Batched Energy-Entropy acquisition for Bayesian Optimization
Authors: Felix Teufel
Carsten Stahlhut
Jesper Ferkinghoff-Borg
Venue: NeurIPS 2024 poster (2024)
Keywords: bayesian optimization
gaussian processes
acquisition
batch
parallel
Primary area: active_learning
TL;DR: A novel acquisition function for batched Bayesian optimization with Gaussian processes that trades off exploration and exploitation with a controllable parameter.

Abstract: Bayesian optimization (BO) is an attractive machine learning framework for performing sample-efficient global optimization of black-box functions. The optimization process is guided by an acquisition function that selects points to acquire in each round of BO. In batched BO, when multiple points are acquired in parallel, commonly used acquisition functions are often high-dimensional and intractable, leading to the use of sampling-based alternatives. We propose a statistical physics inspired acquisition function that can natively handle batches. Batched Energy-Entropy acquisition for BO (BEEBO) enables tight control of the explore-exploit trade-off of the optimization process and generalizes to heteroskedastic black-box problems. We demonstrate the applicability of BEEBO on a range of problems, showing competitive performance to existing acquisition functions.

Batched Energy-Entropy acquisition for Bayesian Optimization
Felix T
Carsten Stahlhut
Jesper Ferkinghoft-Borg
Machine Intelligence, Novo Nordisk A/S *Department of Biology, University of Copenhagen (fegt, ffoiosssoooeeooooorrdii..
Abstract
Bayesian optimization (BO) an attractive machine learning framework for per- forming sample-efficient global optimization of black-box functions. The optimiza- tion process is guided by an acquisition function that selects points to acquire in each round of BO. In batched BO, when multiple points acquired in parallel, commonly used acquisition functions are often high-dimensional and intractable, leading the use of sampling- based alternatives. We propose a statistical physics inspired acquisition function for BO with Gaussian processes that can natively handle batches. Batched Energy-Enten Entrroy acquisition for BO (BEEBO) enables tight control of the explore-exploit trade-off of the optimization process and gen- eralizes to heteroskedastic black-box problems. We demonstrate the applicability of BEEBO on range of problems, showing competitive performance to existing methods.
Introduction
Bayesian Optimization (BO) has since its in- ception [1, 2] made profound contribution to the realm of global optimization of black-box functions through the usage of Bayesian statis- ties. For global optimization problems pursuing a. a (ar), BO has surfaced as premier strategy efficiently handling espe- cially complex and costly unknown functions, (mee While BO traditionally formulated in single-point scenario, where individual points are queried and results are observed sequentially, there are situations where batched acquisition is needed. Such situations arise when five expensive to evaluate in either time cost,
Explore
Exploit
B-UCB
BEEBO
but can be effectively evaluated in parallel by dispatching multiple experiments, reducing the overall optimization time. This is often the case in e.g. drug discovery, materials design hyper- parameter tuning for deep models [3, 4, 5, 6, 7].
Figure I: q-UCB does not allow for controlling its explore-exploit trade-off with large batches. A GP surrogate (background) was initialized with 100 random points of the Ackley function. q-UCB was run with Æ = 0.1 and A = 100, BEEBO with T 0.05 and =0= Batch size Q=100.
The realization that BO could be employed for the training deep neural networks, as sug- gested by [3], sparked renewed research interest, with advancements encompassing a variety of areas, including the generalization accommo-
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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date noisy inputs [8, 9I, heteroskedastic noise [10. 11], multi-task problems [12], multi- muult--fdelityy [13], high-dimensional imput spaces [14]. and parallel methods with batch queries [15, 16). Generally, these desired properties are addressed by customizing one of the two key components in BO, either the surrogate model or the acquisition function. The surrogate model f approximates the black-box function fuve using the available data. In BO, the surrogate is formulated from a Bayesian perspective, allowing us to quantify the model's s uncertainty when evaluating new points. Typically, the model of choice is Gaussian Process (GP) [17]. The acquisition function is responsible for guiding the selection of new input point(s) to evaluate at each optimization step, utilizing the surrogate model to identify promising regions in the input domain and exploring the unknown function further.
Any acquisition process needs to trade off exploration (reducing uncertainty learn better surrogate model) against exploitation (selecting points with a high expected (m) based on the current surrogate). In this work, we are particularly interested in acquisition processes that make this trade-off controllable using a hyperparameter. Controllability can be a desirable property if e.g. domain knowledge relating to the difficulty of the optimization process and the quality of the surrogate model is available, or if the strategy needs to be adjusted depending on future experimental budgets. Similarly, it can be desirable to acquire multiple æ with high fime in batch (as opposed to just finding the optimum I., with the remaining x being considered explorative). This is useful when optima identified in BO can be subject to constraints that are unknown at optimization time, but may render z, intractable [18]. Such constraints arise when the fuve explored in BO is necessary simplification of the actual objective. Practical examples include e.g. the synthesizability of a material at larger scale, when BO experiments are performed at lab scale; or the in vivo activity of molecule with BO experiments performed in vītro.
A wide range of batch mode acquisition functions has been proposed, with approaches often lever- aging random sampling strategies or Monte Carlo (MC) integration, which can adversely affect controllability for large batches (Figure I). In contrast, we here introduce BEEBO (Batched Energy- Entropy acquisition for BO), a statistical physics inspired acquisition function for BO with GP surrogate models that natively generalizes to batched acquisition. BEEBO enables
Parallel gradient- optimization of the inputs, without requiring sampling or Monte
Carlo integrals.
Tight control of the explore-exploit trade-off in batch mode using a single temperature hyperparameter.
* Risk-averse BO under heteroskedastic noise.
We demonstrate the application of BEEBO on a wide range of test problems, and investigate its behaviour under heteroskedastic noise.
2
Related works
Batch variants of traditional strategies Parallel acquisition in BO has seen a variety of approaches, often starting from established single-point acquisition functions like probability of improvement (PI), expected improvement (EI), knowledge gradient (KG) or upper confidence bound (UCB) [2, 19, 20, 21, 22]. Reformulating these to batch mode with Q query points, we obtain q-PI, q- EI, and U-UCB [23, 24]. While the single-poin specifications provide an analytical form and enable gradient-based optimization, batch expressions are more challenging and require different optimization strategies, typically involving greedy algorithms [25] or deriving an integral expression over multiple points.
For instance, in the popular El acquisition function, single point is selected by maximizing the expression agı (z) = E[max (0, f (z) ft)] = max (0, (x) (f) P (f(x) df. Here fi represents the best observed evaluation of fine so far. With a surrogate model in the form of GP. the acquisition function depends only on the predictive mean and variance functions, μ (r) and C (x). Effectively, we need to evaluate the cumulative normal distribution, which quickly becomes intractable for large batch sizes and approximating the gradient of the q-EI acquisition function typically requires MC estimation [26, 27]. However, proper MC integration can be laborious and is sensitive to both the dimension of the problem and the choice of batch size Q. Specifically, MC methods face the curse of dimensionality problem when applied to high-dimensional integrals, as they require an exponentially increasing number of sample points to maintain accuracy, making them
2

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computationally impractical for such tasks [28, 29]. Of particular interest is Wilson et al. [24], in which they adopt the reparameterization trick [30, 31] on acquisition functions integrals, enabling gradient based approaches to the optimization of Pl, EI, and UCB. This demonstrates particular usefulness in modest to higher dimensions.
While EI trades off' exploration and exploitation, users do not have direct control over the balance. To alleviate this, Sobester al. [32] proposed a weighted EI formulation. An alternative strategy with an explicit explore-exploit trade-off is offered by the UCB acquisition function, QUCB (x) ((x:)) + VR. VC(a), which directly expresses exploration and exploitation two terms, traded off by the parameter K. As we are particularly interested in enabling this direct user control, we focus our primary comparison on q- q-UCB in the main text, while more extensive comparison with alternative methods can be found in Appendix B, both theoretically and experimentally.
Greedy strategies As mentioned, popular approach for leveraging single-point acquisition functions is devising batch filling strategies that score candidate points sequentially. Kriging Believer (KB) [33] uses EI to select points and iteratively updates the GP by fantasizing observation with the posterior mean. Likewise, GP-BUCB [34] uses fantasized observations update /C(x) at each step. Local penalization (LP) [35] introduces a penalization function that repulses selection away from already selected points. Contal et al. [36] propose selecting single point using UCB and dedicating the remainder of the batch budget for exploration in a restricted region around the believed optimum. GLASSES [37] treats batch selection as multi-step lookahead problem to overcome the myopia of only considering the immediate effect of selecting point.
Entropy based strategies From an information theory perspective, BO can be interpreted as seeking to reduce uncertainty over the location of optima of the unknown function. This has given rise to entropy-based acquisition functions such as entropy search (ES) [38], predictive entropy search (PES) [39] and max-value entropy search (MES) [13, 40, 41]. MES is distinct in that it seeks to quantify the mutual information between the unknown fuue and the observations y/ D. rather than the location of z.. General-purpose Information eiformattoon-Bsedd Bayesian OptimizatioN (GIBBON) [42] provides an extension of MES that enables application to batched acquisition as well as other challenges such as multi-iidelity BO. GIBBON proposes a lower bound formulation for the intractable batch MES criterion, which then optimized using greedy selection. Despite being formulated to handle large degree of parallelism, Moss et al. [42] reported that GIBBON fails in practice for large batches with Q > 50. Potentially, this behaviour is a consequence of the accuracy of the lower bound approximation. A heuristic scaling of the batch diversity was proposed to improve performance with large batches. GIBBON may also be interpreted as a determinantal point process (DPP) [4, 43). In Appendix B provide detailed discussion of the relationship of the BEEBO acquisition function to GIBBON and DPPs. Note that while we will also make use of the term entropy in BEEBO, the quantity is distinct from the ones leveraged by the aforementioned approaches in the sense that it does not relate to an unknown optimum.
Thompson sampling Given the challenges of generalizing acquisition functions to batch mode, Thompson sampling (TS), which was originally adopted from bandit problems [44, 45, 46, 47, 48), is a popular alternative strategy for guiding batched BO. While being an attractive approach in general, it has been demonstrated that default TS can become too exploitative, motivating the use of alternatives such as Bayesian Quadrature [49], advanced strategies on top of TS that ensure diversity (18). Eriksson et al. [50] demonstrate that overexploration also can be problematic in higher dimensions, and alleviate this using local trust regions in TuRBO. Maintaining such regions with high precision discretization can be
as indicated by [5 I,, who suggest using MCMC-BO with adaptive local optimization to address this by transitioning a set of candidate points towards more promising positions.
3
The BEEBO acquisition function
Assume fave : X + R is some real output associated with the input and a set of data be given D {(2,, W. where y: R represent some noisy observations of fuv( say
y/ fune (2:) + ea
(I)
3

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with e denoting the measurement noise. Let (I1, ,IQ) E X represent collection of test points we wish to assign an acquisition value to. In keeping with the BO framework, we assume given posterior probability distribution over the surrogate function f evaluated at x,
f(x) P([ D)
The lack of knowledge we have of the surrogate function is quantified by the differential entropy H:
P (n
(())D,x) - P(T)
P(TI --
This entropy be contrasted with the expected entropy of the surrogate function if Q observations
y = (V1. HQ) were acquired at x, i.c. if the training data D would be augmented with D' D'(y)= =
[(Tg. 1 to form the joint data set Daug (y) (D))( We refer to this entropy as Huug:
Hong (rI - f Ply (mg))((
where P(y| D, x) represent the posterior predictive distribution at x. The expected information gain, [xx), from acquiring observations is given by the expected reduction of entropy from this process:
T(x) = (() D,x) Haug [
(5)
We propose to represent the explore component of the acquisition function, OBtEBO. by (x). The information gain I (x) is distinct from the quantities exploited by entropy search approaches, as it quantifies global uncertainty reduction, rather than estimating the information over an unknown x.. The information gain is directly applicable to multivariate functions and to heteroskedastic settings where d² = (x). Since large measurement uncertainties imply smaller information gain, óBEEbo exhibits risk-averse behaviour [11] by automatically prioritizing regions of small uncertainties from where more precise information of fine can be obtained, everything else being equal.
The exploit component of BEEBO relies on taking expectation values of scalar function of the random variable f fxx), E R + R. that summarizes the optimality properties of given batch x. Natural choices would be the mean or the maximum of f(x). Of particular interest is expressing the optimality as a softmax-weightt sum over f(x), as this allows us to smoothly interpolate between the two regimes:
E(x) = -E([(x) Q = -E ( Q,
(6)
where 3 is the softmax inverse temperature. At 3 with Q so that both / and E scale linearly with increasing batch size. While the mean provides a closed form expression for its expectation, this is not the case for the general softmax-weighted sum of multivariate normal. Using Taylor expansion, we introduce an approximation of the expectation sum that is fully differentiable and can be computed in closed form. A
recover the mean. We scale the expectation
of the softmax-weight detailed derivation is provided in Appendix A. At 8 = 0, all Q points contribute equally to E(x), whereas at 3 >> O, points that do not compete for optimality are dynamically released. This effect can be quantified as the effective number of points via the entropy of the softmax weights. In the following, we will refer to the (exact) 8 limit BeanBEEBO, and the (approximated) general case as maxBEEBO.
The BEEBO acquisition function then takes the form
(7) where T sets the balance between exploitation (small T) and exploration (large T). As both E and / scale with the batch size Q. a given choice of T would set the explore-exploit balance in an approximately Q-independent manner. This acquisition function bears strong similarity to the definition of (negative) free energies in statistical physics, where F. and correspond to respectively the thermodynamic energy and entropy of the system and T' corresponds to the temperature.
(BEEBO(X) = +T

