Principled Bayesian Optimization in Collaboration with Human Experts
Authors: Wenjie Xu
Masaki Adachi
Colin Jones
Michael A Osborne
Venue: NeurIPS 2024 spotlight (2024)
Keywords: Bayesian optimisation
human-AI collaboration
knowledge elicitation
Primary area: probabilistic_methods
TL;DR: Robust and efficient Bayesian optimisation algorithm that elicits knowledge from human experts to accelerate optimisation.

Abstract: Bayesian optimisation for real-world problems is often performed interactively with human experts, and integrating their domain knowledge is key to accelerate the optimisation process. We consider a setup where experts provide advice on the next query point through binary accept/reject recommendations (labels). Experts’ labels are often costly, requiring efficient use of their efforts, and can at the same time be unreliable, requiring careful adjustment of the degree to which any expert is trusted. We introduce the first principled approach that provides two key guarantees. (1) Handover guarantee: similar to a no-regret property, we establish a sublinear bound on the cumulative number of experts’ binary labels. Initially, multiple labels per query are needed, but the number of expert labels required asymptotically converges to zero, saving both expert effort and computation time. (2) No-harm guarantee with data-driven trust level adjustment: our adaptive trust level ensures that the convergence rate will not be worse than the one without using advice, even if the advice from experts is adversarial. Unlike existing methods that employ a user-defined function that hand-tunes the trust level adjustment, our approach enables data-driven adjustments. Real-world applications empirically demonstrate that our method not only outperforms existing baselines, but also maintains robustness despite varying labelling accuracy, in tasks of battery design with human experts.

Principled Bayesian Optimisation in Collaboration with Human Experts
Wenjie Xu*.1.2, Masaki Adachi 3,4, Collin N. Jones', Michael A. Osborne',
Automatic Control Laboratory, EPFL Urban Energy Systems Laboratory. Empa
Maachnne Learning Research Group, University of Oxford
Toyota Motor Corporation
fwenjie.xu xu, colin. jonnnj)ppps ch, {masaki, mrobotr ox.oxx...ooo
Abstract
Bayesian optimisation for real-world problems is often performed interactively with human experts, and integrating their domain knowledge key to accelerate the optimisation process. We consider setup where experts provide advice the next query point through binary accept/reject recommendations (labels). Experts' labels are often costly, requiring efficient use of their efforts, and can at the same time be unreliable, requiring careful adjustment of the degree to which any expert is trusted. We introduce the first principled approach that provides two key guarantees, (1) Handover guarantee: similar to no-regret property, we establish a sublinear bound the cumulative number of experts' binary labels. Initially, multiple labels per query are needed, but the number of expert labels required asymptotically converges zero, saving both expert effort and computation time. (2) No-harm guarantee with data-driven trust level adjustment: our adaptive trust level ensures that the convergence rate will not be worse than the without using advice, even if the advice from experts is adversarial. Unlike existing methods that employ a user- defined function that hand-tunes the trust level adjustment, our approach enables data-driven adjustments. Real-world applications empirically demonstrate that our method not only outperforms existing baselines, but also maintains robustness despite varying labelling accuracy, in tasks of battery design with human experts.
Introduction
is a successful approach to black-box optimisation that has been applied across wide array of applications. BO is often praised for 'taking the human out of the loop' 80 by automating laborious optimisation processes, such as hyperparameter optimisation thus relieving human Nonetheless, a growing trend involves the opposite direction, which brings humans back into the loop and leverages human expertise as adviser the optimiser 77]. This numan-in-the-looy approach is particularly relevant to scientific and explorative tasks, such as materials discovery [24]] 2 and product design (48 44 77. Experts have accumulated domain knowledge and should be helpful in accelerating the optimisation process, yet their experience and knowledge are often qualitative- can struggle to express their knowledge in a functional form or to pinpoint the best candidates as an absolute quantity [7). At the forefront of science, experts are also in the middle of trial and error; demanding dellede and error-free inputs can limit the applicable range of BO. As such, human-Al collaborative setting in BO has emerged, driven by practical demands, and has been
Bayesian optimisation (BO) 60 65 33
29
[03 and neural architecture search 77 99]
from these tasks.
gaining popularity in the literature
50 24
42
* "Equal contribution
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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A prevalent issue in this domain is the lack of both shared assumptions and theoretical guarantees, making fair comparisons challenging. Our community has yet to reach a consensus on acceptable assumptions, particularly in the following areas. (a) The level of effectiveness of experts' knowledge:
assuming near oracle-like like knowledge, e.g. in [1] 39 12 collaborative settings can significantly surpass vanilla BO. However, if experts are entirely erroneous (yet confident) which can happen
50 24 7 -vverreliance on experts' input cannot guarantee the global optimum convergence. (b) Human interaction method: ideally, humans prefer minimising interaction with machines for convenience. Minimising interaction leads to maximising the information at each query to human, which often ends up requesting error-free and quantitative information for humans 82
43
43 42)
However, accurate knowledge elicitation remains a oongsstaddng quest 79 68 58 Inversely, when assume human belief is also a black-box function and require the elicitation of the belief function through statistical modelling, e.g. 773 34
we will demand excessive queries of the experts.
Contributions. We propose an expert-advised algorithm with the contributions summarised below:
1. Handover guarantee: we model the expert's role as cognitively simple and qualitative- the expert serves as black-box classifier, providing binary labels on the desirability of the next query location. Similar the no-regret property, we establish a sublinear bound on the cumulative number of binary labels needed. Initially, multiple labels per query are needed, but the frequency of querying binary labels asymptotically converges to zero, thus saving both expert effort and computation time.
2. No-harm guarantee: we show that the convergence rate of our expert-advised algorithm will not be worse than that of vanilla BO (i.e. without expert advice), even if the advice from experts is adversarial. Our convergence is achieved through data-driven trust level adjustments, and is unlike existing methods that rely on hand-tuned user-defined functions. empirically, our algorithm provides both fast convergence and resilience against erroneous inputs. It outperformed existing methods in both popular synthetic, and new real-world, tasks in designing lithium-ion batteries.
3. Real-world contribution:
2 Problem Statement
We address the black-box optimization problem,
cross-check expert's 's reliability
x² E arg min f(x),
(1)
reject
Experts
uncertain
ZE
(e.g. human)
while collaborating with an expert, where X' c. Rdd and d is the dimension.
accept
f(x)
no check needed!
confident
direct query
Blackbox function
BO algorithm
Expert labelling model. We model an expert as binary labeller (see Fig. [.. An expert labels a point x E X' as either 'accept' or 'reject', An "accept' label indicates that the point is worth sam- pling, while 'reject' label indicates it not. These labels are binary, with 0 for 'accept' and 1 for 'reject'. In practice, the labelling process can be noisy, since humans may find some points hard to classify. Non-expert incorrect belief may label the optimum x* 'reject'. The distribution of the labels is determined by the expert's prior belief about the black-box function f, and we model the expert's belief through another unknown black-box function g.
Figure 1: BO-expert collaboration framework: The algorithm (red) decides an expert's 's (blue) label is necessary. If rejected, it generates dif- ferent candidate; otherwise, it directly queries.
Assumption 2.1. The notation æ > O denotes the event where x is labelled as 'reject', based on the expert's belief function g. Additionally, the random indicator 1=>,0 {0, 1} takes value 1 if I > 0 and O otherwise. The probability distribution of 1 1x> 0 E 1) follows the Bernoulli distribution
with 1P(10>,0 0 = =) P=r,0 = S(g(z)), where S(u) 1/(1+e-") is the sigmoid function.
Example 2.2. Let us define an example 'synthetic" expert's labelling response as Pa>,0 = where a is the accuracy coefficient and p is the linear scaling function from bound [min fz),, maxxex [x) to 3.. When = 1, p, (x=)) = -3, S(-3) ~ 0.05, resulting in a Bernoulli distribution that yields an acceptance label of 0 with a 95% % chance at the global minimum
xr. In this case, the sharpness of the belief Pzr,0 is influenced by both the shape of fxx and a; if
ff f() ) is peaky or > 1, the expert can nearly pinpoint x".
2

