synthetic)  | 0.00 ± 0.00   | 0.00 ± 0.00   | 0.00 ± 0.00   |\n\nSince we use a fully Bayesian surrogate with NUTS and large batch sizes, all steps are relatively time-consuming. Fully Bayesian models tend to be expensive in general, and HIPE shares similar computational demands. We want to emphasize though that in the few-shot settings considered in this work this cost is typically dwarfed by the cost of running evaluations, which in practice can often be on the order of days or weeks (e.g. training large ML models or running online A/B tests).\n\n\n***&nbsp;***\n\n2. **Am I understanding correct that using the proposed method requires a candidate set of initial data points?**\n\nOur method does not require a candidate set of initial data points. What it does require / allow for is a distribution $p(x_*)$ to inform the method about promising regions (e.g. based on user priors or domain knowledge). In the absence of prior knowledge about promising regions, a uniform distribution provides the best coverage, which is what we used in all experiments. However, users *can* bias initialization toward a-priori promising regions with a non-uniform $p(x_*)$. From an implementation perspective, we represent $p(x_*)$ as an empirical distribution of sample points. Importantly though, these sample points *do not* need to be evaluated on the objective function.\n\n***&nbsp;***\n\n3. **Is there any reasons the authors only show the Sobol results in Figure 5 (right) and not all the other methods?**\n\nOnly to de-clutter the plot. Since there were 40 dimensions to visualize, we thought we would only include the two most relevant comparisons. The other methods are very similar to Sobol.\n\n**&nbsp;**\n\n4. **Did the authors look at performance of their proposed method in the BO setting for iterations > 2?**\n\nYes, we have included the terminal performance for 5 batches with a batch size of 8 (initialization in the first batch only) as a table below; We will also include them in a potential CR. HIPE performs very well in this slightly smaller batch size, but one can naturally expect the importance of initialization to be largest when it constitutes a larger part of the overall budget.\n\n| Function       | HIPE            | LHS-Beta        | NIPV            | Random          | Sobol           |\n| -------------- | --------------- | --------------- | --------------- | --------------- | --------------- |\n| Fashion-MNIST  | **84.65 ± 0.07**| 84.19 ± 0.06    | 84.29 ± 0.08    | 84.17 ± 0.04    | 84.23 ± 0.04    |\n| MiniBooNE      | **85.47 ± 0.02**| 85.25 ± 0.05    | 85.40 ± 0.02    | 85.24 ± 0.04    | 85.21 ± 0.06    |\n| car            | **84.38 ± 0.15**| 83.10 ± 0.36    | 83.74 ± 0.19    | 82.36 ± 0.37    | 82.98 ± 0.33    |\n| higgs          | **65.21 ± 0.08**| 64.56 ± 0.13    | 64.90 ± 0.11    | 64.18 ± 0.14    | 64.29 ± 0.14    |\n| segment        | **83.86 ± 0.18**| 83.12 ± 0.14    | 83.39 ± 0.12    | 82.70 ± 0.18    | 82.95 ± 0.24    |\n\n***&nbsp;***\n\n5. **The method is presented only for GPs, which may be a shortcoming as BNNs and NNs are becoming more used surrogates.**\n\nWe agree that Bayesian Neural Networks (and other types of surrogates) are increasingly used, though less so for Random Forests. HIPE can be applied to BNNs, where uncertainty over *θ* corresponds to uncertainty over all neural network parameters. However, the computation is more involved than for GPs, as BNNs lack closed-form expressions for posterior variance or conditioning on new data.\n\nRandom Forests do not inherently have a Bayesian treatment of model parameters and thus cannot use HIPE directly.\n\n***&nbsp;***\n\n6. **Do the authors have any insights on why the init scheme seems to be relatively problem sensitive?**\n\nYes, we do. Since HIPE’s EPIG criterion minimizes integrated (average) variance over the search space, it tends to avoid points near boundaries, as average variance is rarely reduced by sampling near edges. Sobol and random methods have no such bias, which explains the performance swings between HIPE and random methods. Differences between NIPV and HIPE are less straightforward.