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3.1 BEEBO with Gaussian processes
Gaussian processes offer particular convenient framework for BO, due the availability of closed- form expressions for the inference step [17]. Specifically
P(f D,x) = (f (((x),((x))
u(x) ((x, XD) MD YD
C(x) = ((x,x) (x, xD) MM ((xx,
(8) where (-- μr. C) is the multivariate Gaussian distribution with mean μ. and covariance C, x and YD the and y values of the acquired data, a (xD) diag (o) , is diagonal matrix with the measurement uncertainties in the diagonal and ((:-)) are matrices derived from the GP-kernel, k(' ), i.e [(x, x') X') k zz2)) It is worth noting that C(x) only depends on the input location of the test points and the data points xD with their corresponding measurement uncertainties, (x(x), but not on the actual observations, yD- Consequently, the entropy of the posterior distribution
MD = K(XD,XD) + +²(xx)
x
Q In (2Te)) + In de)(C(x))
(f|
is independent of yD well, with In det denoting the log determinant. Similarly, the expected entropy if observations at x were acquired, simply reads
Q In ln((Te)) + In det (x)),
1
Hang (,)
(10)
2
where
Cang (x)
K(x,x) K K(x, Xmng) M K (Xaug x)
Maug
K (Xaug: Xmug) + (Xaug)
(11)
and Xaug XD
The BEEBO acquisition function is then given by
( = =[{E(x)] + I(x)
(12)
where the expectation is either the mean, 12 " Mg: or the closed form approximation of the weiihte- sum described in Appendix A, and
1(x) 2
In det(C(()) In det (x)). 2
(13)
All operations needed to compute the ac- quisition value ( analytical. Using automatic differentiation, the batch of points can therefore be optimized with gradient-based methods, as laid out for meanBEEBO Algorithm 1, with learn- ing rate y. In the pseudocode, GP denotes trained GP model object that holds the training data and the kernel function. Using the kernel's 's learned amplitude A. can relate BEEBO's s T parameter to the к of UCB. This allows us to configure BEEBO using a scaled temperature T" that ensures both methods have equal gradients iso- surfaces, enabling the user to follow exist- ing guidance and intuition from UCB to control the trade-off. A derivation is pro- vided Appendix B.I.
Algorithm 1: meanBEEBO optimization
Input: model GP, initial batch points x, temperature T repeat
Calculate μ(x),, Cxx) from Equation using GP
E E - 11g
1==
gP sug <- fantasize(GP
Calculate Caug (x) from Equation 11 using GP aug
I + 2 In del (C(x)) In del (Cang(x)
+- F
- x + until converged Output: optimized batch points
5

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Experiments
Table Overview of the test problems used the experiments.
Test problems We benchmark acquisition function performance on range of maximiza- tion test problems with varying dimensions (Ta- ble 1) available in BoTorch [52]. Test problems that are evaluated on multiple dimensions sup- port specifying the respective arbitrary high-dimensional problem with low inherent dimensionalityy, we embed the six-dimensional Hartmann function in d 100 [50, 53, 54). We additionally test on two robot control problems (robot arm pushing and rover trajectory plan- ning) in Appendix D.3 [55, 56].
Function
Dimension
Ackley
2, 10, 20, 50, 100
Shekel
4
Hartmann
6
Cosine
8
Rastrigin
2.
10, 20, 50, 100
Rosenbrock
2.
10,
20, 50, 100
Styblinski-Tane
20, 50, 100
Powell
10, 20, 50, 100
Emhedded Hartmann 6
100
On each test problem, we perform 10 rounds of BO using q-UCB or BEEBO with a given explore-exploit parameter for direct comparison. We use the scaled temperature T (B.1) to ensure that both methods operate at the same trade-off. In round O, we seed the surrogate GP with Q random points that were drawn so that each point has a minimum distance of 0.5 to the test problem's 's true optimum. We perform ten replicate runs for each problem and method, with replicate seeds controlled so that all methods start from the same Q random points in replicate. As evaluate performance in a fixed-round fixed-Q optimization scenario, the explore hyperparameter to 0 in the last round (for maxBEEBO, we also set the softmax /3 to 0). We use Q 100 for all experiments, which is commonly understood to be a large batch size [50]. Additional results on small batch sizes (5, 10) are provided in Appendix D.2. All experiments use BoTorch's 's default utilities for acquisition function optimization and GPyTorch [57] GP training (C.1).
set
Heteroskedastic noise We investigate performance when optimizing under heteroskedastic noise on the 2D Branin function with three global optima. To construct a heteroskedastic problem, we specify noise that the noise level is maximal at optima and 3, decaying exponentially with distance to any of the two noised optima (C.4). No noise maximum is added at optimum 1. Therefore, while all three optima share the same (z) (Figure A1), only optimum 1 is favorable in terms of heteroskedastic risk. We perform BO for ten rounds with 3 = 0.1 and Q = 10 using a heteroskedastic GP that learns surrogate models for both (ame(a)) and o² (x)) We report results five replicate runs.
Metrics We report the mean best observed objective value after 10 rounds over the five replicates. As test problems have highly varying scales, we normalize the results each test problem using min-max normalization. Typically, the minimum of a maximization problem is not known explicitly. We therefore set the minimum for normalization to the highest value observed among the random seed points. The maximum is given by the fime(*)) of the problem. The metric thus directly quantifies how much progress has been made the optimum from the random starting configuration 0-1 scale.
As not only interested in identifying single with good fu))(z), additionally quantify the overall quality of the final (exploitative) batch. We compute the batch instantancous regret R ExEc fxme(x*)- Sime (x.g) of the last, exploitative batch. To bring results all test problems to similar scale, we divide it by the batch instantaneous regret of batch of Q random points each problem. We refer to this metric as the relative batch instantaneous regret, Riel = R==10/Rrammom
For BO under heteroskedastic noise, we wish to quantify the preference of given method for different optima. As optima share the same Jmme(x"), ),"", metrics operating on fime are inherently unsuitable, and preference needs to be evaluated on æ directly. For each acquired point we compute the distances || x>]l22 l| to the J individual optima. We report the mean distance to each optimum over all points in a batch.

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Table 2: Highest observed value after 10 rounds of BO with Q = 100. The best value at each indicated in blue. BEEBO is configured with T' = 1/2/R. R. Full BO curves are provided in D.6, confidence intervals and statistical tests in Tables A2, A3 and A4
Problem
v/= 0.1
v 1.0
V/= =
10.0
meinBEEBO
maxBEEBO
qCCCB
mcinBEEBO
maxBEEBO
qCCCB
meunBEEBO
maxBEEBO
q.UCB
Ackley
0.973
0.967
0.988
Levy
1,000
1,000
1,000
0.999
1,000
1,000
0.999
0.996
0.998
Rastrigin
0
0.991
0
0.989
0.988
0.963
0.983
0,993
0.933
Rasenbrock
2
0,976
0,982
0.949
0.956
0.979
0.943
0.955
0,938
0.962
Styblinski -ang
0
1,000
1,000
1,000
0.999
Shikel
0.540
0,300
0.244
0.915
0.378
0.330
0.698
0:111
0.264
Hartmann
6
1,000
0.894
0.918
1,000
0.976
0.950
0.986
0.974
0.889
Cosine
1,000
0.999
0.934
0.999
0.972
0.924
0.619
0.895
0,621
Ackley
0.915
0,800
0.908
0.736
0.772
0.822
0.513
Levy
10
0.989
0.966
0.904
0.966
0.953
0.904
0.966
0.914
0.560
Powell
10
0.987
0,951
0.920
0.970
0.949
0.916
0.861
0.909
0.283
Rastrigin
10
0,463
0.558
0.420
0.536
0.573
0.522
0.595
0.590
0.311
Rasenbrock
0.9144
0,991
0,966
0.991
0,986
0.971
0.903
0.975
0.645
Styblinski-Tang
10
0.837
0.822
0.309
0.835
0.6338
0.492
0.289
0.229
0.049
Ackley
20
0.827
0.818
0.741
0.851
0.781
0.753
0.777
0.404
0.474
Levy
0.949
0.926
0.943
0,914
0.899
Powell
0.955
0,939
0.948
0.965
0,913
0.913
0.872
0.915
Rastrigin
20
0,359
0.484
0.423
0.473
0.472
0.480
0.508
0.522
0.401
Rosenbrock
20
0.993
0.992
0.973
0.995
0,9983
0.982
0.5077
0.933
0.924
Slyblinski-Tang
0.667
0,203
0.689
0.391
0.561
0.330
0.054
Ackley
50
0.235
0.623
0.638
0.342
0,594
0.759
0.823
0,465
0.730
Levy
50
0.940
0.971
0.948
0.965
0.958
0
0.943
0.879
0.941
Powell
50
0
0.975
0.970
0.982
0,969
0.961
0.950
0.938
0.980
Rastrigin
0.322
0,431
0.432
0.481
Rosenbrock
50
0.971
0.976
0.968
0.984
0.981
0.986
0.983
0.962
0.98]
Styblinki-Taam Tang
50
0.584
0.509
0,312
0.675
0.342
0.694
0.393
0.325
0.356
Ackley
100
0,277
0.417
0.708
0,190
0.540
0.645
0.863
0.682
0.844
Hartmann
0.987
0.928
0.916
Levy
100
0.837
0.966
0.950
0.961
0.950
0.934
0.944
0.940
0.964
Powell
100
0.810
0.985
0,982
0.952
0.980
0.98.
0.983
0,979
0.984
Rastrigin
100
0.497
0.441
0.446
0.401
0.455
0.442
0.459
0.455
0.44%
Rosenbrock
0.972
0,971
0.969
0,980
0.978
Stybllinki-Tangg
100
0.537
01423
0.308
0.474
0.353
0.532
0.401
0.296
0.278
Mean
0,795
0.811
0.758
0,828
0,790
0.801
0,788
0.744
0.678
Median
0,940
0,951
0,920
0
0.913
0.889
0.819
5 Results
5.1 BO on test problems
We benchmark BEEBO against q- at three rates of K. Overall, we find that the simpler mean- BEEBO variant outperforms BaxBBEBO in terms of mean performance on all but the lowest rate of K (Table 2). As we consider the configuration with the lowest rate to be exploit-dominated, this can be understood further exploration, with the low explore rate seeming sufficient to induce the necessary diversity.
consequence of maxBEEBO effectively releasing non-contributing points for
While results individual test problems vary, mcanBEEBO shows improved performance over q-UCB especially in the medium dimension range up to 50. For d=100, we find mixed performance, with MeanBEEBO gradually becoming more competitive with increasing k. This is due to the fact that with increasing dimensionality, more exploration is beneficial for learning a good surrogate model before an actual BO process becomes effective. As we find that q-UCB inherently performs more random-like sampling at large Q, irrespective of k, it benefits in such situations.
On average over all 33 performed experiments, Bean improves upon q-UCB for large batches at any of the three rates (Table A3). We additionally benchmarked BEEBO against other popular BO strategies without explore-cxploi hyperparameters. Interestingly, we found that the Kriging Believer (KB) iterative heuristic [33] can perform very competitively for large batches when using LogEl [58] as the acquisition function (Table AI), especially on the two robot control problems (Appendix D.3), but can be slower to optimize than BEEBO (Table A9).
When evaluating Ri in the ultimate round of BO, we find that BEEBO allows us to effectively acquire a batch with high fime( highlighting the controllability of the acquisition function (Table 3). The REd of q-UCB is only slightly better than the R of a random batch in many cases, even though the explore component was explicitly set to We note that this is not due to the surrogate function being unsuitable- the results in Table 2 indicate that in most cases the location of fine (xme(z)) ) is approximately
7