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However, in reality, the expert does not know the exact true f and therefore, we consider g to be "subjective' belief function representing f. This differs from typical surrogate model f of f, which infers an 'objective' belief function from oracle queries. If g has better predictive ability than the surrogate model f. exploiting g can accelerate convergence; otherwise it may decelerate the process. In the optimisation process, 9 may act regularizer function in addition to the objective function f. For simplicity, we use this Ex. 2.2 as synthetic human feedback. Readers interested in other examples encouraged to refer Appendix [.]
Assumption 2.3. is compact and non-empty.
Assumption 2.3 is reasonable because in many applications (e.g., continuous hyperparameter tuning) of BO, are able to restrict the optimisation into certain ranges based on domain knowledge. Regarding the black-box function f and the function g, we assume that,
Assumption 2.4. f 9 E Hig where k Rd E R. representing ky or ky. is symmetric, positive-semidefnnte kernel function, and Hy is its corresponding reproducing kernel Hilbert space (RKHS, see (77)). Furthermore,
assume Il,flk,, I B and ||9||k, I Bg where || || is the norm induced by the inner product in the corresponding RKHS Ha. We use By to denote the set (ā E Hk, | |k S Bg).
Assumption 2.4 that the objective and the function g are regular in the sense that they have bounded norms in the corresponding RKHS, which is a common assumption.
Assumption 2.5. k(2, x²) I 1, x, E X, and k(z, x²) is continuous on R x Rd
Assumption 2.6. At step if query point 2; is evaluated, we get noisy evaluation of f (we refer to an oracle query), = (x)) + &t. where & is i.idd o-sub-Gaussian noise with fixed o > 0.
Notation. We refer to 1, as data realisation of 1z, 00 at step We denote the following sequences iff is queried step r}. and expert queries as Q respectively (t 2 2/1. 2 1 We use capitals, e.g. o for the set (x+) reef
of steps: iterations as [4] : {1,2. ..). queries as of := {T E [ 1] [1-1]
3
Confidence Set of the Surrogate Models
We introduce surrogate models for the objective and the function g. We opted for Gaussian process (GP; 86 100) for f and the likelihood ratio model 667 27 for g.
3.1 Surrogate Model of the Objective f: Gaussian Process
Definitions, We employ a zero-mean GP regression model, with predictive posterior ft Df. ~ GP()
Mg (X ²) (Ko! +r1) (K!++l) Yof:
(2a)
of. (z) ky (z, ki (X of T)T (Ko, +r1) ky (x),
(2b)
where K = (k5(F,, Fr.)
neef Df = (o:), r is the regularisation term 61 The
maximum information gain 84 for the objective is,
ioft
max
log 1 ++- K5,X,, where K5,x = x')
(3)
2
2.4 and 2.6 hold. For any d (0, 1), with probability at least - 8/2, the following holds for all x. E X and S St
Lemma 3.1 (Theorem 2, (22)). Let Assumptions 2.3
(1fee_.. + 1 In(2/8)
(x)(()) f(x) (,))
B = Bj + t
where (z) and o | are as given in Eq. 22 and Eq.
and 76
We follow the definition from 22
3