\n\nFor some problems, HIPE’s more centered queries are beneficial early on, while for others (e.g., SVM), where optima lie near boundaries, this is less so. However, HIPE compensates by correctly identifying active dimensions and catches up in later batches.\n\n***&nbsp;***\n\n7. **I think the title is a little confusing...**\n\nDoes the reviewer refer to the term “*Informed”?* We are open to suggestions on what a more representative title would be.\n\n***&nbsp;***\n\n8. **...whilst I think the abstract is overclaiming.**\n\nHIPE is designed to produce the most informed model (in terms of point spread and hyperparameter inference) to enable successful subsequent BO batches, rather than to achieve the best performance after batch 1. From this perspective, we believe the statement is accurate but agree that we can make it more precise to improve clarity.\n\n***&nbsp;***\n___ \n\nMinor comments have been addressed and will be included in the camera-ready version:\n\n1. The missing citation related to BO surrogate quality and calibration has been added.\n2. We agree that the surrogate model is equally important as the acquisition function in BO; this is now reflected in line 80.\n3. The claim about Scrambled Sobol sequences not achieving desired coverage anecdotal. We will avoid general statements and discuss Fig. 1 specifically.\n4. We have rewritten the repeated content in lines 163-166.\n5. The discussion around Figure 1 has been expanded to better highlight qualitative differences between NIPV and HIPE, including a modified figure clarifying axis alignment.\n6. The grammatical error in line 306 has been corrected.\n\n&nbsp;\n___\n\nWe once again thank the reviewer for their constructive feedback and appreciate their recognition of the paper’s clarity, motivation, and the relevance of the problem setting. We hope that our responses have addressed the reviewer’s questions regarding runtime analysis and ablation studies.\n\nIn light of these additions, we hope the reviewer finds the additional results compelling and the overall submission strengthened."}, {"review_id": "jPlZqTuWDH", "reviewer": "Reviewer_XxAJ", "summary": "The authors introduce a novel acquisition function (HIPE) for Bayesian Optimization (BO) that aims for an effective initialization of BO algorithms. HIPE incorporates surrogate model hyperparameters into the selection of new candidates to account for different importances of dimensions while aiming to be space-filling (i.e., achieving a good coverage of candidate configurations). They empirically evaluate the proposed acquisition function on BO tasks, showing HIPE's effectiveness.", "questions": "- in Fig. 2, the rank of BALD and NIPV tremendously changes when considering RMSE instead of NLL. Can the authors explain why this is happening?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n- The work tackles the important problem of making batch BO more sample-efficient by better initialization/selection of query points\n- mostly clearly written\n- The proposed approach is simple and easy to adopt\n- extensive empirical evaluation is provided \n\n**Weaknesses**\n\n- as far as I understood, the novel acquisition function proposed is a linear combination of EPIG and BALD. Therefore, it is not necessary to discuss NIPV in Sec. 2\n- Fig. 1: it is not clear what the authors consider to be \"optimal\" in this example. It is not easy to see why HIPE selects better points than NIPV since both have broad coverage.\n- The authors do not explain why strategies like Sobol are not able to accomplish a space-filling selection (see Fig. 1 caption).\n- Prop. 1 introduces a connection between EPIG, BALD, and HIPE. However, it is not clear to me what this connection means for the proposed method. For example, does this influence the convergence properties of the BO? It would be good if the authors could comment on this.\n- A theoretical consideration of HIPE in terms of convergence would be good in general.\n- in line 209, the authors claim that \"entropy in Eq. 10 is Gaussian\". Can the authors provide proof or a reference showing that this is the case?\n- in the experimental section, it would be great if the authors could provide an ablation varying the number of batches and the batch size to see if HIPE performs well for different settings as well (esp. for Sec. 5.2, 5.3, 5.4).