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Table 3: Relative batch instantancous regret Ryel in round 10 (K = 0) with Q = 100. The best value at each is is indicated in blue. BEEBO configured with T = 1/2v/.. R. Lower means better.
Problem
-==10
v==
memBBEBB
maxBEEBO
e-UCB
mcmmBEbB0
naxBEEBO
e-UCB
mammBEEBO
maxBEEBO
GUUCB
Ackley
0.292
0.259
1,006
0.268
0.245
0,999
0.257
0.165
1.002
Levy
0.134
0.092
0.102
0.102
0.11]
Rastrigin
0.455
0.578
1,010
0.425
0.454
0.999
0.407
0.500
1,020
Rosenbrock
0.001
0.004
0.992
0.001
0.004
1.094
0.002
0.002
1.014
Styblinski-Ting
0.166
0.172
1.024
0.169
0.170
10077
0.170
0.170
1.051
0.810
0.776
0.688
0.730
0.688
0.695
0.988
Hartmann
0.060
0.229
0.968
0.078
0.086
0.971
0.100
0.098
0.862
Casine
0.045
0.001
0.953
01.011
0.016
0,975
0.222
0.061
0.922
Ackley
0.478
0.338
0,931
0.314
0.345
0.943
0.253
0.452
0,950
Levy
0.041
0.830
1,188
0.023
0.048
1.011
0.361
0.103
1.111
Powell
10
0.016
0.027
1,037
0.009
0.067
1.101
0.067
0.151
1:215
Rastrigin
10
0.629
0.563
0,920
0.523
0.541
0,907
0.567
0:402
0,905
Rosenbrock
0.002
0.004
0.074
0,052
Slyblinkk-TTng
10
0.196
0,220
1,174
0.223
0.337
1,126
0.559
0.196
1.219
Aokley
20
0.629
0.219
0.945
0.282
0.292
0.50
0.226
0.586
0.917
Levy
20
0.128
0.241
0,839
0.063
0.113
0.914
0.140
0.182
1.056
Powell
0.093
0.010
0.074
0.028
0.110
Rastrigin
20
0.686
0.600)
0 866
0.610
0.635
0,838
0.541
0.555
0,852
Rosenbrock
20
0.017
0.105
0.591
0.001
0.048
0,578
0.036
0.051
0,903
Styblinski-Tang
0.426
0.398
1.113
0.378
0.501
1.107
0.691
0.578
1,177
Ackley
0.895
0.738
0.177
0.530
Levy
0.055
0.029
0.611
0,033
0.085
0.68]
0.051
0.268
0.892
Powell
50
0
0.021
0.542
0.014
0.078
0.499
0.018
0.064
0.785
Rastrigin
0.793
0.573
0.813
0.653
0.592
0,810
0.795
0.585
0,768
Rosenbrack
0.016
0.049
0.010
0.048
Styblinski-Tang
0.463
0.478
1.012
0.676
0.574
L,196
0.681
0.727
0.81
Ackley
100
0.718
0.636
0,948
0,900
0.466
0,935
0.137
0.321
0.863
Emhh. Hammmnn 6
100
0.068
0.144
0.035
0.086
0.863
0.175
0.172
0.692
Levy
0.119
0.044
0.164
0.042
0.056
Powell
100)
0.054
0.011
0.465
0.0027
0.141
0.495
0.013
0.018
0.524
Rastrigin
100
0.506
0.501
0,759
0.604
0.557
0.832
0.540
0.544
0.780
Rosembrock
0.114
0.027
0.014
Stybiliski--Ting
0.389
0.522
0.924
0.503
0.582
1,2003
0.562
0.742
0.9%0
Mean
0,291
0.256
0.872
0,257
0,265
0.889
0,261
0,292
0.904
Median
0,134
0,219
0,931
0.092
0,161
0,943
0.175
0,172
0,917
known by round 10. Rather, we assume that this a consequence of the challenges of MC-based optimization of the acquisition function large Q.
5.2 BO under heteroskedastic noise
We compare performance of meanBEEBO and q-UCB on the 3-optimum Branin function. Under heteroskedastic noise, we find that BEEBO preferentially optimizes towards the low-noise optimum I at the expense of the noisy optima 2 and 3 (Figure 2) and is therefore risk-averse. In the homoskedastic case, BEEBO does not exhibit this preference and optimizes for multiple optima. As expected, q-UCB, which only uses the model posterior variance vC(x) instead of quantifying the actual information gain, does not display any preference for low-noise optima, showing similar behaviour under heteroskedastic and homoskedastic noise and remaining risk-neutral.
6 Discussion
We introduce BEEBO, an acquisition function for BO with GPs that can be optimized analytically and that scales natively to batched acquisition. By exploiting the independence of the information gain l(x) on measurements when using GP surrogates, BEEBO models the interdependence of unknown points x in a batch and can optimize their positions jointly using gradient descent.
BEEBO enables full control of its explore-exploit trade-off using a hyperparameter T that directly balances two terms, akin to UCB. Unlike in the reparametrization-baasee q-methods, BEEBO's T has predictable behaviour also at increasing batch sizes.
The numerical complexity of BEEBO is dominated by the need to compute the inverse of Maug in Equation 11, which in a plain implementation scales as O(NN + Q) However, this can be reduced to ((N specifically, the Cholesky decomposition of Mang can be expressed as Q rank-1 updates of the pre-computed Cholesky decomposition of MD. where each update will have the complexity of O N2). The calculation of the energy, F., and the information gain, 7, scales as O(N Q) and ((²*), respectively, and are thus sub-dominant to the update needed for M For large N this
8

---

Heteroskedastic noise
Homoskedastic noise
15
BEE8O
30
Optmum Opömum
Optmum
15
q-UCB
10
Bound
Figure 2: Mean distances of acquired points to the different optima of the Branin function. Under heteroskedastic noise, BEEBO risk-averse and preferentially optimizes towards the low-noise optimum 1. Under homoskedastic noise, there is no preference. q-UCB does not adapt its behaviour noise, remaining risk-neutral. The means and standard deviations over five replicates are shown.
approach may nevertheless become prohibitively slow. To overcome this limitation, methods for scalable GPs and fast predictive covariances such as LOVE [59] can be considered. The LOVE method allows further reduction of the complexity of the Cholelsky update of Maug to O(N r Q) [60], where r the rank of the LOVE approximation for Mp. typically r < N.
As opposed to g- JCU, BEEBO can take heteroskedastic noise into account when computing the information gain, and preferentially acquires more informative low-noise points. We note that when the noise function o² (a:) is unknown, and needs to be explored at the same time with fine (z). it is critical that the initial random points sufficiently capture the noise landscape well enough for the information gain component to be useful, as the uncertainty of the surrogate on o² (z) is not used. This would require a fully Bayesian approach that integrates over the distribution of a² (zx). The problem does not arise if the heteroskedastie noise of experiment is known beforehand by e.g. instrument calibration. While not the focus of this work, we note that using the information gain could also be beneficial in sequential single-sample BO on heteroskedastic problems.
7 Outlook
In our experiments, we have focused on maintaining consistent explore-cxploit ratios throughout the optimization rounds to ensure an equitable experimental comparison with q-UCB and demonstrate the effect of the hyperparameter choice. However, more dynamic approach involving variable ratios could be more effective in real-world applications with predetermined number of rounds [61]. Adopting a fully Bayesian perspective, one could consider the temperature hyperparameter T as a random variable. This opens up an intriguing avenue for BEEBO, where T could be drawn from a prior distribution that e.g. varies across optimization rounds, depending on the specific application. By tailoring this distribution, one could encourage high level of exploration in the initial rounds, gradually transitioning towards a more exploitation-foct focuused approach towards the end. In the presented experiments, we have implemented this strict constraint, maintaining fixed T until the final round, at which point we shift to full exploitation, i.c., T 0.
While not explored in this work, we note that the BEEBO expression could naturally be extended to multi-objeecive optimization problems by capitalizing on GPs that handle vector-valued functions, such as multi-tase task GPs [12, 62). Through e.g. the usage of the intrinsic model of coregionalization. we obtain a covariance function k and thereby covariance matrix C(x), over all input-task pairs. As the multi-task covariance matrix is jointly Gaussian, the expression of the information gain remains unchanged and can computed like the single-task case. The energy F(x) becomes vector-valued, providing an energy term for each of the tasks. This would allow for the introduction of task-specific

---

weights in the acquisition function. As the extension only affects the surrogate model, the scaling remains cubic in the number of imput-task observations.
Beyond GPs, BEEBO could be generalized to work with any probabilistic model. However, GPs unique in that Huu is available in closed form and be used to compute T(x) analytically, without solving the integral over y in Equation 4. Other models may require approximations and sampling-bes basse approaches for computing the information gain.
Availability
A BoTorch implementation of BEEBO is available at
https://github.. httppss//ggthuub.ccm/vvvoorrrrd conm///oooorrddikkkeseeeerereEE
Acknowledgments and Disclosure of Funding
We thank Christoffer Riis, Jan C. Refsgaard and Kilian W. Conde-Frieboes for helpful discussions. FT was funded in part by the Novo Nordisk Foundation through the Center for Basic Machine Learning Research in Life Science (NNF200C0062606).
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Appendix
Table of Contents
A Approximating the expectation of the softmax weighted sum
15
A.1 Motivation
15
A.2 Derivation
16
A.3 Practical considerations
17
A.4 Number of effective points
18
B Relationship to other acquisition strategies
18
B.I Relationship of BEEBO 7 and UCB x hyperparameters
18
B.2 GIBBON
19
B.3 Determinantal Point Processes
20
B.4 Local penalization
21
B.5 RAHBO
21
Implementation details
22
C.I Acquisition function optimization.....
22
C.2 Benchmark BO methods
................................
22
C.3 Test problems
23
C.4 Heteroskedastic noise
Extended results
24
Results including additional baselines
24
Results for batch sizes and ................................
26
D.3 Control problems
28
D.4 Run time
28
D.5 Results with random initialization in round
29
D.6 BO curves for all experiments in Table and Table Al
30
A Approximating the expectation of the softmax weighted sum
A.1 Motivation
We are free to choose any energy function E in BEEBO, the only requirement being that we are able to compute an expectation of in order to obtain the scalar summary = =f)]] Of particular interest is the softmax weighted sum,
B(f)
softmax(3f).ftt
(AI)
where 8 is the softmax inverse temperature. The softmax weight vector computed
exp(3f.)
Wi
(A2)
1 exp( + exp((,
where exp(3gMmaz) is an optional reference threshold value, as in expected improvement, which set to 0 if not used (B, is either simply 3 dynamically scaled value that ensures É(f) does
15

---

not become O, see Equation A28). The parameter / allows us to interpolate between two extreme regimes,
E(f) = ((() max
for - 0
(A3)
Q
for 00
(A4)
In the first regime, all points of a batch equally contribute the energy, whereas the second regime only the single point "responsible" for the maximum is controlling the energy. Note that for numerical reasons, operating towards the 3 - oo limit impractical, will lead zero gradients for all but one point, preventing optimization. We of the GP kernel. Since A represents the expected energy fluctuations for points far from data, this weighting scheme will reflect natural compromise between Equation A3 and Equation A4.
3 A-1/2, with A being the prior uncertainty scale
As opposed to the mean, in the general case, the expectation of the maximum of a Q Q-dimensional multivariate normal is not available in closed form. To our best knowledge, this is also the case for the softmax weighted sum. In the following, we derive a closed-fore approximation of the expectation of the softmax of 3f that can be used for gradient-based optimization.
A.2 Derivation
Consider the softmax denominator
d(f) exp( exp(2y)ma))
(A5)
We will Taylor expand In(d) to the second order, using
i (n((d) (d)
3 exp(Bf)) = Bw:
(A6)
Of: d
8 In(())
1 0f.ff = 3² (- (- exp(B(fi + fz) + d
(A7)
d2
(w - w(ws),
(A8)
where di is the Kronecker delta. So
32
In(d) = In(d,) + Sw Af AAT W Aff
(A9)
2
where a is the Taylor expansion point, d, is d evaluated at a, Af a, w is the Q-dimensional
vector w evaluated at a, and W is the Q x Q matrix W = diag w) wwT T Inserted into Equation A2
we have
B 2
(f); exp(3Af: -Bw Af APT W. Af)
(A10)
With this approximation we can calculate the expectation value
Blw(f); * fl] W/y (ex
exp(A, f.df
(A11)
(2T)
xx(f) BAʃ Bw APT
AFT W. Af - 1¹" .0(x) -1 .. -f- 11)
2
(A12)
= 0 1 (f v T C(x) C(x)aal softmax (f - (())
())T
(A13)
2
16

---

Where c (0 and C(x) softmax are defined as follows:
((x)softmn (C(x) + 3W
(A14)
( = e - -w
(A15)
( C(x)
(ab()) + -4 BBW
(A16)
( (1 CC( 2 BBbb())
(A17)
2
8² aaa W. a zu² .C(x)-1
(A18)
and ( is the i'th basis vector with components of dig. We can avoid the explict use of the precision matrix by rewriting the updated covariance matrix
((x)sax (C(x)-+ (1+ Cx(() w)) U(x) C(x),
where we have defined ('(x) = B3+ w) The updated mean vectors can conveniently be expressed as
(+)_ sxx)stl
(Bb ((x))= μ 3 a)
(A19)
(3) = 3C(x)softmax softmax e + C(x) softmax (-3w + C(x) # B a)
(A20)
x+) 3C(x) softmax
( +v',
(A21)
where v' is constant vector for all (i). Similarly
( = -aa + /
is + Bv/ + = 3w a + "T C(x) v'
(O(x))) (Cx
(A22)
8² W
2 a
C(x)-1 μ
(A23)
2
with d' again being a constant for all (x). The expectation the softmax weighted summary given by
BIE] K Σ wi exp(c(0) * vi
(A24)
where K (det(C(x)-)
(det(C(()) +,99W) v (u))( The most natural choice for the expansion point is in which case (1) and c(() reduces to
(i)
3C softmax (el -w) +
(A25)
g2
2 (e()) -w) w)²
T C(x))sffmmx (e(#-w) -w) w)
(A26)
A.3 Practical considerations
Linear algebra For numerical reasons, we avoid computing C(x) softmax explicitly, and instead use the Uxx). C(x) factorization compute solutions with U
(xC()) W.
sox)sxtmax (e()) w) = Ux() (C(x) - (t) C(x) w)
(A27)
Following GPyTTorch practices, we make use of the LinearOperator package to exploit the structure of (xx) an AddedDi agL inearOperator when solving. For determinants, we find that Linear- Operator's logdet implementation gives nondeterministic results, and we therefore perform dense cast before computing K using default Pytorch.
While the factorization numerically advantageous, it is still limited with regards to 3. We find that at 5, numerical emors prevent reliable calculation of the expectation. In practice, -12 lies in range that allows numerically accurate solutions.
17