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For brevity, we denote the lower/upper confidence bound (LCB/UCB) functions f,(2) and fc(x) as.
[(x))= - (1,,0)))) f((2) = P (x) + (T)
3.2 Surrogate Model of the Expert Function gi Likelihood Ratio Model
While a GP classifier 63 is popular choice, we opted for likelihood ratio model [67 27]. The combination of a Gaussian prior with Bermoulli likelihood in GP models presents challenges in estimating the posterior confidence bound both theoretically and computationally. Moreover, GPs assume strong rankability 38 23 presuming humans can rank their preferences accurately in all cases, which often leads to inconsistent results 20). To address these issues, we drew inspiration from classic expert elicitation methods using imprecise probability theory TO 41). Instead of estimating the predictive distribution, we estimate the 'interval' of the worst-case prediction only. This approach does not assume any distribution within the interval, thereby relaxing the rankability assumption 78]. This method is particularly well-suited to the GP-UCB algorithm [83), which only requires a confidence interval. We developed kernel- based method to provably estimate the predictive interval.
Definitions. First, we introduce the function, P3 (T 1,) 1-5 (g(z)) +) (1-1,) 1,) [1 ((+())))
which is the likelihood of I over the event when 1z,> 0 = 1, under the Assumption 2.1 and Ò is an
estimate function of g E Hk, under the Assumption 2.4 We can then derive the likenhood function
of fixed function I over the historical dataset D Zyy 1r)]rE which becomes the product, Pg ((x,, 17)))))) = IrEo PT 1,). The log-likelihood (LL.) function,
LL value: (())) ==
reduces to ==())= - re 2, Er log (1+ where 2, = ((x,)) (this equality can be checked as correct for either 1, = or 1, 0). We then introduce the maximum likelihood estima- tor (MLE), MML arg maxieB, log 1
=
(())((((__)))))) (I 1r))))) Similar to [54] 27 [07), the confidence set
can be derived as shown in Lemma 3.2
Lemma 3.2 (Likelihood-basse confidence set). Ve, 8 > O, let,
B= {0- Bg ((())
LB on an 12).
where an (e, (21. - 3321 log
2et. We have,
64
P (g E B Vt 2 1) 2 1
The proof is in Appendix A As introduced in Assumption 2.4 while the function g was originally in a broader set of RKHS functions E By. it is now in a smaller set defined as g E B++1 conditioned on the expert labels D. Intuitively, with limited data, the MLE may be imperfect. Hence, it is reasonable to suppose that B bounded by LL values 'slightly worse' than the MLE, contains the ground truth with high probability.
(e, 6, 1Q .) depends on small positive value e. It will be seen that can be selected to 1/r in Appendix where T is the running horizon of the algorithm.
Remark 3.3 (Choice of e). In Lemma 3.2
Remark 3.4 (Confidence bound). By Lemma 3.2 we define the pointwise confidence bound for unknown g Hag (x) & g(x) S (e(x), where g (x) := infges; g(x) and ge(x)
sup ((x).
Remark 3.5 (Pointwise predictive interval estimation). At a given prediction point x, the predictive
interval [ (z). [=(x)] can be estimated through two individual minee-dimensional optimisation for for details). Subsequently, applying the sigmoid function yields the
problems (See Appendix B.3
predictive interval in probability space (z(g))))), ((G( (see Fig. 2 for visualisation).
4 Algorithm and Thcoretical Guarantees
4.1 Mixing Two Surrogate Models f and via Primal-Dual Method
Primal dual. We introduce the following primal-dual problem (5) as our acquisition policy,
Primal xf E arg min f
L (x) + Aeg, (x), Dual A [Xr Cg, (z+)++,
(5)

---

vanilla LCB
expert-augmented LCB
D
D
primal
7, g,
far
close
Figure 2: Visual explanation: While the vanilla LCB returns xh a far point from global minimum x², expert-augmented LCB can successfully navigate to closer point of by mixing fi and ge with L + Aıg, where X is the dual variable. In the figure, Df is the set of the sample points of the objective function f and D is the set of human feedback.
where X is the primal-dual weight at the f-th iteration and Ç is the step size for dual update. See Fig. for the intuition: we prioritise the sample in the expert-preferred region (i.e., the region with small g, (z)). The primal-dual method a classical algorithm for constrained optimisation [64] and has recently been applied to, for example, the constrained bandit problem [110]. In terms constrained optimisation, Prob. ( can be understood as solving min,ex f() s.t. g, S O. Interestingly, the primal-dual approach is also roughly analogous to Bayesian inference 25 Just as the prior acts as a regulariser to the LL maximiser [94]. expert belief g, (I) regularises the (z( minimiser. More specifically, the weight Ac+1 increases when g, (=r) > O; otherwise, At+1 decreases. The condition g, (1ff) > 0 indicates that the primal solution or is more likely to be rejected Under such risk of rejection, increasing the weights A--1 is natural because it more strongly regularises the [, minimiser to enhance feasibility in the next round, and vice versa.
Level of frust. Note that the primal-dual method is not the primary reason we achieve the no-harm guarantee. Indeed, its proof (detailed in Appendix B) does not rely on the primal-dual formulation. Therefore, technically speaking. our algorithm could employ a more aggressive exploitation of g (e.g., simply minimising 9t). Nevertheless, the primal-dual approach is our recommended policy for generating the expert-augmented candidate z to enhance resilience to erroncous inputs. The initial level of trust Sr is determined by the initial weight λo where larger λ values correspond to greater trust in the expert. We compared the effect of Ao in the Fig. 3 of the experimental section.
Efficient computation. Leveraging the representer theorem 77 107 due to the RKHS property, we further reformulate Prob. (5) to a ( + didieennio tractable optimisation problem (6).
min
[(x)) + X12
7 E ER TEX
[ Zo
Ze
(6)
subject to
K 00.0
< B3.
(Zo0
DV) N MLB)
a. a1, 8, 8,111t))),
where Ko = (kg oxgg and 8 {(Z
D.)) = Σreor 7, 1,
ErErp log (1+ e2r) is the LL value when g(r+) = Z,, Vr E Oi We update det1 = de + Ç=*,
where 2, (=f)
is the optimal of Prob. (
Key hyperparameter estimation. A key hyperparameter in Prob. (6) is the norm bound B, in the first constraint. Another hyperparameter, a1, in the second constraint, also scales with By. (see Lemma 3.2 However, B, may be unknown in practice, and its mis-specification leads to mis- calibrated uncertainty. We estimate B, by starting with a small initial guess (e.g., 1) and doubling when the following condition is met based on newly observed expert labels: a (c, 8, 1/ t 22 < l, (O (M.E.. where B, is our current guess. Intuitively, if the likelihood (, is
Recall that S(g(2{)) S(0) 0.5 implies higher chance of rejection than random (=0.5).
5