\n- to further strengthen the evaluation, considering other HPO-benchmarks like [1] and [2] could be beneficial.\n\n**References**\n\n[1] HPO-B: A Large-Scale Reproducible Benchmark for Black-Box HPO based on OpenML. Arango et al. 2021.\n\n[2] HPOBench: A Collection of Reproducible Multi-Fidelity Benchmark Problems for HPO. Eggensperger et. al. 2021.", "was_revised": true, "final_justification": "The authors have addressed my main concerns and have added additional results as requested. Overall the authors provide an incremental, but still valuable contribution.", "rebuttal": "We thank the reviewer for their detailed and thoughtful review. We appreciate the recognition of the importance of the method, as well as its simplicityekecnerbflrktggenuicujfrfttluvnj, the clarity of our presentation, and the extensive empirical evaluation. Below, we address the specific concerns raised.\n\n&nbsp;\n\n1. **As far as I understood, the novel acquisition function proposed is a linear combination of EPIG and BALD. Therefore, it is not necessary to discuss NIPV in Sec. 2.**\n\n   It is correct that our method is a combination of EPIG and BALD. We introduced NIPV because it is strongly related to EPIG and is arguably the more well-known method in the broader active learning literature.\n\n&nbsp;\n\n2. **It is not clear what the authors consider to be ‘optimal’ in this example (Fig. 1). It is not easy to see why HIPE selects better points than NIPV since both have broad coverage.**\n\n   We try to motivate this in the text accompanying the figure but agree it should be clearer in the figure itself. The coverage is basically equally good for NIPV and HIPE, but HIPE additionally has the hyperparameter-informative axis alignment like BALD. Thus, by design, HIPE intuitively should be better for hyperparameter learning than NIPV while visually retaining equal coverage.\n\n   We will add helper lines to the figure to emphasize the axis alignment in HIPE.\n\n   Figure 1 is meant primarily as an illustration to provide intuition of the typical behavior of the different initialization approaches; in general it is not straightforward to draw conclusions on the performance without actually evaluating performance (in terms of model fit and/or optimization performance).\n\n&nbsp;\n\n3. **Prop. 1 introduces a connection between EPIG, BALD, and HIPE. However, it is not clear what this connection means for the proposed method. For example, does this influence the convergence properties of the BO?**\n\n   Proposition 1 shows that, for a certain choice of beta, HIPE is equivalent to jointly maximizing both EPIG and BALD, justifying the linear combination from a theoretical perspective. However, this has no direct influence on subsequent BO convergence. A formal proof of convergence for BO with dynamically updated hyperparameters (e.g., MAP or fully Bayesian estimates) is still an open problem and beyond the scope of this work.\n\n&nbsp;\n\n4. **In line 209, the authors claim that ‘entropy in Eq. 10 is Gaussian’. Can the authors provide proof or a reference showing that this is the case?**\n\n   Our original formulation was imprecise. More precisely, $y(x)∣X,\\theta,{D}$ is the predictive posterior at input $x$ for a Gaussian Process with data $D$. This predictive posterior is Gaussian, and the entropy of a Gaussian has a closed-form expression. Since the entropy depends only on the covariance, it is easily computable. This is a standard result in Gaussian Process literature (e.g., Rasmussen and Williams, 2005).\n\n&nbsp;\n\n5. **It would be great if the authors could provide an ablation varying the number of batches and the batch size to see if HIPE performs well for different settings as well (esp. for Sec. 5.2, 5.3, 5.4).**\n\n   We have included the terminal performance for 5 batches with a batch size of 8 (initialization in the first batch only) as a table below; We will also include them in a potential CR. We can add additional ones as plots in the Appendix, similar to the active learning ablations. HIPE performs very well in this slightly smaller batch size, but one can naturally expect the importance of initialization to be largest when it constitutes a larger part of the overall budget.