---

Softmax When Ymax grows much larger than the softmax input vector f a situation that can arise easily when initializing with random points for gradient- based optimization the softmax weights w can become numerically zero for all "real" points, thus leading to E(() = O, and vanishing gradients. As we always wish to preserve minimal energy contribution from the real points, we parametrize the inverse temperature applied to Vmax. Sy. using hyperparameter a that denotes the minimal fraction of probability mass pertaining to real points. This parametrization resembles the LogEI version the expected improvement acquisition function [58] to address the problem of vanishing El-gradients.
Let N denote the softmax denominator excluding 1/max N
2ز=1 exp(3Afi). We define
min -1-
N, eexp(3A)0laa)
(A28)
α
We used a = 0.05 as a default all our experiments.
A.4 Number of effective points
We can interpret the softmax as the number of effective points contributing to the energy of the batch. The entropy H of the softmax is given by
H(w)
Σ wi In(w;),
(A29)
and the number of effective points, Dett is exp(H so that
Der exp
Σ w, In(wz)
(A30)
Dew bounded by (approaching the maximum) and Q (approaching the mean). Note that if include 1/max in the softmax denominator, we add wiyna In(w)am) to H (w), and the resulting number becomes bounded by and Q + 1.
B Relationship to other acquisition strategies
In the following section, we will discuss how BEEBO is related to UCB, GIBBON, Determinantal Point Processes (DPP), the Local Penalization heuristic and RAHBO. We will base our analysis on mean as the softmax-mediated interdependency of points in maxBEEBO prevents simple interpretation the objective in a single-point stepwise manner and does not allow for the same direct analogies to other strategies.
B.1 Relationship of BEEBO T' and UCB hyperparameters
BEEBO bears some resemblance the UCB acquisition function, which in the single particle mode, Q = 1, reads
(x) = m(x) + vk)C(z).
(A31)
where the parameter x controls the balance between exploitation and exploration and (=x) and C(x) respectively the mean and variance of the posterior distribution, P(f xx, D). before. We note that avcB does not account for the uncertainty the measurement at æ, and therefore remains risk-neutral under heteroskedastic noise [11]. To understand the relationship between BEEBO and UCB, we will therefore limit ourselves to the homoskedastic case and furthermore assume that measurement variances 2² much smaller than the typical prior variance of the GP surrogate, A, of f. e.g. A N N-'Tr(K) Tr(K), so x² < and M (K + o²) x K-1 In this limit, the variance of f (x:) after measurement (indexed at n, say) reduces 2
C(x) ((K-) + to -1) nn (K(K + +02) R0²
(A32)
18

---

and the information gain becomes
((x) In(C(x)) log
Consequently, the gradient of the two acquisition functions reads
VaucB(() = Vp(z) + VK
VC()) VC(x)
T
VABEEBO(I)
Vu(z) + 2- C(x)) VC(x).
The two gradients will be identical at points a where the posterior uncertainties satisfy vC(x) T For comparison, we may desire equal gradients at iso-surfaces corresponding to a given fraction, of the prior uncertainty scale VA, by setting I' accordingly as T' = v. vA /k. In our experiments, we use / = 2/2 and configure BEEBÒ using dimensionless T" explore-exploit parameter, defined as T" VA and set T" 2 VE for given benchmark experiment.
Vk²
B.2 GIBBON
GIBBON [42] approximates the (intractable) General- purppsse max-value Entropy Search acquisition function, which quantifies the mutual information MI y D) of a batch of measurements and the unknown optimum fine- It does so using a lower bound on the information gain and MC estimation of the expectation over fine. It can be written as
(GIBBON(X) 1
det
Σ ME i=1
Σ In (1- po ((T(mm)) ((7((m)) [((m) + ((7((m))
M|M
{(>i(m))
(IGIBRON(X) 2
det) + åGIBRON (x;;)
(A33)
C(x)s;
M is set of samples for where R is the correlation matrix with entries Ra CC()),, the max-value fime. and pÚ the correlation of ys and fune (z;). & and ø are the standard normal cumulative distribution and probability density functions, and (mm) = m=
The definition of BEEBO introduced in Equation 7, with the scalar summarization function set to the expected mean, (xx) - - Dd p(x.). gives
OBEEBO = T (en (C(x)) II det (Caug (x))) +
Σ m(xx).
(A34)
From the second formulation of GIBBON, it becomes obvious that although being distinct their motivation and derivation, BEEBO and GIBBON implement acquisition functions with similar structure. Taking an information theoretic and multi fulti-fidelity BO standpoint, GIBBON refers to this trade-off as diversity against quality, whereas in BEEBO we follow the intuitions of UCB, and use exploration and exploitation.
Quality Exploitation: GIBBON employs an MC estimate of the lower bound approximation of the information gain provided by each point, whereas BEEBO directly summarizes the optimality of all points in closed form, either as their mean approximated softmax weighted sum.
Diversity Exploration: In GIBBON, the diversity derived from the differential entropy H(f)D, x) is the entropy of the posterior correlation 7 In det(R).. In BEEBO, we em- ploy the reduction of entropy, the information gain T(x). Under homoskedastic noise, T(x) In dct((x))) Since R(x) diag diag(C(x))--22 C(x) diag(C(x)))+ -1/2, have that In det (R) = In det (C(x)) - D (n((()):)) Therefore, maximizing the log determinant of R. penalizes points that have high variance.
19

---

Therefore, while GIBBON presents an attractive approximation of max-value Entropy Search for batched acquisition, BEEBO an alternative that avoids approximating quality criterion using MC. Moreover, GIBBON's diversity criterion implicitly penalizes points that have high variance, whereas BEEBO's criterion maximizes the reduction of variance. We find that BEEBO is orders of magnitudes faster to compute than GIBBON (Figure A3).
In the context large batches (Q >> 10), modification of GIBBON exists that is further similar to BEEBO. Departing from the strict max-value entropy search derivation, scaling factor Q- is introduced to counteract growing dominance of the diversity term:
AscaledGiBBON (x)
1 2Q
In de + Σ ÔMES (z+).
(A35)
This scaling is motivated by the fact that R contains Q2 elements. However, we note that R is summarized by its log determinant, which scales linearly in Q: As the determinant is the product of the eigenvalues, the log determinant the sum of the log-eigenvalues. The number of eigenvalues scales linearly with matrix size Q, and so does the log determinant.
B.3 Determinantal Point Processes
Determinantal Point Process [43] specifies a probability over a set of points, or a "configuration of points" drawn from a ground set. Specifically, the probability of set of Q points x is given by
P(x) x det (Ix),
(A36)
where Lxx is a Q x Q symmetric matrix. Kulesza et al. [43] provide a decomposition of the general DPP kernel I, that makes quality and diversity components explicit, that
Lij
(A37)
with A being Rdd x Rd - RR+ similarity kernel, and q being a unary Rd - R scalar quality function. This framework is naturally amenable to batch BO, as we seek to select a collection of points that trade off quality (optimality) and diversity. Note that both ks and g are distinct functions that need to be specified by the user, leading to the practical complication that they must be chosen very carefully so that their scales do not dominate each other, which limits the utility of this decomposition in practice [42].
In the following, show how BEEBO is equivalent to DPP, and derive the necessary k and q. Again, we consider BEEBO
(BETBO(()) - -x(()) + + +
(A38)
with the scalar summarization function set to E(x) = - - f(x)). We will first focus the information gain term [xx), which we can rearrange as
1 1(x) = In det(C(())) In det (x)) = In det (C(x) Crng (x))
(A39)
2
2
Our similarity kernel k is therefore given by the entries of the matrix S = C(x) Caug (x) 11 that k(,,, 22.)) = Sty. Note that due to the augmented covariance term, the implied k also depends on all other currently selected points in x, and Lx is not submatrix of an all-sample L. Therefore, BEEBO does not implement a DPP under heteroskedastic noise. However, we only consider homoskedastic noise, BEEBO's T [xx simplifies to the posterior entropy [63], and therefore S C(x). As C(x) can be accessed as a submatrix of an all-sample C, this permits a DPP.
Given the choice of Eff),
rewrite BEEBO as
20

---

1
O dnd (5) * T
Pi
2
Q
2
* (BBEBBO = In (5) +
T
T
1=1
2
2
OBEEBO (x) Indet (5) + In det (D))
with D (exag 11))
2
(BBEEBO( (x) de (D+ S. 1²)
2 D V /exp( (exp( exp( ( 1
( dn (Dž. DJ T
1
(x) In det (L) TT+
(A40)
where is matrix with entries Lay Sig xxp(} exp(( 2 pÿ). BEEBO therefore uses the DPP quality function ((x;) exp( 7 (4;), and, like proven previously for GIBBON, batch with maximal OBEEBO corresponds to the MAP of a DPP.
B.4 Local penalization
Local penalization (LP) is a greedy batch selection strategy that given any arbitrary single-point acquisition function, ensures diversity by applying penalization function vO(x, x') that downweights the acquisition value candidate locations based on their proximity to already selected points. The criterion for selecting x; is given by
= arg max (( Il (x,
(A41)
Note that in this formulation, the product includes all previously selected points, just the current batch. The penalization function v0 may in principle be chosen freely. Gonzalez et al. [35] propose exploiting the fact that fwne is Lipschitz continuous in order bound the position of the unknown optimum and penalize accordingly. The Lipschitz constant L is inferred from the GP surrogate and used parametrize v. In LP, acquisition function optimization proceeds iteratively. After an x; chosen, the corresponding penalizing multiplier is added to the objective before optimizing for the next 1+
While BEEBO enables optimization to proceed in parallel, is of course possible to also optimize BEEBO greedily (under homoskedastic noise, is submodular). In this case, it implements an LP strategy where a(x) μxx)). Rather than product of individual Rd x Rd + R function evaluations, the penalizer implied by BEEBO is the information gain [(x) : Rixd + that we evaluate by concatenating candidate point to the already acquired each iteration. Like in GIBBON, this constitutes an LP strategy that does not require estimation of any properties fine beyond learning the GP surrogate.
B.5 RAHBO
Risk-averse Heteroskedastic Bayesian Optimization (RAHBO) [11] is a UCB-derived single-point acquisition function that avoids heteroskedastic risk, preferentially selecting points with low noise. While it is not applicable to batched acquisition directly. we here compare it to single-sample BEEBO to highlight different ways of addressing noise. Given heteroskedastti surrogate model that learns an additional GP for the noise, the variance proxy, RAHBO reads
21

---

(R)H ORAHBO(Z)
Lar)
My + By of (0)
Byar * J (x))
(A42)
where 2uf and of are the posterior mean and variance of the surrogate model and 9 is the standard UCB trade-off hyperparameter, yielding the standard upper confidence bound UCB,. a is the chosen risk tolerance, and LCB is the lower confidence bound of the variance GP with posterior mean Pvar and variance var traded off using Bvar-
At Q = 1, BEEBO can be expressed as
1
OBEEBO = 1s((z) + In(ar(z)) T in(or"(())
OBEEBO my(x) +T In 0f(x)
(A43)
where the variance proxy at is considered via the augmented posterior variance of
While RAHBO penalizes risk on an absolute scale, subject to a, BEEBO optimizes for high uncer- tainty reduction, quantified as the log ratio of the variance before and after making measurements.
Moreover, RAHBO differentiates between known and unknown variance proxies, and uses the LCB term discount the predicted variance according to its uncertainty. In its closed-form analytical expression, BEEBO does not permit for the uncertainty of the variance proxy to be taken into account, being more similar to the known variance RAHBO
@RAHBO (x)
ny (z) + By (z(2x) (x))
(A44)
where ylaa is noise-free free proxy. Either sampling-basse approach, approximations to [(x) would need to be introduced to handle variance proxy uncertainty in BEEBO.
C Implementation details
C.1 Acquisition function optimization
BEEBO was implemented for full compatibility with the BoTorch framework (version 0.9.4) [52] as an AnalyticAcquu wewuuuistionunnnniiic Standard Bo utilities for initializing and training GPs, initializing q-batches and performing gradient descent optimization of the acquisition function are used. We trained GPyTorch (version 1.11) [57] GP models with KeOps [64] Matérn 5/2 kernels (following BoTorch defaults with a separate length seale for each input dimension, and Gamma priors on the length and output scales). Log determinants for the information gain were computed using singular value decomposition for numerical stability.
GPyTorch provides a get_ training data of GP with set of points, as done in BEEBO. However, we observed that GPy- Torch's 's implementation suffers from GPU memory leaks when used with automatic differentiation enabled. We therefore instantiate augmented models explicitly, not making use of the (more efficient) augmentation strategy.
method that allows for the efficient augmentation of the
All experiments were performed with double precision. Sobo SMMCNormaaSSmppp was used for acquisition functions making use of the reparametrization trick. Experiments were run on individual Nvidia RTX 6000 and V100 GPUs. Five replicates for the benchmarking experiments required total of approx. 5,000 RTX 6000 GPU hours, with the majority of the run time dedicated to the GIBBON baseline, rather than BEEBO itself (Figure A3, Table A9).
C.2 Benchmark BO methods
All methods were benchmarked in BoTorch. For El, we used LogEl [58]. For TS, 10.000 base Sobol samples were drawn and sampled with MaxPosteriorSamplll using the Cholesky decomposition
22