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Algorithm COllaborative Bayesian Optimization with Labelling Experts (COBOL).
1: Input and Initialization: function space ball By. trust weight 77, and uncertainty threshold 9tr
2: Set By = =g. Q 0, and Q - 0.
3: for E [T] do
4: Solve Prob. (5) via Prob. (6) to generate zi- Solve the unconstrained problem, 2x E arg min (x). if [ (xf) S minnex ft( and oy (x+) S (xf) then
D Expert-augmented LCB
5:
D Vanilla LCB
6:
b No-harm guarantee
7:
Set r, zt.
8:
if Ge (3t)) g (at) > Ahr then
b Handover guarantee
9:
Query the expert's ' label to get the feedback 1,- Update Q 0 U {f} and the posterior confidence set 1++ 1, then
11:
Set O = o 1. and continue the loop at line 4
12:
else
14:
Set 0 = Ø and x.--
Evaluate the black-bor function at the point Tv. and set 0/ = 0 (t). Update the posterior mean/variance of the objective f.
15:
16:
significantly larger, then 2B, is more likely valid bound. We iterate this estimation online during optimisation and in pre-training with the initial dataset (see details in Appendix F.
4.2 Algorithm and Theoretical Guarantee
Algorithm. Our algorithm Alg. T generates two candidates: the vanilla LCB x and the expert- augmented LCB z.. (See App. 1.3 on extension other acquisition functions.) Always selecting the vanilla LCB guarantees no- harmhamm but misses the chance to accelerate convergence using the expert's belief. Intuitively, this can be corresponds to the handover guarantee, stating that our algorithm stops asking the expert once our model g becomes more confident than the predefined Othr- Line the conditions for achieving the no-harm guarantee by assessing the reliability of the condition ensures x/ is least possibly better than the worst-case estimation of the optimal value. The second condition acts as active learning of human belief, exploring uncertain points to avoid inaccurate yet confident expert beliefs. The hyperparameter 7) 2 represents the initial level of trust in the expert. A larger indicates greater priority in exploring expert-preferred regions.
bandit problem regarding which arm to select. Line
candidate x;- The first
Theoretical guarantee. For Alg. I we mainly care about two metrics: cumulative regret Re
Eieol (f(x)) - f(xx)) and cumulative queries Q := 107 R Ro captures the cunulative regret over the query points to the black-box function. Q4 captures the number of queries to the expert. Since intuitively each query to the expert causes inconvenience, ideally, the frequency of query to an expert should be low (e.g., 0% grows sublinearly T).
Theorem 4.1. Under Assumptions 2.7 to 2.6 with probability at least 1 8, Alg satisfies,
oof s s0
Q7 < 500 (17))) log TW(B, l/r, lloe) (7b)
10/ V |
(7a)
8
See Appendix B for the proof of Thm. 4.1 Intuitively, Eq. (7a) shows the no-harm guarantee, since it provides a cumulative regret bound independent of the latent function g. Eq. (7b) shows the handover guarantee, since the bound cumulative queries to the expert is sublinear for commonly-used kernel functions (See Ta.le This means that the frequency of querying the expert asymptotically converges zero. We do not query human label for x to reduce human effort. Since
2t U Of [T]. of grows linearly in T. There is a trade-off accelerate convergence when feedback is informative, but it may also cause the worse convergence rate for adversarial feedhack (see Appendix B which includes an additional constant factor (2+1)/4 compared to the original UCB). In practice, setting 17 3 is sufficiently effective (see Figure
1) selection. A larger 1 can

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Table I:
Kernel speciffc bounds (fixed 77 is hidden) where v is the smoothness parameter of the Matérn kernel that is assumed to satisfy v > (3 + d vd² + 14d 17) ((d)).
Metric
Linear
Squared Exponential
Matérn
(Veefllog) 10g1
O (log |log| @4)
104
2rrt-dd Au4aa log 2r (12f1)
Q4
O (log
(llog ( ())(++1))
201d+3) OT 20+dla+ty T (log T)*)
By plugging in the maximum information gain bounds 84 93 and covering number bounds [104] [05 [8 [09], we apply Thm. 4.1 to derive the kernel-specific bounds in Table In practice, kernel choice and scalability to high dimensions are common challenges for BO. Existing generic techniques, such as decomposed kernels [4)] can be applied in our algorithm choose kernel functions and
achieve scalability in high-dimensional spaces.
4.3 Related Works
Human-Al Collaborative BO. There are two primary approaches: the first approach assumes that human experts can express their beliefs through quantitative labels, such as well-defined distributions
69
82 43
24
42 or pinpoint querying locations [1] 39
50 12 70]. While this strong
assumption is valid in specific cases, such as physics simulations 39), many experimental tasks- such as chemistry, which lacks the consensus numerical representations of, e.g. molecules- require more relaxed assumptions [24] 46). The qualitative approach, on the other hand, involves human experts providing pairwise comparisons (7) or binary recommendations (ours). The algorithm trains a surrogate model from experts' labels, thereby expanding applicable scenarios. Ours is the first-of its-kind principled method with both no-harm and handover guarantee on a continuous domain.
Related BO tasks. Eliciting human preference from labels has been explored in preferential BO 28 37 59
11 9 [177] However, this approach treats human preference as the main objective of BO, whereas our work uses experts' belief as an additional information source. Constrained BO 88, 87 110, 106 62] 44 96 (77 is another line of research that investigates BO under unknown constraints, placing another surrogate model on the constraint inferred from queried labels. However, our approach does not treat human belief as constraint that must be satisfied a reward to maximise, given that expert knowledge can sometimes be unreliable (see details in Appendix
32 35
5 Experiments
We benchmarked the performance of the proposed algorithm against existing baselines in collab- orative setting with human experts. We employed an ARD RBF kernel for both f and g. In each iteration of the optimisation loop, the inputs were rescaled to the unit eube [0, 1] and the outputs were standardised to have zero mean and unit variance. The initial datasets consisted of three random data points sampled uniformly from within the domain, and in each iteration, one data point was queried. Additionally, we collected initial expert labels by asking an expert to label 'accept' (= 0) or 'reject' (= 1) for 10 uniformly random points. All experiments were repeated ten times with different initial datasets and random seeds. We tuned hyperparameters online at each iteration. The GP hyperparameters were tuned by maximising the marginal likelihood on observed datasets using a stlti-start L-BFGS-B B method [53] (the default Bo'Torch optimiser
[14)). The key hyperparameters of the confidence set, B, and a1, were optimised via the online method in Appendix (F) Other hyperpa-
rameters were set as 1) = 3, An = 1, and Ohr = 0.1 by default throughout the experiments, with their sensitivity discussed later in Fig. Bl (see also Appendix p.Ip. The constrained optimisation in Prob. (6) was solved using the interior-point nonlinear optimiser IPOPT (95), which is highly scalable for solving the primal problem, via the symbolic interface CasADi [8. The unconstrained optimisation (line 5) was solved using the default Borh optimiser [1). More details for reproducing results
are available on GitHub The models were implemented in GPyTorcl (31]. All experiments were conducted on laptop PC Computationall time is discussed in Appendix In addition to cumulative regret and queries, we also consider simple regret defined as SR, := min min,ee/ (f(x+) - fx(=))
/attpa://github. ctapttuub..omm//a21//00000
"MacBook Pro 2019, 2.4 GHz 8-Core Intel Core 19, 64 GB 2667 MHz DDR4
7