\n\n   | Function       | HIPE            | LHS-Beta        | NIPV            | Random          | Sobol           |\n   | -------------- | --------------- | --------------- | --------------- | --------------- | --------------- |\n   | Fashion-MNIST  | **84.65 ± 0.07**| 84.19 ± 0.06    | 84.29 ± 0.08    | 84.17 ± 0.04    | 84.23 ± 0.04    |\n   | MiniBooNE      | **85.47 ± 0.02**| 85.25 ± 0.05    | 85.40 ± 0.02    | 85.24 ± 0.04    | 85.21 ± 0.06    |\n   | car            | **84.38 ± 0.15**| 83.10 ± 0.36    | 83.74 ± 0.19    | 82.36 ± 0.37    | 82.98 ± 0.33    |\n   | higgs          | **65.21 ± 0.08**| 64.56 ± 0.13    | 64.90 ± 0.11    | 64.18 ± 0.14    | 64.29 ± 0.14    |\n   | segment        | **83.86 ± 0.18**| 83.12 ± 0.14    | 83.39 ± 0.12    | 82.70 ± 0.18    | 82.95 ± 0.24    |\n\n   Hopefully, this provides evidence that HIPE does well even with smaller batch sizes.\n\n&nbsp;\n\n6. **To further strengthen the evaluation, considering other HPO-benchmarks like [1] and [2] could be beneficial.**\n\n   We thank the reviewer for these references and will consider including these benchmarks in future evaluations to broaden the empirical validation.\n\n&nbsp;\n\n7. **in Fig. 2, the rank of BALD and NIPV tremendously changes when considering RMSE instead of NLL. Can the authors explain why this is happening?**\n\n   Certainly. Since NIPV spreads out points to maximally reduce integrated variance, hyperparameter estimation gets difficult. On the contrary, BALD tends to stack axis-aligned queries (Fig. 1) making hyperparameter estimation simple at the cost of poor coverage.\n\n   With great coverage, the mean prediction (as with NIPV) is generally fairly accurate in most places, but poor hyperparameter estimation simultaneously can lead to drastic over- and underconfidence that punishes the MLL. BALD is the opposite: rarely over-or underconfident, but the mean predictions are not very accurate.\n\n&nbsp;\n____\nWe hope these responses address the reviewer’s concerns. We appreciate the constructive feedback and hope that, if satisfied with our responses, the reviewer might consider raising their score to better reflect the strengths and contributions of our work."}, {"review_id": "7w7ZCk572b", "reviewer": "Reviewer_gQSM", "summary": "The paper introduces a new acquisition function for Bayesian Optimization and Active Learning in the case of hyperparameter uncertainty modeled via fully Bayesian Gaussian Processes. This acquisition function is designed to perform especially well in the large-batch and few shot case. It balances hyperparameter uncertainty reduction, as well as space filling designs for uniform test space distributions.", "questions": "- Why don't you take Query-By-Committee and its variant (Riis et al., 2022) as a baseline since it outperforms BALD in the settings used in their paper ?\n- Would Statistical distance-based Active Learning from Hvarfner et al., 2023 be another choice as a comparative acquisition function?\n- I am wondering whether there are any theoretical guarantees to proof for your acquisition function", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- The topic of hyperparameter uncertainty in practical examples is under-adressed in the current GP literature and therefore of great interest\n- The paper is well written and gives a good intuition for the new acquisition function\n- The experiments are extensive and support the superiority of the new acquisition function against baselines\n\n- I am not sure, whether the authors compare to the current state of the art in their experiments (see questions for suggestions)", "was_revised": true, "final_justification": "Increased score from Borderline accept to accept due to the clarifications and additional experiments given by the authors (see rebuttal for more information).", "rebuttal": "We thank the reviewer for their insightful and constructive feedback. We appreciate the recognition of the importance the topic, and the quality of the presentation. Below, we address the specific questions and suggestions raised.\n\n1. **Why don't you take Query-By-Committee and its variant (Riis et al., 2022) as a baseline since it outperforms BALD in the settings used in their paper?