---

of the covariance matrix. GIBBON was optimized using sequential optimization following the BoTorch tutorial. We additionally implemented custom version of GIBBON that applies the Q 2 scaling factor to the diversity term, as proposed in GIBBON's supplementary material. We used 100,000 random discretized candidates for max-value sampling. In few iterations, optimizing GIBBON seemed challenging, with BoToorc reporting that no nonzero initialization candidate could be identified. KB was optimized using a custom greedy optimization loop with fantasized observations, using (single-sample) LogEI as the underlying acquisition function. TuRBO-1 was optimized following its BoTorch tutorial. None of the methods use hyperparameter for controlling their explore-explo trade-off. The results therefore based 10 iterations at defaults.
C.3
Test problems
Test functions All test functions were used in their BoToorc implementations. As done previous work, the embedded Hartmann function was created by appending all-O dummy dimensions to the original six dimensions [53, 54, 50].
Control problems We consider two control problems from previous work: A 14-dimensional parameter tuning task for controlling robot arms pushing two objects to target location [55], and 60-dimensional trajectory planning task for rover navigating through a maze of obstacles [56]. Instead of converting the problem objectives into rewards as in the original work. we operate on the actual minimization objectives directly (distance to target, navigation loss), and follow BoTorch's approach simply inverting the objective in order to yield maximization problems. Both problems were adapted from their available implementations in Wang et al. [56] to follow the BoTorch test problem API.
C.4 Heteroskedastic noise
The (inverted) Branin function has three global optima f(x") = -0.397887 at oं (9.42478, 2.475), z. = (-T, 12.275) and zzz- (T, 2.275). We define heteroskedastic noise so that the variance is maximal at 222 and zz. The noise decays exponentially with the distance from any of the two noised optima at a rate
(xx = o maa exp(-X min z1l12, || 25l12)
(A45)
For our experiments, we set o2 maz 100 and À = 0.05. As the surrogate function, we use TaskGP provided in BoTorch. This model learns two GPs simultaneously, one for the function fzx and one for the (also unknown) variance function o² (x). When querying the oracle with a batch of points, noised observations of fxx are provided together with the true o² at each point. The homoskedastic control experiment uses SingleTaskGr with inferred noise level. The homoskedastic noise set to o² = 77.5, which is the average noise level of the heteroskedastic function over the whole domain.
23

---

f(x)
0²(x)
14
12-
90
10
100
150g
70
200
*
250
300
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
Figure A1: The Branin function with added heteroskedastic noise following Equation A45. o 100, 0.05.
D Extended results
D.1 Results including additional baselines
Table A1: BO on noise-free synthetic test problems. The normalized highest observed value after 10 rounds of BO with 1=100 shown. Colors are normalized row-wise. The BEE-BO and q- qCUCB columns are equivalent to Table 2. Higher means better. Results are means over five replicate runs.
000%
400%
642 1040
L40 2010
0401
844
C10
1448-12
2311
1790 225
1100
By
10229 3880
5:00
4.044-12
1307 02000
120
L10
030 253
E400 40330
4.304
420
55010000
650 42.0
0200
2008
6312 C04C
6422
5401
140 8.00
24

---

Table A2: BO on noise-free synthetic test problems. The relative batch instantaneous regret of the last, exploitative batch is shown. Colors are normalized row-wise. The BEEBO and q-UCB columns are equivalent to Table 3. Lower means better. Results are means over five replicate runs.
1880 380
676 42mh
890
0000000000
400 1100
880 00/0
84h
10120 380
11:00 3941
0.64
154h
Table A3: Paired t-test p-values for the results of meanBEEBO in Table 2. The combined p-value was computed using Fisher's 's method. P-values smaller than 0.05 are indicated in bold.
18.03
14:01
(1:0)
$6.43
$1.60
11.44
16:45
66:45
18:41
30-31
18:45
16.41
61-6
48:4
10-91
$1-41
18:03
07:11
01
18:38
15:15
15:41
05.33
18:10
15/45
68.33
15:00
11:41
38/10
18:08
16:36
35.55
16.00
18:10
15:43
15:15
1641
03.4
15-0
36.11
16/01
03.03
15.33
1840
17:18
18.00
19.44
18-33
19:00
1EA6
12-33
18-0
1940
15.00
33.44
1940
17:11
05.31
19.00
19.0
15.0
10-0
15.0
35.45
18:10
15:31
15:00
19:00
0.10
1--0
19:1
15/00
13.99
33.00
1946
15:1
16:0
19:00
16-09
06-8
16:15
1900
1645
51-0
11.00
1646
16-8
16-0
1940
51-11
48.00
1846
18-10
01:44
15.4
15:05
11.09
11.44
95-0
01:11
15.27
75/0
16-11
11.81
14
15
21.41
16:31
1934
3545
Dog
18/8
15:41
13:41
1946
61.00
11.44
18/6
16.43
19400
1940
15:48
4-40
16-0
1840
15:00
61.04
11.40
(846
44/1
2642
13:43
65-41
15:40
2013
16.3
2012
15.66
1540
2043
36:41
305
363
16-0
1840
1640
65:44
45.07
8/3
Table A4: Paired t-test p-values for the results of maxBEEBO in Table 2. The combined p-value was computed using Fisher's 's method. P-values smaller than 0.05 are indicated in bold.
15:41
6541
23-44
01
14.04
18.18
18:0
18-19
18:11
64.00
19:04
13:14
17:01
18.00
19:44
$7.44
15:00
09.00
15:31
15:48
48.00
11.81
48:40
15:40
1810
011
35-01
1541
11400
15.50
12-0
19:50
19.30
13-44
12.00
18-10
18:03
19-1
15.00
1900
1926
19.64
19.31
19-0
18:1
15:03
1644
16-04
1900
1900
0.4
1846
1940
1940
07-08
19.00
18455
13
35.00
15:4
15.09
1840
1940
12:09
12-9
4241
15.00
1940
1500
38:08
18:00
19.55
05.5
01.41
1946
0.5
19:
16:30
1540
19:08
15:14
15-13
04-00
11.0
18-0
016
15.05
19m
17:40
(2.66
1141
1519
15/
1241
15.6
1541
15:0
154
1546
1141
15.8
Bonk
48:1
18.0
0945
30.41
13.6
16:4
1540
18.00
4520
1440
01.4
19.66
16/
16-0
(F41
11:00
47.41
1548
16.8
12:37
1841
(15)
15.40
9:40
18:00
170
1940
19.5
9:14
16.0
18-0
(H40
18:00
17:31
(8.15
9.30
25

---

D.2 Results for batch sizes 5 and 10
Table A5: BO noise-free synthetic test problems. The normalized highest observed value after 10 rounds of BO with 5=5 is shown. Colors are normalized row-wise. Higher means better. Results are means over ten replicate runs.
Falla
V-OCE
GlOoS
50
aell Tag
615
E.00 2.0
04.00
3512 830
3175 6.10
5.13
03012
1902
010m1 115
0000
0227 5.15
5.77 -0.00
5106
07704
02:5 5/15
0379 1.31
600 -001
1603 6013
3111
114
6000 20.21
3108 6.00
0601
110
15.3 21-0
1:th
1214
0135
61/00
9544
0.11
6117
8.554 3333
6441
Table A6: BO on noise-free synthetic test problems. The normalized highest observed value after 10 rounds of BO with q=10 is shown. Colors are normalized row-wise. Higher means better. Results are means over ten replicate runs.
FIRMBTS
1999
B40
CORU 134
1X0
EON
M15
8.90
Tung
8407
11441
01001
834
130
8.10
88188 2111
0090
1.000
2215
2420 3.10
0131
8.104
0120
L12 -0.0
313
61108
132
5464
000
335- 416
0332
1-45
14/21
1820 515
L13 -0-1
330
5160
1101
119 -0.5
134-
D0N0 1530
510
110
515 01/87)
116- 1113
0111
26

---

Table A7: BO on noise-free synthetic test problems. The relative batch instantaneous regret of the last, exploitative batch with q=5 is shown. Colors are normalized row-wise. Lower means better. Results are means over ten replicate runs.
2710
8.23
K27/ 2303
L/1 1.31
0.81 1127
020
0506 A-8
118
4180
5000 3066
0381
9-107
853 8.40
2.00
onl- Togg
18
D309 0133)
3220 3000
1.00
0,81
01-01
6178 0.23
012
-222 1253
0.500 01.1
040 5013
3431
Lung
0102
14211 0180
3710-
ailiddi Dog
1M
220 8108
E-ST 20
114
Table A8: BO on noise-free synthetic test problems. The relative batch instantaneous regret of the last, exploitative batch with 9=10 is shown. Colors are normalized rise. Lower means better. Results are means over ten replicate runs.
CIRSTT
2010
6,10
136
20.00
100 1144
3118 K.1
Tang
0279
840
1816
1181
1179
154 0255
0221
1000 3005
6070 1030
34.5 E00L
18200
5113
2010
0367 9475
1600 20517
0800
9212
012 41100
6100
andh Ting
070
5700
057221 1300
530 C0MT
1573-111
2219 URC
EL/00
134
2811- 5175
2015
1900 5200
Lx0 4031
1126
11070 2111
1::0
B17
1177
01911 3114
2113 Em
18.08
325
1038
6/38
27

---

D.3 Control problems
pushing
GIBDOV
10
10
30
Rover trajectory (d-60)
<11
vill 2.0
VE Tx=100 10.0
14
14
5-
5-
Figure A2: Experiments on the 14D robot arm pushing and 60D rover trajectory planning control
problems. 10 replicates each. GIBBON (s) refers to the scaled larged-batch variant of GIBBON.
D.4 Run time
meanBEEBO
P=0.5
meanBEE80 1=5.0
maхθEEBO T=0.05
maxBEFB0 T=0.5
max6EE60 T=5.0
e-UCB =
qUUC vx=
(-UCB 1k.10.0-
GIBBON (default)
GIBBON (scaled)
150
Time Eminutes!
Figure A3: Example run times for the 10-round BO experiment on the 6D Hartmann problem with Q=100. Error bars are over 5 replicate runs. Run times vary depending on the test problem, with GIBBON appearing especially sensitive, becoming e.g. 10x slower on the 50D Ackley problem.
28

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Table A9: Total run times for five replicates of the experiments presented in Table Al. We sum over all test problems.
Method
Configuration
Total time [h]
MeanBEEBO
T = 0.05
66.12
MeanBEEBO
T 0.5
47.08
meanBEEBO
T' = 5.0
37.13
maxBEEBO
T' 0.05
54.85
maxBEEBO
T' = 0.5
44.20
maxBEEBO
T 5.0
47.63
Q-UCB
VK = 0.1
3,33
q-UCB
VE K 1.0
3.70
q-UCB
VE 10.0
4,41
g-El
24.33
TS
6,56
KB
223.78
GIBBON
default scaled
3380.48
GIBBON
1055.93
D.5 Results with random initialization in round
Table A10: BO with random initialization on noise-free synthetic test problems. The normalized highest observed value after 10 rounds of BO with q=100 is shown. Colors are normalized row-wise. Higher means better. Results are means over five replicate runs.
2041
JOMP BORT
0183
SyMtao6 Tag
-000
312
1,000 800
10000 8.00
0098 1 B065
0309 0.95
0.99
0.99 1000
8 0300
L000
Calla
1008
0.0
11
1109
11
5.94
1198
28
0248
20
5:45
0.15
33577 525
0796
3.45
0 1000
036 1000
570
23
3453 384
0315
0314 1:00
500
31
050
0mmh
2m0
2300
Burtia
2197 1.2
211
0509
MAy mil
0121
5m
0:110
Erb Tranc am
148
1272-11
1000
0311
0315
Burga
148
222
3200
1000
0411
OATS
049
1.07
0.65
046 160
2-0 0214
6.30
0331
1689
6034
3.090
6412
29

---

Table A1l: BO with random initialization on noise-free synthetic test problems. The relative batch instantaneous regret of the last, exploitative batch shown. Colors are normalized row-wise. Lower means better. Results are means over five replicate runs.
BTET
1238
1111
210 1000000
2888
800
0183
8.00
000
8000
080
8.00
199 6134
038 0.333
131 0234
030
2963 0000
0.00
0.44
0000
0700
0090
131 51.00
8140
040 0-003
4800
0.20
11
2700
11:30
224
0399
1007
0339 1011
14
0.79 5.85
000 LXTT
4000
035 .00
00000
0100
3100
130 -m
0002
1/0
100
0555 1.00
110
2480
110
1110
1110
0180
01314 11:00
230
8:10
Syy
100
1.11
1100000
(145
0354
0674
8-3000
(087) 0706
0313 1004
1:41- 154110801
8:331
D.6 BO curves for all experiments in Table and Table A1
Shekel (dm4)
Hartmann (d=6)
3.5
10
3.0
objective value
2.5
mesnBEE30
Best objective
2.0
mextittako
1.5
TS
- GtI
0.5
(x)
TuRBO
2
10
0
2
10
(d-8)
3.5
3.0
objective value
Best objective value
2.5
0.0
-0.5
1.5
-1.0
1.0
0.0
Round
Round
Figure A4: Experiments on the Shekel, Hartmann, Cosine and embedded Hartmann test functions with VK 0.1 for BEEBO and q-UCB.
30