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Foedback accuracy
trust weight y
primal-dual weight
Query threshold Ser
vanila LCB
vanilla LCB
vanilaa LCB
vanill LCB
reguet SR,
Simple
(ondom)
-b.1 -.1 A
(nformativ))
80
Cumulative
nogres Ro!
40
20
Function evaluation [2!
Cumulative
QF
80
100 200
100
200
100
200
100
200
iteration
Figure 3: Robustness and sensitivity analysis using the Ackley function. Lines and shaded areas denote mean standard error. The no-harm guarantee ensures the convergence rate is on par with vanilla LCB even in adversarial cases. Handover guarantee ensures that O plateau, allowing optimisation without expert intervention once sufficient information has been elicited.
Ackley
Hölder Table
Rastringin
Michalewiez
x10
Rosenbrack
Simple regnet SR,,
0
10
20
30
40
10 20
40
20
30
10
30
40
0
30
Fanction ovaluation
[2]
Function evaluation
12
Function evaluation
@1 Function evaluntion (())
Function evaluation (2))
Vanilla CCB random
sampling
Ours
Figure 4: Ablation study on five common synthetic functions with synthetic expert labels (a = 1).
Robustness and sensitivity. First, tested the robustness of algorithm the accuracy of the expert's labels using the 4-dimensional Ackley function [I]. We modelled the synthetic agent response according Example 2.2 In particular, we examine the impact of feedback accuracy, denoted as a. Fig. illlustrates the robustness our algorithm. When labels are informative (a = 1, 2). the convergence rate for both simple and cumulative regrets is accelerated in accordance with the accuracy. Even if the feedback is completely random (a 0) or adversarial (a = -1,- the no-harm guarantee ensures that the algorithm converges a rate par with vanilla LCB by adjusting the level of trust to be lower over iterations. Refer to Appendix J.2.3 for additional confirmation of the no-harm guarantee based on more extensive experimental results. Handover guarantee ensures that our algorithm stops seeking label feedback once sufficient information has been elicited, as indicated by the plateau in the cumulative queries Q% We also tested the sensitivity the optimisation parameters n. Åo. and fthr- The change in convergence at those parameters were varied mostly within the standard error, indicating that our algorithm is insensitive to these hyperparameters and that feedback accuracy is more dominant. For the primal-dual weight, Ào O corresponds to starting optimisation without primal-dual mixing objective, which performs worse than mixing cases (Ao = 1,2), demonstrating the efficacy of incorporating the primal-dual mixing objective.
Synthetic dataset. We compared our algorithm against five common synthetic functions 89
(sce
details in Appendix (.2). using simple baselines for an ablation study: random sampling, vanilla LCB (unconstrained optimisation), and expert sampling. Expert sampling involves direct sampling from the expert belief distribution Pu-,u- We employ rejection sampling by generating a uniform random sample over the domain and then accepting with probability P=>-,0- We fixed the feedback accuracy at a = 1 (as in Example 2.2 The efficacy of expert labels is roughly estimated by how much faster expert sampling converges compared to random sampling. all synthetic experiments, algorithm outperformed the baselines. While expert sampling is least more effective than

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Effective advise
Unreliable advise
Li standard design
Li" methyl-acetaal
Li" polymer-nanocompssie
Li" lonic liquid
Best Observed
2
0.6
-1
-1
1
0.3
-2
-2
0
0
10
20
30
40
10
20
30
40
10
20
30
40
0
10
20
30
40
Function evaluation
Q
Function evaluation
12
Function evaluation
[2)1
Function evaluation 10
Vanilla LCB random expert sampling Ours Ouus-[[1]] NeurlPS 2022-[50) ECML 2023- [7] AISTATS 2024
Figure 5: Real-world experiments with four human experts of lithium-ion batteries.
random sampling, it is not always better than vanilla LCB. For functions with a very sharp global optimum, such Rosenbrock 171), Pa>;" nearly pinpoints the global minimum. Still, our algorithm performs slightly better than expert sampling. See Appendix [.2.2 for computation time and query frequency. The overhead of our algorithm is comparable to that of other baselines.
Real-world experiments with human experts. We conducted real-world experiments in collabora- tion with four human experts who possess post-doctoral level knowledge lithium-ion batteries. In this experiment, human labelling costs vary among experts but typically range from a few seconds to several minutes. In the real-world development of lithium-ion batteries, creating and testing prototype cell requires at least a week, making the labelling cost negligible by comparison.
Lithium-ion ion batteries are crucial for realising a green society, a rapidly growing field where knowledge continuously updated at an unprecedented rate. This field typically suffers from data scarcity 46 due to the ongoing development of new materials synthesised by chemists. Consequently, transfer learning approaches, e.g., [90, 101 21,, are not effective in this setting. We prepared four cases for the experiments: the first is a standard task where we optimise the standard electrolyte composition [26] 33) and the second involves a slight modification the first setup by changing one solvent material (6)] We expect the experts to have informative knowledge on these two tasks. The
remaining two cases involve emerging new categories of materials: one is a polymer-nanocomposs electrolyte [108), and the other is ionic liquid (72). We anticipate that the experts' knowledge these new materials will not be as effective as in the first two tasks (see more details in Appendix (.3] Given the scarcity of real experts, we conducted pre-experimental step to elicit their knowledge for a fair baseline comparison. We asked them to label 50 random points uniformly from the domain, for all experiments before seeing the results. Then fit the confidence set model to these results and used g MLE as the estimated human response. Additionally, we asked the participants to manually select the next query point without any assistance from BO, which refer
'expert sampling' in the baseline. We also compared against state-of-the-a art algorithms [1] 55 77. These methods have predefined levels of trust, roughly ranked from strong weak: [I] the level of trust based on data, so we expect it to perform well in both effective and ineffective cases.
50) Ours can adjust
Fig. [ summarises the results. For the first two tasks, our algorithm outperformed all baselines. Particularly in the second task, human sampling was better than vanilla LCB, indicating that we should trust their advice aggressively. Our algorithm in significantly accelerated convergence. On the other hand, expert sampling for the new materials tasks was, although unintentionally, worse than random, thereby discouraging trust. While trustful algorithms [1] 7 struggled to converge, the distrustful algorithm (50] was able to converge on par with vanilla LCB, Our no-harm guarantee worked in this situation, gradually equating to LCB, and showed identical performance to the distrustful algorithm [50). See also Appendix J.4.1 for the complete experimental results the number of queries and computation time.
adapt to trust them over time, resulting
6 Discussion
Feedback form. Other forms of feedback, such as pairwise comparisons 7 or preferential rankings [2). can be incorporated into our algorithm with slight modifications. However, empirically
Thii slight change makes optimal design challenging enough [36]. See Appendix for details.