**\n\n   While BQBC outperforms BALD in the Riis et al. paper, it does not do so in the Hvarfner et al. paper, where BALD performs substantially better on identical benchmarks. BQBC performs identically across the two papers. We suspect there may have been a bug in the Riis et al. implementation of BALD, since it has subsequently been more competitive. Therefore, we consider our choice of BALD baseline appropriate.\n\n&nbsp;\n\n2. **Would Statistical distance-based Active Learning from Hvarfner et al., 2023 be another choice as a comparative acquisition function?**\n\n   Statistical distance-based Active Learning (SAL) is indeed a strong candidate in the sequential setting but as far as we know, _it has not been tested in a batch setting_. However, we have run the version that is publicly available in BoTorch. This variant is runnable as a batched acquisition function. Below, you can see updated AL task performance with SAL included:\n\n   MLL performance (larger is better): HIPE performs best, followed by BALD and SAL, which are both competitive.\n\n   | Function   | BALD            | HIPE            | NIPV            | Random          | SAL             | Sobol           |\n   |------------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|\n   | Australian | -2.66 ± 0.08    | **-2.61 ± 0.22**| -7.47 ± 0.80    | -3.00 ± 0.17    | -4.11 ± 0.73    | -3.40 ± 0.24    |\n   | car        | -3.07 ± 0.45    | **-2.68 ± 0.02**| -6.09 ± 0.65    | -3.09 ± 0.14    | -4.16 ± 0.82    | -3.30 ± 0.27    |\n   | hartmann6  | -0.48 ± 0.02    | **-0.31 ± 0.02**| -0.43 ± 0.06    | -0.57 ± 0.07    | -0.48 ± 0.02    | -0.63 ± 0.07    |\n   | ishigami   | **-1.74 ± 0.09**| -2.59 ± 0.23    | -6.94 ± 1.26    | -2.45 ± 0.17    | -2.50 ± 0.42    | -2.32 ± 0.12    |\n\n&nbsp;\n\n   RMSE performance: NIPV generally performs best. Batched SAL does not perform well on RMSE.\n   | Function   | BALD            | HIPE            | NIPV            | Random          | SAL             | Sobol           |\n   |------------|-----------------|-----------------|-----------------|-----------------|-----------------|-----------------|\n   | Australian | 6.40 ± 0.15     | 5.29 ± 0.11     | 4.96 ± 0.12     | **4.70 ± 0.10** | 6.42 ± 0.14     | 4.91 ± 0.11     |\n   | car        | 8.67 ± 0.20     | 6.99 ± 0.17     | **5.99 ± 0.15** | 6.06 ± 0.13     | 8.96 ± 0.22     | 6.23 ± 0.15     |\n   | hartmann6  | 0.37 ± 0.01     | 0.34 ± 0.01     | **0.34 ± 0.01** | 0.37 ± 0.01     | 0.37 ± 0.01     | 0.37 ± 0.01     |\n   | ishigami   | 2.78 ± 0.05     | **2.25 ± 0.04** | 2.25 ± 0.06     | 2.53 ± 0.05     | 2.93 ± 0.05     | 2.47 ± 0.04     |\n\nThanks for the suggestion to include SAL, we will happily include it in the results for the CR.\n\n&nbsp;\n\n3. **I am wondering whether there are any theoretical guarantees to proof for your acquisition function.**\n\n   Our acquisition function is designed to produce a well-informed initialization to improve subsequent Bayesian Optimization performance rather than to minimize regret immediately. As such, regret bounds are not currently available. While theoretical guarantees on model predictive performance might be possible, such results are rare in the broader Bayesian active learning literature. We acknowledge this as a limitation and an interesting direction for future work.\n\n&nbsp;\n___\nWe hope these clarifications address the reviewer’s questions and concerns. We appreciate the constructive feedback and hope that, if satisfied with our responses, the reviewer might consider raising their score to reflect the strengths and contributions of our work."}, {"review_id": "L6TYxv0inO", "reviewer": "Reviewer_Qt15", "summary": "The authors propose HIPE, a method for “cold start” initialization of the hyperparameters of surrogate models. The method is intended for application to surrogate models that will be used in the context of sequential learning settings like Bayesian optimization or Bayesian active learning. The proposed acquisition function is an additive combination of two existing acquisition functions, EPIG and BALD.", "questions": "- The motivation for HIPE is to balance between hyperparameter learning and space-filling design. Yet the EPIG objective targets reduction in predictive uncertainty, not space-filling design. Under what conditions are these equivalent? If and when they disagree, should the user prefer the design that reduces predictive uncertainty, and why? If not, why not use an acquisition function that optimizes for space-filling design directly?