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Shekel (d-4)
Hartmann (d-6)
3.5
10
3.0
objective value
maindEEso maxéEEBO
Best objective
2.0
1.5
Q-E1
1.0
0.5
TURBO
2
8
10
0
2
4
10
Cosine
Embedded (d=100)
1.0
3.0
Best objective value
2.5
objective
0.5
1.5
-1.0
1.0
-1.5
0.5
0.0
Round
Round
Figure A5: Experiments on the Shekel, Hartmann, Cosine and embedded Hartmann test functions
with V 1.0 for BEEBO and Q-UCB.
Shekel (d=4)
Hartmann (d=6)
3.5
10
objective value
Best objective value
2.5
maanBEE30 MaXBEEBO
2.0
1.0
KB
0.5
TURBO
2
8
10
0
2
4
6
10
Cosine
Embedded (d=100)
3.5
1.0
3.0
Best oblective value
2.5
objective
-0.5
1.5
-1.
1.0
-1.5
0.5
0.0
Round
Round
Figure A6: Experiments on the Shekel, Hartmann, Cosine and embedded Hartmann test functions
with / 10.0 for BEEBO and g-UCB.
31

---

Ackley (d-2)
Ackley (d-10)
-5
-5
objective
-10
Q-ICB
-10
-15
-15
GIBBON
GIUBON
-20
10
10
Ackley (d=20)
Ackley (d=50)
objective value
Best objective value
-5
-5
-10
-15
-15
-25
-25
Round
Round
Ackley (d=100)
-5
objective
-15
8
10
Figure A7: Experiments on the Ackley test function with / 0.1 for BEEBO and -- QCB..
32

---

Ackley (d-2)
Ackley (d-10)
-5
-5
objective
-10
Q-ICB
-10
-15
-15
GIBBON
GIUBON
-20
10
10
Ackley (d=20)
Ackley (d=50)
objective value
Best objective value
-5
-5
-10
-15
-15
-25
-25
Round
Round
Ackley (d=100)
-5
objective
-10
-15
-20
6
8
10
Figure A8: Experiments on the Ackley test function with / 1.0 for BEEBO and -- QCB..
33

---

Ackley (d-2)
Ackley (d-10)
-5
-5
objective
-10
Q-ICB
-10
-15
-15
GIBBON
-20
GIUBON
-20
10
10
Ackley (d=20)
Ackley (d=50)
objective value
Best objective value
-5
-5
-10
-15
-15
-25
-25
Round
Round
Ackley (d=100)
-5
Best objective
8
10
Figure A9: Experiments on the Ackley test function with v 10.0 for BEEBO and - UCB.
34

---

Levy (d=2)
Levy (d=10)
5.0
2.5
Best objective value
0.0
-2.5
Best oojective
-5.0
UCB
-7.5
TS
-10.0
GIBION GIBBON (s)
-12.5
1,R80
-15.0
-100
10
Levy (d-20)
Levy (d=50)
200
objective value
-50
objeccive
-400
-100
-150
-600
-200
-800
-250
-1000
10
10
Round
(d-100)
objective
-1000
-1500
2000
0
10
Figure A10: Experiments on the Levy function with v 0.1 for BEEBO and q- QCB.
35

---

Levy (d=2)
Levy (d=10)
5.0
2.5
Best objective value
0.0
-2.5
Best oojective
-5.0
UCB
-7.5
TS
-10.0
GIBION GIBBON (s)
-12.5
1,R80
-15.0
-100
10
Levy (d-20)
Levy (d=50)
200
objective value
objjective
-400
-100
-150
-600
-20
-800
-250
-1000
10
10
Round
(d-100)
objective
-1000
-1500
2000
0
10
Figure Al1: Experiments on the Levy function with v 1.0 for BEEBO and q- QCB.
36

---

Levy (d=2)
Levy (d=10)
5.0
2.5
Best objective value
0.0
-2.5
Best oojective
-5.0
UCB
-7.5
TS
-10.0
GIBION GIBBON (s)
-12.5
1,R80
-15.0
-100
Levy (d-20)
Levy (d=50)
200
objective value
-400
-100
-150
-600
200
-800
-250
-1000
10
10
Round
(d-100)
objective
-1000
-1500
2000
0
10
Figure A12: Experiments on the Levy test function with V 10.0 for BEEBO and g UCB.
37

---

Rastrigin (d=2)
Rastrigin (d= 10)
5
20
0
Best objective value
-5
-20
-10
oojective
-15
UCB
-60
-20
TS
-80
-25
GIBION
-100
-30
GIBBON (s)
-35
1,R80
-120
Rastrigin (d-20)
Rastrigin (d-30)
50
Best objective value
0
-200
-100
-400
-150
-500
-200
-250
8000
-300
-1000
10
10
Round
Round
Rastrigin (d (00)
objective
-750
-1000
-1750
-1500
-1750
-000
10
Figure A13: Experiments on the Rastrigin test function with vie 0.1 for BEEBO and Q-UCB
38

---

Rastrigin (d=2)
Rastrigin (d= 10)
5
20
0
Best objective value
-5
-20
-10
oojective
-15
UCB
-60
-20
TS
-80
-25
GIIION
-100
-30
GIBBON (s)
-35
1,R80
-120
10
Rastrigin (d-20)
Rastrigin (d-30)
50
Best objective value
0
-200
objeectvve
-100
-400
-150
-500
-200
-250
8000
-300
-1000
10
10
Round
Round
Rastrigin (d 100)
objective
-750
-1000
-1750
-1500
-1750
-000
10
Figure A14: Experiments on the Rastrigin test function with v 1.0 for BEEBO and Q-UCB
39

---

Rastrigin (d=2)
Rastrigin (d=10)
5
20
0
Best objective value
-5
-20
-10
oojective
-15
UCB
-60
-20
TS
-80
-25
GIIION
-100
-30
GIBBON (s)
-35
1,R80
-120
10
Rastrigin (d-20)
Rastrigin (d-30)
50
Best objective value
0
-200
objeccive
-100
-400
-150
-500
-200
-250
8000
-300
-1000
10
10
Round
Round
Rastrigin (d 100)
objective
-750
-1000
-1750
-1500
-1750
-000
10
Figure A15: Experiments on the Rastrigin test function with v 10.0 for BEEBO and q-UCB.
40

---

Rosenbrock (d=2)
Resenbreck (d=10)
100
50
objective value
-20000
MEOnBEEBO
maxBE:BO
T5
60000
-150
9-EI
-80000
-200
T,RBO
-250
100000
10
1e-6
2
10
1e6
Rosenbrock (d=20)
1e6
1e6
Rosenbrock (d-50)
1e6
0.0
Best objective value
objective
-0.4
-066
-3
-0.8
-1.0
10
Round
Round
1e7
Rosenbrock (d-100)
1e7
0.0
Bezt objective value
-0.4
-0.6
-0.8
1e-8
Figure A16: Experiments on the Rosenbrock test function with / 0.1 for BEEBO and g-UCB.
41

---

Rosenbrock (d=2)
Resenbreck (d=10)
100
50
objective value
-20000
MEOnBEEBO
maxBE:BO
40000
T5
60000
-150
9-EI
80000
-200
T,RBO
-250
100000
10
1e-6
2
8
10
1e6
Rosenbrock (d=20)
1e6
1e6
Rosenbrock (d-50)
1e6
0.0
Best objective value
objective
-0.4
-066
-3
-0.8
-1.0
10
Round
Round
1e7
Rosenbrock (d-100)
1e7
0.0
Best objective value
-0.4
-0.6
-0.8
1e-8
Figure A17: Experiments on the Rosenbrock test function with / 1.0 for BEEBO and g-UCB.
42

---

Rosenbrock (d=2)
Resenbreck (d=10)
100
50
objective value
-20000
MEOnBEEBO
maxBE:BO
40000
T5
60000
-150
9-EI
-80000
-200
T,RBO
-250
100000
10
1e-6
2
10
1e6
Rosenbrock (d=20)
1e6
1e6
Rosenbrock (d-50)
1e6
0.0
Best objective value
-0.4
objective
-066
-3
-0.8
-1.0
10
Round
Round
1e7
Rosenbrock (d-100)
1e7
0.0
Best objective value
-0.4
-0.6
-0.8
1e-8
Figure A18: Experiments on the Rosenbrock test function with V 10.0 for BEEBO and g- q-UCB.
43

---

Powell (d=10)
Powell (d=20)
-200
Best objective value
objective
-400
MconBEESO Q-JCB
-4000
-600
-6000
e-E1
-800
TURB
-1000
-10000
0
10
1e-5
2
10
1e-5
Powell (d=50)
Powell (d=100)
50000
Best odjective value
20000
objective
-50000
-100000
-20000
-40000
200000
1e-6
1e-6
Round
Round
Figure A19: Experiments the Powell function with vFc 0.1 for BEEBO and q-UCB.
Powell (d=10)
Powell (d=20)
objective value
-200
-400
meaelEr30 max8EEBO
Best 04 Bcove
-4000
QUCB
-600
-6000
(+E)
-800
TARBO
-1000
-10000
0
10
1e-5
2
10
1e-5
Powell (dm50)
Powell (d=100)
50000
Best odjective value
20000
objective
-50000
-100000
-20000
-150000
-40000
-200000
1e
16-5
Round
Round
Figure A20: Experiments on the Powell function with v 1.0 for BEEBO and q- QCUC.
44

---

Powell (d=10)
Powell 201
objective value
-200
Best ocjecciv value
-2000
-400
MoIBEEBO
-4000
-600
-6000
-800
URRO
-1000
-10000
0
10
1e-5
2
10
1e-5
Powell (d=50)
Powell Id=1000 100)
50000
Best obsective value
20000
objective
-50000
-100000
-20000
-150000
-40000
200000
10
Round
Round
Figure A21: Experiments the Powell test function with v 10.0 for BEEBO and q-UCB.
45

---

NeurIPS Paper Checklist
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Question:
Answer: [Yes]
Justification: As claimed, we experimentally demonstrate a) the controllability of the acquisition strategy. b) competitive performance on 33 test problems compared q-UCB, g q-BI, Thompson sampling, GIBBON, TuRBO and Kriging Believer, and c) behaviour under heteroskedastic noise.
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3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [NA]
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47

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(d)
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Justification: We include full BO curves with standard deviations over five replicates for all quantitative experiments in the appendix. These detailed curves are referenced in the main text at the appropriate place.
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Answer: [Yes]
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The paper does not make of human participants or datasets. To the best of our understanding, there are no potential harmful consequences and wider negative societal impact expected from the proposed method.
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The authors should make sure to preserve anonymity (e.g., if there is special consid- eration due to laws or regulations their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification: The paper introduces method for Bayesian optimization (BO). While BO has widespread applications in the sciences and engineering, there is no direct societal impact expected from this contribution.
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If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
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Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
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Answer: [Yes]
Justification: We credit the GPyTorch and BoToorc packages that our codebase builds upon. The packages used as dependencies, and as such are not included directly assets.
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The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
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Answer: [Yes]
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53