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found that the binary labelling approach performs best (see Fig. } and therefore, we recommend using binary feedback as the primary choice. For those interested in using alternative feedback forms, detailed instructions on how to adapt them to our algorithm are provided in Appendix H
Time-varying human knowledge. We assume that expert knowledge is stationary, although it can be time-varying, e.g., experts' knowledge often evolves as more data is gathered. A simple extension to accommodate this is the use of windowing, where past queried data is forgotten. This can be easily implemented in our algorithm by removing old data beyond a predefined iteration window. However, our initial trials did not show significant performance gains from this approach, so was not included in the main text. We suggest a dynamic model as potential future direction, which is discussed in Appendix [.I] with additional experimental results. Similarly, we kept the trust weight 1) fixed throughout the optimization process, Since human knowledge can improve over time, an adaptive could be employed to enhance both convergence and robustness. Nevertheless, our no-harm guarantee remains valid even without this adaptation. Further details are provided in Appendix 1.2
Acceleration vs. Robustness. One might seek to derive theoretical guarantee on the acceleration of convergence when the feedback is helpful. However, we want to emphasize that theoretically guaranteeing both acceleration and robustness may be incompatible. From a theoretical perspective, they are in a trade-off relationship [92]. This can be intuitively explained by the no-free-lunch theorem [02] : if algorithm A outperforms B, it does so by exploiting 'biased' information. The "bias' inherent in the acceleration is contradictory to robustness. Our setting is unbiased, meaning we do not have prior knowledge of helpful or adversarial human expert. Therefore, we must make a design choice between prioritizing robustness or acceleration as theoretical contribution, depending on whether we assume that expert input can be adversarial (weak bias) or that it will always be helpful (strong bias). Indeed, there are lower bound results for the average-case regret of Bayesian optimization in the literature (e.g., see [76)). GP-UCB is already nearly rate-optimal achieving this lower bound. This means theoretical acceleration is obtained in the price of worse robustness. In Appendix we present a slightly modified version, Algorithm 2 which offers an improvement guarantee based on strong bias. Our Algorithm can be seen as relaxed version of this algorithm (soft constraint), which helps explain the empirical success in accelerating convergence.
7 Conclusion
Our algorithm, with its data-driven adjustment of the level of trust, successfully accelerated conver- gence from effective advice while ensuring a no-harm guarantee from unreliable inputs. The handover guarantee also ensures that the BO can automate the optimisation process without assistance from human experts at later stage. These features are particularly valuable for scientific applications, where researchers often face trial and error, making it challenging to determine the effectiveness of their prior knowledge before starting experiments. Our flexible and robust framework is also expected to be effective in collaboration with large language models (LLMs), which demonstrate remarkable sample-efficient performance by exploiting encoded priors [55 75 666. and can be regarded as "expert knowledge'. Our safeguard features would be particularly effective for shared challenges, such as difficulty in eliciting knowledge
[45 [6] and varying accuracy of advice due to hallucinations 81 98 85 Although ours is the first-of-its-kiee algorithm with a general theoretical guarantee in the experf-collaborative setting, it is still based the GP-UCB algorithm and shares its limitations (e.g.. high dimensionality). One future direction is combining our approach with the high-dimensional BO methods [5]. Additionally, our current setting does not consider the batch setting, yet one can easily extend with existing approaches, e.g. 4 6 3 5). Multiple expert scenario is also a promising future extension. While a simple expert aggregation approach (e.g., majority vote, adding multiple experts g) could work without modifications to the current algorithm, more advanced methods, such as choice functions [5]. present promising directions for future work. Explainability also key. 7 showed that Shapley value-basee explanations improve human feedback accuracy, and this can be casily integrated into our framework. Our method can positively influence human experts by empirically demonstrating the value of their expertise, even amidst concerns about job security in the Al era [3). On the negative side, more powerful L.MMs may eventually replace the expert role in our algorithm in areas where data is sufficiently shared on websites or in papers, such as hyperparameter tuning (55).
Maximization formulation is adopted in GP-UCB paper (4 while we consider minimization. So LCB in paper essentially corresponds to UCB in GP-UCB algorithm.
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Acknowledgments and Disclosure of Funding
We would like to thank Ondrej Bajgar, Juliusz Ziomek, and anonymous reviewers for their helpful comments about improving the paper. Wenjie Xu and Colin N. Jones were supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF40 51NFF40180545. Masaki Adachi supported by the Clarendon Fund, the Oxford Kobe Scholarship, the Watanabe Foundation, and Toyota Motor Corporation.
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17