\n- I find the notation for the information theoretic measures a bit confusing. In line 79, $y(\\boldsymbol{x})$ appears to be defined as a given value of the measurements (a scalar or vector). In eqs. 1, 2 and 4-8, is $y(\\boldsymbol{x})$ intended to be a fixed value of the measurements, or the random variable characterizing measurements that arise from a particular value of $\\boldsymbol{x}$? The latter is consistent with standard definitions of the EIG. If the latter, should the expectation in eq. 1 be w.r.t. this random variable? If not, could you please clarify how to interpret $\\mathrm{H}[y(\\boldsymbol{x})]$; what does it mean to take the entropy of a fixed value?\n- How are the batches selected? Especially in the large-batch setting, enumerating over every possible batch of locations is generally computationally intractable. It’s common to approximate the optimal batch by greedily optimizing the acquisition function over candidate locations one-at-a-time. How would this practice affect the performance of HIPE?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The paper tackles the important problem of hyperparameter initialization in a principled way. The proposed method is intuitive, and the authors provide thorough empirical evaluation of HIPE against a range of benchmarks and in a variety of settings. Excepting some instances of unclear notation (see Questions below), the paper is clearly written.\n\nThe technical contribution strikes me as somewhat limited: HIPE is a relatively straightforward combination of two existing methods (EPIG and BALD). The paper nevertheless carefully considers and specifies the objectives of the initialization phase, which is important and not always straightforward. I find the proposed hyperparameter-free method for setting $\\beta$ quite interesting and creative.", "was_revised": true, "final_justification": "The paper tackles an important problem, and provides thorough empirical evaluation of the proposed method. My main concerns are (1) limited technical novelty (HIPE is a relatively straightforward combination of two existing methods), and (2) that as currently written, the paper does not make clear for the reader the effect of the EPIG objective in HIPE (to reduce predictive uncertainty, rather than identify space-filling designs). Concern (2) would be resolved by revising the manuscript to include the discussion and clarifications in the authors’ rebuttal. I lean towards acceptance given the value to the Bayesian Optimization community of developing principled approaches for hyperparameter initialization.", "rebuttal": "We thank the reviewer for their review of our work. We appreciate the recognition of the importance of hyperparameter initialization and the clarity of our presentation. Below, we address the specific questions and concerns raised.\n\n&nbsp;\n\n1. **(...) the EPIG objective targets reduction in predictive uncertainty, not space-filling design. Under what conditions are these equivalent? If and when they disagree, should the user prefer the design that reduces predictive uncertainty, and why? If not, why not use an acquisition function that optimizes for space-filling design directly?**\n\n   Great question. Assuming space-filling refers to criteria like minimax-distance or similar, the EPIG objective and true space-filling designs are most likely never equivalent. Referring to our designs as space-filling is a misnomer on our part, we will adjust this.\n\n   Nonetheless, space-filling or random designs are often employed to globally reduce uncertainty, a view supported by many related works [we will add citations]. If this is indeed the purpose, then such a criterion should be employed directly.\n\n   It is certainly possible—and advisable—to prefer a true space-filling criterion (e.g., minimax) if strict input space diversity is desired. However, this would, by definition of the criteria, be less helpful for *prediction*, which we consider the more important measure for Bayesian Optimization performance.