Reviews:
[{"review_id": "b2TSOfbPMJ", "reviewer": "Reviewer_mXVT", "summary": "This work introduces a batched acquisition function that balances exploration and exploitation by using a weighted sum of mutual information and expected value, with the weights defining the trade-off. The discussion links the proposed algorithm to UCB and asserts that it naturally addresses heteroskedastic noise.", "questions": "One concrete example of the second aforementioned weakness is the criticism of MC approximation in high-dimensional applications. Recent advancements in applying MCMC address this issue in a principled manner and might be of interest. \n\n***Reference:***\n\nYi, Zeji, Yunyue Wei, Chu Xin Cheng, Kaibo He, and Yanan Sui. \"Improving sample efficiency of high dimensional Bayesian optimization with MCMC.\" arXiv preprint arXiv:2401.02650 (2024).", "limitations": "Discussed above.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The proposed acquisition function and its optimization and approximation methods are straightforward and practical. \n2. The paper provides extensive empirical results to illustrate the proposed algorithm's efficiency.", "weaknesses": "1. The introduced parameter controlling the trade-off lacks interpretation as in previous methods.\n2. The completeness of the related work discussion is concerning. This is potential because the summarization lacks high-level extraction of the design of the algorithm, and the focus of the paper is, to some extent, scattered。", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments! We have incorporated the feedback in the updated manuscript, and look forward to further discussion.\n\n> The introduced parameter controlling the trade-off lacks interpretation as in previous methods.\n\nWe fully agree that an entropy quantity is in principle harder to intuitively interpret than a univariate variance, as used in (single-point) UCB. However, we note that when working in batch mode, the $q$-UCB parameter also becomes harder to interpret, as illustrated in Figure 1. In supplementary section B.1, we provide a derivation showing how BEEBO’s $T$ parameter can be set to match the interpretation of the (single-point) UCB parameter.\nIn the experiments as well as in our available implementation, we always make use of this derivation, operating with the UCB-matched $T’$, rather than a raw $T$.\n\nWe have reworked main text section 3.1 to more prominently explain the relation of $T$ to UCB.\n\n\n\n> The completeness of the related work discussion is concerning. This is potential because the summarization lacks high-level extraction of the design of the algorithm, and the focus of the paper is, to some extent, scattered。\n\nThank you for pointing out the recent Yi et al. paper on MCMC for BO, which we have now included in the related works section. Could you provide further guidance as to what you would find missing from the related works section? \n\nGiven the space constraints and the focus of our paper, our intent was to offer a concise summary of batch BO methodology, rather than delving further into MC methods. As suggested by another reviewer, we have now also added TurBO to the related works section as well as the experiments.\n\nThe related works section now ends in \n>Eriksson et al. demonstrate that overexploration also can be problematic in higher dimensions, and alleviate this using local trust regions in TuRBO. Maintaining such regions with high precision discretization can be memory-expensive, as indicated by Yi et al., who suggest using MCMC-BO with adaptive local optimization to address this by transitioning a set of candidate points towards more promising positions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2lJifqDPkW", "reviewer": "Reviewer_x4Ge", "summary": "The paper introduces a new approach to batch Bayesian optimization that explicitly trades off between exploration and exploitation via energy and entropy terms. The method is able to efficiently generate large batches for evaluation that outperform comparable methods for Bayesian optimization.", "questions": "The issues raised above can be addressed by running more experiments (batch sizes 5 and 10; TURBO; more replicates to get reasonable CIs for the results tables). But it will probably require more experiments that can be run in the rebuttal period. Do we expect the method to work well for Q=5 and Q=10?", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The method is novel and well-motivated. \n\nI found the analysis in Appendix B to be especially strong, in which the proposed method BEEBO is compared to other methods for batch Bayesian optimization. This analysis shows the originality of the method and gives strong context for it.\n\nThe paper is clearly written and easy to read. The appendix was especially useful and had many valuable parts.\n\nThe analysis of heteroskedastic noise was great to see and shows a useful and understudied setting where the method is especially valuable.", "weaknesses": "I think the method is interesting and will be of value to the field. However, I do not find that the experimental evaluation of the method provides full support for the claims of the paper.\n\n**Issue 1: Evaluation limited to large-batch setting**\n\nThe major issue is that the paper claims the method is for general batch Bayesian optimization problems, without any qualifiers that I can see. The experiments all use q=100, which is a large batch size. Smaller batch sizes are often of interest too, e.g. batch sizes of 5, 10, and 50 in the GIBBON paper. The setting q=100 used here is also used in the TURBO paper (Eriksson et al. 2019) where it is described as a \"large batch size.\" \n\nGiven the experiment results in the paper, I don't know if this method will perform well for small- or medium-sized batches. Thus, either the experiments need to be expanded to include experiments with batch sizes such as 5 and 10, or the framing of the paper needs to be adjusted to emphasize that the method is specifically for large-batch Bayesian optimization, not general batch BO problems.\n\nThis issue also relates to the choice of baselines. The experiments only explore large-batch settings, where GIBBON (as the paper notes) is known to perform poorly. If the paper wants to claim that it performs better than GIBBON in general, then it needs to make that comparison on batch sizes of 5 and 10. If the paper wants to claim superiority only on large-batch settings, then that's fine to only use q=100, but then it needs to compare to state-of-the-art for large batch. Thompson sampling is a popular method for large-batch settings which is included as a baseline, but to my knowledge the state-of-the-art for large-batch BO is TURBO (Eriksson et al. 2019). In fact the q=100 and the robot pushing and rover trajectory problems are all exactly as in the TURBO paper, so its inclusion as a baseline is pretty obvious and, I think, necessary.\n\n**Issue 2: Lack of statistical significance**\n\nThe results of the experiments do not appear to be statistically significant. The main results given in the main text are tables, and these tables do not have confidence intervals. The only place where uncertainty in the results is show are the figures in the appendix, and there the confidence intervals appear to overlap in most cases. This is due to the use of only 5 replicates. I appreciate that these experiments are costly to run since they are using 1000 iterations, nevertheless the lack of statistical significance in most of the results provides weak support for the claim that BEEBO is actually better, vs. what we're seeing just being noise in the experiments. The paper needs some form of statistical analysis to convince the reader that what we're seeing is not just noise in the experiments. The best way to do this would be to include confidence intervals in tables 2 and 3, and then increase the number of replicates as necessary to achieve statistical significance in the differences in means. I do not feel it appropriate to highlight the method as being \"best value\" when it is possible that the confidence interval for that value contains the values of the other methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their encouraging comments! Having a compelling experimental evaluation is of key importance to us, and we have performed the requested experiments to further demonstrate BEEBO's performance.\n\n> **Issue 1: Evaluation limited to large-batch setting**\n\nThank you for the helpful suggestions! We have now ensured that \n- TurBO is also discussed in the related works section. \n- TurBO is included in the experiments.\n- Experiments on batch sizes 5 and 10 were added. \n\nPlease see the PDF of the global response for the results of the additional experiments.  \nIndeed, it was our focus to improve performance in the “large-batch” setting, as this is where we found $q$-UCB to underperform, and GIBBON to not be applicable. We have now clarified this in multiple places in the paper to be more precise.\n\n> **Issue 2: Lack of statistical significance**\n\n\nWe appreciate that we need to be more rigorous with our reporting of results. We have now doubled the number of replicates and performed statistical testing on the differences in means. We can confirm that on aggregate over all experiments, BEEBO’s performance gain over the baselines is statistically significant. We have updated all results tables accordingly, and include the results of the statistical testing in the appendix. Please see the global response for combined p-values.\n\n(For the time being, performance of GIBBON will still be based on nine replicates on five Ackley test problems, as one replicate can take multiple days. This should not affect the findings, as we already achieved statistical significance, but we will update the manuscript once the last runs finish for consistency.)\n\n\nWe agree that the best way to present results would be in principle to include the confidence intervals in Tables 2 and 3. However, we found that this makes the tables very dense and extremely hard to read.. As a compromise, we added the confidence intervals to the extended tables in the supplementary material (and the rebuttal PDF), and mention this in the captions of Tables 2 and 3.\n\n\n>The issues raised above can be addressed by running more experiments [..] Do we expect the method to work well for Q=5 and Q=10?\n\nWe really appreciate that the reviewer is mindful of computational requirements - thankfully, we managed to run all critical experiments in the rebuttal period, as outlined above. We find that also at Q=5 and Q=10, BEEBO is competitive with TuRBO and GIBBON. Especially at Q=10, q-UCB shows strong performance. Overall, both BEEBO and q-UCB benefit from higher exploration rates at low Q.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hExOAwba9l", "reviewer": "Reviewer_4oKm", "summary": "Proposing a new acquisition function inspired by statistical physics, which allows explicit control of exploration-exploitation trade-offs in a batch BO setting.", "questions": "Questions are written in the above weakness section.", "limitations": "Limitations are discussed in the discussion section.", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Drawing inspiration from statistical physics is a promising direction, as it naturally aligns with Bayesian approaches.", "weaknesses": "**Major Points**\n1. **Lack of Unique Selling Point:** \n   The method does not appear to solve any unique cases that other methods cannot. While the related work section outlines many similar approaches, this work only compares itself to q-UCB. Without a theoretical study, such as convergence rate analysis, there is insufficient motivation to adopt another heuristic approach like this. To demonstrate efficacy, a comprehensive empirical study with extensive comparisons to existing works is necessary, given the unclear advantages.\n\n2. **Review of Claimed Selling Points:**\n   - **Not Based on MC Integration Like BoTorch:** \n     While this is true, it is unclear if it is beneficial. MC approaches are approximations but have convergence guarantees (refer to the BoTorch paper's appendix). This work lacks such guarantees.\n   - **Tight Control of Exploration-Exploitation Trade-off:** \n     The proposed method is not the only solution. UCB theory can bound the feasible region by [max LCB(x), UCB(x)] with 1 - $\\delta$ probability (e.g., see [1]). This region can be controlled by $\\beta$ hyperparameters, corresponding to $\\delta$. Constrained batch BO within this region would yield similar results with theoretical guarantees.\n   - **Heteroskedastic BO:** \n     There are no comparisons with existing methods. UCB variants can address this problem. Vanilla UCB theory does not differentiate between epistemic and aleatoric uncertainty. Therefore, UCB with heteroscedastic GP can serve the same purpose. For example, training a heteroscedastic GP with the observed dataset and replacing the inner GP on noise variance with normal isotropic likelihood variance (pure epistemic uncertainty) will yield similar results as risk-averse batch BO, without needing the change in modelling the acquisition functions regardless of hetero-/homo-scedastic cases like this work.\n\n3. **Setting k in Practice:** \n   Setting  $k$ is not an easy task for users. In UCB, $\\beta$ presents a similar challenge, but there are theoretical guidelines and algorithm to compute this (e.g., [2]).\n\n4. **Constrained Uncertainty Sampling:** \n   This work can be understood as a variant of constrained uncertainty sampling. As [3] explains, variance reduction can be written similarly to the entropy proposed in this paper (see section 2.2). It also shows that the variance-only approach is inferior to UCB both theoretically and empirically. The batch setting may lead to model misspecification, particularly when hallucination (aka fantasization) is applied. The concerns and approach are notably similar to ([49] in your citation number), making a comparison with their method unavoidable.\n\n5. **Data Augmentation Procedure:** \n   The explanation is unclear. Is it fantasization (aka hallucination) or simply observed points? How does this differ from Eq.(3) in [3]?\n\n- [1] Fengxue Zhang, Jialin Song, James C Bowden, Alexan- der Ladd, Yisong Yue, Thomas Desautels, and Yuxin Chen. Learning regions of interest for bayesian optimiza- tion with adaptive level-set estimation. In International Conference on Machine Learning, pages 41579–41595. PMLR, 2023.\n- [2] Kihyuk Hong, Yuhang Li, and Ambuj Tewari. An optimization-based algorithm for non-stationary kernel bandits without prior knowledge. In International Conference on Artificial Intelligence and Statistics, pages 3048–3085. PMLR, 2023.\n- [3] Srinivas, N., Krause, A., Kakade, S., & Seeger, M. (2010). Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design. In Proceedings of the 27th International Conference on Machine Learning (pp. 1015-1022).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed comments. We have answered them individually below, and are happy to discuss further.\n>**Lack of Unique Selling Point**\n\nWe would like to point out that as mentioned in line 275, supplementary section D.1 provides a comprehensive benchmark beyond q-UCB, including GIBBON, TS, KB, q-EI and now, as suggested by reviewer x4GE, also TurBO.\n\nMoreover, we provide an extensive theoretical analysis of how BEEBO compares to other strategies in supplementary section B, highlighting how BEEBO can be understood in light of existing works and improve upon them, tying our method to theory such as DPPs.\n\nBEEBO does in fact address issues that other methods face: q-UCB lacks the controllability of acquisition behavior in practice, as quantified in Table 3, and GIBBON is known to scale poorly to larger batch sizes, whereas BEEBO does so.\n>**Not Based on MC Integration**\n\nWe believe that Figure 1 and Table 3 highlight a serious practical problem with large batch q-UCB in BoTorch. Moreover, the MC q-EI also failed to outperform BEEBO in our extended benchmark.\n\nAs BEEBO is an analytical expression without sampling, we are unsure what kind of convergence guarantees are meant by the comment. We optimize BEEBO using gradient descent, and while it is, as most acquisition functions, multimodal, it can be optimized efficiently using multiple restarts, as is common in BO practice (and done in BoTorch).\n>**Tight Control of Trade-off**\n\nThank you for the reference. BALLET uses local regions to shrink high-dimensional search spaces, similar to TuRBO. However, unlike TuRBO, the paper is exclusively about single-point BO. It is not directly obvious to us how batched acquisition is meant to be done using their acquisition function (Eq. 7) without further generalizing BALLET. The sampling-based acquisitions (Eq. 8,9) may permit batch mode, but no mention of this is made in the paper.\n\nWe would like to note that the primary focus of our work is batch BO, rather than high-dimensional BO. We now include TuRBO as an exemplary method for trust region approaches.\n\nWe think it would be inadequate to benchmark single-point works that make no mention of batched BO in their own experiments. It is beyond the scope of our paper to generalize existing single-point works to batch mode. If there is a batched BALLET that we are not aware of, we would be happy to add it to our benchmark.  \nWe have modified our claims to explicitly state “Tight Control of the explore-exploit trade-off *in batch mode*”.\n>**Heteroskedastic BO** \n\nWe fully agree that existing (single-point) methods also address heteroskedasticity. Hence we compare to Makarova et al.’s RAHBO [11] in supplement section B. We are happy to include any other relevant UCB variants that handle batch BO. A key limitation of existing work is that to our knowledge, e.g. RAHBO has not been generalized to batches. The core focus of our work is batch, with heteroskedasticity as a secondary consideration.\n\nAs you mention, UCB does not differentiate between epistemic and aleatoric uncertainty. Thus we are unfamiliar with the concept that we can get risk-averse BO from a heteroskedastic GP without modifying UCB as done in e.g. Makarova et al. Could you provide a reference?\n>**Setting k**\n\nIt is true that setting parameters in BO can be challenging. However, in our work, we considered UCB-like controllability a desirable feature. In supplementary section B.1, we show how BEEBO’s $T$ and UCB’s $\\beta$ (or $\\kappa$, as used this work to avoid confusion with the softmax $\\beta$) are related. We always use $T’$, a parameter scaled for UCB equivalence, to allow users to follow existing guidance and allow fair experimental comparison.   \nAs outlined in the main response, we have edited section 3.1 to feature this more prominently.\n>**Constrained Uncertainty Sampling** \n\n[3] indeed also discusses the information gain. However, we understand that it does not use it for batch BO.  \nWe are unsure how the “variance-only approach” comment is meant to be understood in light of our work. We would also expect only using the variance of a point to be a poor strategy for BO - however, we do no such thing.\n\nCould you provide a reference regarding misspecification, and why batch BO in general causes more misspecification vs. single-point? We understand that [49] considers misspecification of GPs irrespective of the BO strategy, with no causal relation to batch BO.  \nNote that BEEBO does not use iterative fantasization - we immediately discard fantasized GPs after computing H_aug. The model itself remains unchanged for further gradient descent iterations, and no hallucinated observations are used.\n\nWe attempted including SOBER [49]. Following the quickstart guide in the official repository, and plugging that into the experimental loop that we used for all other experiments, we encountered problems such as linalg errors, invalid gaussians, and OOM errors on some test problems. We have thus decided to exclude SOBER, rather than reporting results that may not be valid due to technical limitations in the available code. We are open to add a statement on the exclusion due to these problems in the manuscript.\n> **Data Augmentation Procedure**\n\nThe terms in BEEBO and [3] do not differ - they both refer to the information gain of a GP (and call it so). We avoided using the term fantasization, as it is often understood as fantasizing (and using) $y$ values at $x$, as done in e.g. the KB baseline. As also pointed out in [3], when using the entropy (and not the posterior mean), $y$ is not used, so no hallucinated $y$ values affect the result.\n\nAs we introduce the concept already in Eq. 4, the generic form that marginalizes over $y$, we use the term augmentation, as in the that case it would not be called fantasization.  \nWe have edited Alg. 1 to use the term “fantasize” instead of \"augment\" to make clear that we use fantasization to compute $C_{aug}$ on the implementation level.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rPwQjNwafk", "reviewer": "Reviewer_CMG8", "summary": "This paper introduces a new acquisition function BEBBO for batched BO. BEBBO tries to build (negative) free-energies-like acquisition function, enabling gradient-based optimization, tight exploration-exploitation control, and risk-averse BO under heteroskedastic noise. It tries to improve existing parallel acquisition functions in the following ways:\n1.\tuses a hyper-parameter T to directly balance exploration and exploitation by separating these two parts clearly;\n2.\tkeeps the behavior predictable by scaling E and I with batch size;\n3.\tenables the optimization of gradient descent by holding the availability of closed-form expressions for GP.\nThis paper demonstrates several experimental comparisons and shows its effectiveness.", "questions": "- How to set the parameter T properly? Do you have some instructions?\n\n- In line 158, why does I(x) also scale linearly with batch size? \n\n- The Equation A5 is not shown.\n\n- What are the advantages of multiplying batch size Q in E(X)?\n\n- How does the behavior of BEBBO change with batch size?", "limitations": "Yes", "rating": 3, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.\tThis paper shows an enlightening acquisition function method inspired by statistical physics. \n\n2.\tThe experimental results show the effectiveness of BEBBO on problems without noise or with heteroskedastic noise.", "weaknesses": "1.\tThe idea is straightforward: simply combine two common components—entropy reduction and the weighted sum of values. In my opinion, the novelty is not very strong. \n\n2.\tThe article doesn't discuss the situation when BEEBO is used with other surrogate models. It seems that if BEEBO is not used with GP, it loses the advantages of closed-form expressions and gradient descent optimization. \n\n3.\tIn control problems shown in Figure A2, the performances of meanBEBBO and maxBEBBO are not outstanding. Especially, these two variants are surpassed by KB in Robot pushing problem.\n\n4.\tAlthough BEEBO performs well on many synthetic test problems, its versatility and effectiveness require more experimental validation in specific applications.\n\n5.\tIn the experiments in main text, the authors only showed the comparison with q-UCB. It would be better to show comparisons with other batched baselines and provide a thorough analysis. The comparison with q-UCB shows the advantage on the balance between exploration and exploitation. But other advantages emphasized in the paper, such as the benefits of gradient descent optimization and the tight control of the explore-exploit trade-off, are not fully demonstrated. I suggest that the authors can re-organize the paper to move the comparison with other baselines to the main paper. \n\n6.\tThe theoretical analysis is not deep. No regret bound is analyzed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments and questions. We respond to them individually below, and are looking forward to further discussion.\n\n> 1. The idea is straightforward [..]\nWe are convinced that BEEBO is a novel acquisition function. In section B.2, we extensively compare BEEBO to the, to our knowledge, most similar existing work, GIBBON.  We show how BEEBO’s entropy term can have advantages over GIBBON’s, and empirically show that the closed-form summarization term works better for large batches than GIBBON’s lower bound approximation. \n\nMoreover, BEEBO’s closed-form softmax-weighted sum expectation is novel. It is of course not our intent to claim that we have discovered entropy reduction. However, we are not aware of any existing work that proposed an acquisition function of this form for batch BO.\n>2 . [...] other surrogate models.\n\nIndeed, BEEBO was designed to work primarily with GPs, as closed-form expressions and multivariate normal posteriors enable efficient computation. We have edited the abstract to already mention the GP focus there. In the outlook section, we now provide a discussion of how BEEBO could be generalized:\n\n>Beyond GPs, BEEBO could be generalized to work with any probabilistic model. However, GPs are unique in that $H_{\\textup{aug}}$ is available in closed form and can be used to compute $I(\\mathbf{x})$ analytically, without solving the integral over $\\mathbf{y}$ in Eq. 4. Other models may require approximations and sampling-based approaches for computing the information gain.\n>3. In control problems [...] the performances [...] are not outstanding [...]\n\nWe agree - on these problems, KB turns out to be very strong, not just in comparison to BEEBO, but all baselines including TuRBO, which was originally found to be state of the art for these problems. We think that this is a consequence of using LogEI. Hence, we also mentioned in the results section of the main text that KB using LogEI performs very competitively for large batches. We have now updated this statement to explicitly point to the control problems also. \n>4 . Although BEEBO performs well on many synthetic test problems [..] more experimental validation in specific applications.\n\nWe are happy to include additional experiments. Are there any specific ones that we should consider? Overall, we are convinced that our empirical section is comprehensive and well aligned with common practices in BO research, such as e.g. Wilson et al, NeurIPS 2017 [24], Ament et al, NeurIPS 2023 [57], Gonzalez et al. AISTATS 2016 [37].\n\n> 5. [...] comparisons with other batched baselines [...]\nThank you for the recommendation. We had tested multiple layouts for the results and ultimately decided that given the number of experiments and methods, including the baselines in the main text tables makes them too hard to read and interpret. We have thus maintained the focus on q-UCB, and discuss additional baselines in the text, ensuring the supplement is referred to at numerous places.\n\nWe believe that both Figure 1, as well as the 33 batch regret experiments in Table 3, fully demonstrate the controllability of the trade-off in BEEBO and highlight the key problem with q-UCB not having tight control with large batch sizes. \n> 6. The theoretical analysis is not deep. No regret bound is analyzed.\n\nWe provide an extended analysis of the BEEBO algorithm in the supplementary section B, tying BEEBO to the rich theoretical framework of existing methodology, such as UCB, DPP and local penalization.   \nA theoretical analysis focusing on regret bounds, along the approach outlined for B-UCB’s bounds in [34] would indeed be very interesting, and we believe this deserves its own paper. \nGiven the provided theoretical linkage to existing methodology, we consider the manuscript to be quite comprehensive, focusing on empirical demonstration of BEEBO's efficiency. With 33 experiments conducted, three different batch sizes (q=5,10,100), eight different acquisition strategies, several of them operated at different scales (GIBBON default & scaled, q-UCB & BEEBO at different T), and 10 replicates each (corresponding to more than 12,800 recorded BO trajectories total), we consider our benchmarking approach thorough. We believe that including an analysis of regret bounds in the manuscript as well will only challenge the focus of our paper for a reader even more, as already indicated by reviewer mXVT.\n> How to set the parameter T properly\n\nThis is indeed very important - we do not want the controllability (which we consider an advantageous feature in general) to become a nuisance. As outlined in our main response, we now feature the relation of T to UCB more prominently.\n> In line 158, why does I(x) also scale linearly with batch size?\n\nThe linear scaling of $I(x)$ is a consequence of I being a log determinant (or more precisely, the difference of two log determinants). Considering the log determinant as the sum of log eigenvalues, we see that the scaling is in fact linear, as we add an eigenvalue for each increase in Q.\n> The Equation A5 is not shown. \n\nWe removed this trailing newline.\n>What are the advantages of multiplying batch size Q in E(X)?\n\nTo maintain a Q-independent T parameter, we wish for both $I(x)$ and $E(x)$ to scale equally. As we define $E(x)$ to be the mean or the softmax-weighted sum, this expression does not scale with Q itself, and we therefore multiply it. \n>How does the behavior of BEBBO change with batch size?\n\nAs just outlined, the explore-exploit parametrization of BEEBO is invariant. That being said, it is correct that there will always be an effect from Q. Specifically, at large Q compared to the search domain, one might expect to see the exploration (=repulsion) term grow larger, as points will compete to occupy the same regions of the domain. We believe that our additional low-Q experiments in the updated manuscript support this, showing that at low Q higher explore rates are beneficial for BEEBO as well as for q-UCB.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for their helpful feedback! We are excited to hear that they find BEEBO
- *Helpful and practical* (mXVT)
- *Novel and with strong context* (x4Ge) 
- *Effective with heteroskedastic noise* (CMG8, x4Ge)
- Has a *promising Statistical physics motivation* (4oKm, CMG8)