---

Partl I
Appendix
Table of Contents
Proof Lem. 3.2
B Proof of 4.1
21
Bound Error over Historical Evaluations
21
B
Bound Point-Wise Error
24
B.3
Elficient Computations of Confidence Range for the Latent Expert lunction g
25
4
Bound Cumulative Standard Deviation over Sample Trajectory
26
B.5
Bound Cumulative Regret
26
B.6
Bound Cumulative Queries Labeler
28
C
Detailed Discussions on The Significance of Thm 4.1
29
Proof of the Kernel-Specific Bounds in Tab.
29
E Theoretical improvement of convergence rale
30
F Estimating norm bound online
31
Related Work
31
H Comparison and Generalization to Other Feedback Forms.
32
Other feedback forms
32
H.2 Adaptation
33
H.3
Comparison
33
Potential Exiensions for Future Work
34
L.I
Extension Time- varying Human Feedback Model
................................ :o..............................
34
1.2
Extension to Adaptive Trust Weight 7
35
1.3
Extension Different Acquisition Function
35
Experiments
35
J.1
Hyperparameters)
................................
35
J.2
Synthetic Function Details
................................
36
J.3
Human experiment details
37
J.4
How Do Human Experts Reason?
38
A Proof of Lem. 3.2
To prepare for the proof of the lemma, we first prove several preliminary lemmas. Lemma A.1. For any fixed By. we have,
IP (oggP(((E7) 1r)re))) - log ((77, 1, 14)re))) & /81081B3 log 5, 2 6r
(8)
18

---

Proof. We use to denote g(2+), to denote ((x,), and pr to denote (x(x)))
P (log Py (2(( 17)r log P. ((zr, 1,) 1r)r)))) 5
(9)
Σ rE
((zr - 1r)1 - log(1 + e ) + log(1 + E²r)) S
(10)
=P
(2+ u-)1, TEQ!
Σ
E (at (22-2:0 ur ur) 5 TE
(11)
where the probability taken over the randomness from the feedback expert/oracle and the randomness from the algorithm, and &' + Eroo log (1 e²x) Eree: log (1 + e*rr
Ere (3, u,)p,. Let the function (2,) = log (1 + et)) log (1 + e (2r u, up)p can be checked that (27) e*r "atte+>² 2 V2, E and of (u,) 0. Therefore, 3 is convex function and achieves the optimal value at the point un- Hence, ()-(=t) 2 (h(()) 0, which implies 5 2 &. Therefore,
Σ (2+ - u4 u1)1, TE
Ee, - up) E TE
P
E
(z+ - u,)1,
Σ (2r - u Pr S
TE
TE
(12)
Furthermore, it is easy to see that (z, - ur)1, E [-2B,, 2By,, and thus, by applying Azuma- Hoeffding inequality, have,
E²
P
Σ TEQ
(2r
Ur ur
Σ TE
(zr - ur ur)p; SE 21 exp
eee
(13)
[ B
Let exp
22 8
j
S Sc we need,
2 8 1 B3 log &
(14)
is sufficient to pick & /810 log A, Therefore,
P
log Pr ((x+; 1x)reg)) - log ((T (1T,,1,,,e)))) 17)))0)) 8122 B3 f log 5e
>P
Σ (zr - 1r)1, Σ (zr u-)p- S 1 2t B2 log
I 8e
TE
TEQ!
>1 dz,
where the first inequality follows by combining Eq. (11 and Eq. 12
We then have the following high probability confidence lemma.
Lemma A.2. For any fixed ğ that is independent of ((xr, 17)re0)); we have, with probability at
least 8, V
242
log (p 17)re0)) log (r), < / B12 log
(15)
65
19

---

Proof.
We use & to denote the event log Pg(Tr 1))rE0)) - log P,((e,, (t),, 2 By log b. We pick òe ()) and have,
s
I P log P. ( (( r)))) log 1O 1))E)))) S 1800)133 log De
Vt 2 1)
=1 P [ &
=1 P (U
21
P (())
=1 Σ
(1 P (EE)
1=1
=1- Σ
- PP
log (( ( - log Po((ET, 17)= s 88021133 log 5.
21
8
65
=1 12
1=1
=1 - 6.
We then have a lemma to bound the difference of log likelihood when two functions are close in infinity-nomm sense.
Lemma A.3. Ve > O, Vgl, 92 E Bg that satisfies || 22 S. we have,
log g. ((z,, 1,) 17)r)))) log 192(((1,))))) 1,)reg)) s 2e)
(16)
Proof.
log P,, (1I,, 1, 1r)= log p0 ((TT; 1, 1=)))
$ Σ ((21, 22,r log(1 eflir) log( c)2r))
TEQ
<< +
max
V log (1 + + 1,,,- -2,7
=E|- Byl
rEQu
2cl09 + Σ
TE
<2e
where 21, 91 (x+) and 22,r = (2())
We use NBB,, e, || |loo) to denote the covering number of the set By, with (gf) )= (B be set of c-covering for the set B,- Set the '8' Lem. A.2 as 6/ and applying the probability union bound, we have, with probability at least 0, Vgf.
T2 e, 11,o)
log at ((r,, 1))) log ((r,, 1)) S V / B3 log
(17)
By the definition of e-covering, there exists E [VB,, e, || [loo)], such that,
MLE
981lo S.
(18)
20