\n\n&nbsp;\n\n2. **How are the batches selected? Especially in the large-batch setting, enumerating over every possible batch of locations is generally computationally intractable. It’s common to approximate the optimal batch by greedily optimizing the acquisition function over candidate locations one-at-a-time. How would this practice affect the performance of HIPE?**\n\n   All acquisition functions we consider are quasi-Monte Carlo (qMC) acquisition functions, meaning the joint utility (information gain or improvement) is computed as an expectation jointly over samples from \\( y(X) \\) over a batch of \\( q \\) candidates. This allows optimization over the full \\( q \\times D \\)-dimensional space, enabling arbitrary batch sizes without enumeration. This general qMC-methodology is described in detail in [1, 2], which we cite in Section 4.2 to explain how HIPE fits into this framework.\n\n   We conducted the full joint optimization using the MC formulation since it was feasible, but a sequential approach to acquisition function optimization would be feasible as well. In low-dimensional ablations, we found that joint optimization yielded slightly larger acquisition function values than sequential, but we do not expect that there would be substantial performance differences between them.\n\n&nbsp;\n\n3. **I find the notation for the information theoretic measures a bit confusing. In line 79, $y(x)$ appears to be defined as a given value of the measurements (a scalar or vector). In eqs. 1, 2 and 4-8, is $y(x$) intended to be a fixed value of the measurements, or the random variable characterizing measurements that arise from a particular value of $x$? The latter is consistent with standard definitions of the EIG. If the latter, should the expectation in eq. 1 be w.r.t. this random variable? If not, could you please clarify how to interpret $y(x)$; what does it mean to take the entropy of a fixed value?**\n\n   Thank you for carefully reviewing the notation. $y(x)$ and $y(X)$ always denote the random variable characterizing measurements arising from $x$ (or $X$ in the multivariate case). We acknowledge that the notation was overloaded and will clarify this in line 79. The expectation in Eq. 1 is a typo and should indeed be over the random variable $y(x)$. We appreciate the reviewer for spotting this.\n\n&nbsp;\n____\n\nWe hope these clarifications address the reviewer’s outstanding questions and concerns. If not, do not hesitate to ask further clarifying questions. We appreciate the constructive feedback and hope that, if satisfied with our responses, the reviewer might consider raising their score to reflect the strengths and contributions of our work.\n\n____\n&nbsp;\n**References:**\n\n[1] James T. Wilson, Riccardo Moriconi, Frank Hutter, Marc Peter Deisenroth. *The reparameterization trick for acquisition functions*.\n\n[2] Maximilian Balandat, Brian Karrer, Daniel R. Jiang, Samuel Daulton, Benjamin Letham, Andrew Gordon Wilson, Eytan Bakshy. *BoTorch: A Framework for Efficient Monte-Carlo Bayesian Optimization*."}]

Author rebuttal:

Meta-review: The paper proposes HIPE, a hyperparameter-free initialization strategy for large-batch, few-shot Bayesian Optimization and Active Learning, combining BALD and EPIG in an information-theoretic framework. The method is simple, well-motivated, and easy to implement, addressing an underexplored but practically important problem.

The main point of disagreement among reviewers is novelty: while some consider the contribution incremental (a linear combination of known acquisitions), most acknowledge that identifying and automating this combination for hyperparameter initialization is new in this context. Reviewers also note the method’s strong practical appeal, solid empirical performance, and thorough presentation.

I examined the paper and find the experimental coverage sufficient to demonstrate the method’s merit. Although broader benchmarks and theoretical guarantees could strengthen the work, the majority of reviewers (including several strong accepts) find it interesting and novel enough for NeurIPS. The contribution is practically relevant, well-supported by experiments, and of interest to the NeurIPS community, with novelty concerns outweighed by its usefulness.
Decision: Accept (poster)