We have responded to each reviewer separately. In summary, we would like to highlight the following key improvements of the manuscript:
- **TuRBO was added** to the benchmark (x4GE) and the related works (mXVT). as an exemplary method of batched trust region BO methods such as also BALLET (4oKm)
- The **number of replicates** was **doubled** to demonstrate **statistical significance** (x4GE)
- Benchmark experiments at **batch sizes q=5 and q=10** were added (x4GE)
- Highlight the scaling of the T parameter for **UCB-like interpretability** more prominently (mXVT, CMG8)

As also pointed out by many reviewers, we consider it crucial that BO methods are benchmarked thoroughly against existing methods. This is why in our experimental section, we have now recorded more than **12,800 batch BO trajectories** in total, across 33 problems, multiple acquisition strategies, batch sizes and replicates. We believe that this is adequate empirical evidence of BEEBO's performance, which is complemented by Appendix B, which connects BEEBO to the theoretical background of existing work. 

Please see the attached PDF for:  
- an updated Table A1 including TuRBO, based on 10 replicates  
- the q=5 benchmark  
- the q=10 benchmark  

As the PDF is limited to one page, we include the p-values for the meanBEEBO results in the table here. These were computed using a paired one-sided t-test for each test problem over the 10 replicates, followed by aggregation using Fisher's method.

In the updated manuscript, we include this table in extended form - showing the p-value of each t-test, and the aggregation, for meanBEEBO as well as maxBEEBO. Please just let us know if you wish to see individual p-values that below results are based upon, we will provide them as comments as needed (The full table would be 33x21 cells and poorly suited for the available markdown formatting).

|meanBEEBO |Method | Fisher p-value|
|-----|---------|-----------------|
|$T'$ = 0.05 | $q$-UCB |	2E-31 |
 |  | $q$-EI	|1E-13 |
 |  | TS |	2E-30 |
| | KB |	6E-06|
| | GIBBON |	8E-58 |
| | GIBBON (scaled) |	5E-47 |
| |TuRBO |	4E-29 |
| $T'$ = 0.5 | $q$-UCB |	6E-28
| |$q$-EI$ |	1E-18
||TS|	2E-34
||KB|	6E-10
||GIBBON|	1E-78
||GIBBON (scaled)|	2E-58
||TuRBO|	1E-44
|$T'$=5.0| $q$-UCB |	1E-28
|| $q$-EI |	1E-03
||TS|	4E-20
||KB|	2E-06
||GIBBON|	9E-56
||GIBBON (scaled) |	8E-36
||TuRBO|	2E-46

As for the $T$ parameter scaling, section 3.1 "BEEBO with Gaussian processes" in the updated manuscript ends in 

>Using the kernel's learned amplitude $A$, we can relate BEEBO's $T$ parameter to the $\kappa$ of UCB. This allows us to configure BEEBO using a scaled temperature $T'$ that ensures both methods have equal gradients at iso-surfaces, enabling the user to follow existing guidance and intuition from UCB to control the trade-off. A derivation is provided in section B.1

We believe this will make it easier for the reader to find the relevant derivation, as opposed to the previous version where this was only mentioned in the experimental section.

We are looking forward to further discussion.

Meta-review: This work introduces a new acquisition function inspired by statistical physics for batch Bayesian optimisation with GPs. The work includes a theoretical analysis and an extensive set of experiments to support the claims. While reviewers raised concerns regarding missing baselines and other experimental aspects, all were addressed during the rebuttal. Overall, this work is well executed and of interest to the community.
Decision: Accept (poster)