---

Hence, with probability at least - 8,
log P (ET,, 1, r) log p ( 1, re
9111
= llg aMLE ((I,, 1r)r log yg ((x (re;) + log Po; ((7,, 1+)re02) log 1e)e)
1B12AN(B,, 1lxo)
<2c 8 0 B3 log
60
where the inequality follows by Lem.
A.3 and Lem. A.2
B Proof of Thm.
B.1 Bound Error over Historical Evaluations
gives high confidence set based on the likelihood function. The following Lem. B.1 further gives error bound over the historical sample points. Lem. B.1 highlights that with high probability, all the functions in the confidence set have values over the historical sample points that lic in a ball as the radius. Before we with the ground-truth function value as the center and Va(c, 6/2, 10/ t) proceed, we first introduce several constants that we will use,
Lem. 3.2
Bg S S S max S(u) 1 c uc| Bg Bgl
min S(u) + BB weI- B,l
(19)
S = min wEI-B,,Br Bal
S'(u) B te +2 +2'5'=
S'==
max S'(u) =
(20)
uEl- uEl-Ba,te B.
Hs 252 Bp S(BB)
S(-Ba) S(-B,) S(Bg)
(21)
Lemma B.1. For any estimate Hc+1 E Bb+1 that measurable with respect to the filtration Ft. we have, with probability least 6/2, 41 1,
Σ T
(0++1(x+)) g(xx))2 & a(c, f/2, @1),t),
(22)
and
g DEB"
(23)
where
a(c, 8/2, 2% ()
He (az(c, 6/2, 111,
2 log
WVB, c. 11l) 8
+ et + 2t).
with
= 8 22t + 4et
O B3 log
T' [x)
38
Proof. For any fixed function I, we have,
log IP ( l))regg) log Po ((r,, 1))re0)
Σ (log Py P3((E,))))))) 1,)) - log ( 17)))
Σ
(1, (log P. - log + - (1-17) (log (1 - ---) - llog(1 - pr))),
T
where P. S(g(x,)) and p. S(g(z+)). )). We have,
log < log
(y (y-x) x) Hs(y 2), [S,
(24)
x
21

---

where Hs 25a.
1. Hence,
log PogPy(Er rr)r) log ((z 1.ff.11))rs re)))
= Σ (1, (log Pr logg-) (1 17) (log (1 -pr) -log(1 - -p+)))
r2Q7
$ Σ rE ?
1,
Py Pr Dr
Hs (p- p.)² +(1 (11-1 (1-1,)) 1+)
Pr Pr-rr- Dr
Hy (pr p.)²) ))
Pr
1- Pt
Rearrangement gives,
Hs
Σ (0r - -pr)² (o (ot, log (( 1e
TEQY
Σ
1, Pr Pr
+ (1- 1,) Pr - pT
1,
Pt
Pr
TE
Since E
1, Pr + (1 - 1,) 1r) [ = E
pr
Pr Pr
+ (1 pr) pr [ = 0 and with
Pr
Pr.
probability one,
1 Pr Pr
+ (1 1,) Pr Pr
Pr Pr
Pr Dr
< 1,
+ (1 -1,)
(25)
Pr
-
Pt
1 - Pr
p7
D7
1
+ (1 1,)
(26)
Pr
1 - Pr
$ S(g))
S(-B,)
Bp.
(27)
S(-B,, B,)
S(B,)
By Azuma-Hoefiding inequality, we have Ag > 0,
Σ TE
1,
Pr
Pr
+ (1-1111 1,) Pr - Pr
SE
21 exp {
-
2E
(28)
Pr
1 Pr
We set exp
{-
2²²
1
>0, 0. and derive
[ [B
O log 1
P
Hg
Σ (p- -pp) + log P P3((Er) 1+1r,))))))) - log CogPP((a,)) s
b.
(29)
Dr Pr
Pr Dr
O BB log to
>P
1
(1 1,)
(30)
Pr
p+
TEQ
>1
d,-
(31)
22

---

We use & to denote the event Hs Hs (p+ - -p+)
log pg 1+)=
log Pa( 1))r + 1 log
We pick o, = (66)/ We have,
2
12 12
log 1 by
P
Hs Σ
(pr -p+)² 1 log (( 1()r) log Po 1, x))))
Vt 1
2
T
=1
P (NE-r
=1 P 1
21 Σ P ε.)
-1- Σ
(1 (E))
1
O B& log ét
=1-
1 1 PP
Hs Σ
(fr - s logg log
10))) log JogPo((er,)) 1))e))
1=1
T 22
21 Σ
1=1
66 20
1
t
=1
Resetting the '8' to be lll we can guarantee the inequality 32) holds for all the functions in e-covering of By.
For any Ic+1 E B6++1 there exists ğ in the e-covering of By, such that || rlloo S notations p 9t+1 (I.T), and pr g(x+). Thus, we have,
We use the
Hs Σ
(0t+1 p,)) 2
TEQ
-2Hss
Σ
1++1 -r))² 2 +2HS Σ (p, pp))²
T
TEQ.
=2H
(g (z,) - -(((z)) ))² + 2HH
Σ (pr p
TEQR
=EQ
<8H
+ 2Hs Σ (p. -pr)
T
TEQ.
58 Σ
2 2Hg Σ
(fr - PT)
TE
2 0 B? log T222N( 11o0)
58 21
+2 (log (o (,,, 17)re0))
log Po((,,, 1+)
65
5C((,,, 121,1) + loo P. mmm ((x,, 1, 1)) r)))) - log Pig ((or, 1r)re(;))
(log ((zr, 1, 17)))0)) log Po ((x ( 1,)re::)
6, + del + 2m (e. 6, 11 = + (c, (1
where C(c, 111,t) S 2 + / /2| log
r', and az 11
66
C((e, 6, 11 4et.
23

---

Furthermore,
Σ (p+t p.)²²
Σ (5)² (x)+ -
The conclusion then follows.
B.2 Bound Point-Wise Error
Lemma B.2 (Point-wise Error Bound). For any estimate 9 E B measurable with respect to Ft.
we have, with probability at least 1 5, vt 2 1 E
2Bg [(x) g(x)) S 2 (2B, +r 1/2 2(l. 09 (v).
(32)
where ge (x) Vhy(e,,)) xx) (K +rl) -kk)-kkg (k 2).
Proof. We use ((z) to denote the function (g -), where ø Rd - His maps finite dimensional point x e gd to the RKHS Ha.- For notation simplicity, we set k(.,-) = (;,-) in this proof. For simplicity, we use I h2 to denote the inner product of two functions h1, l2 from the RKHS Ha, Therefore, h(z) = (h, k(x, 1)),, = h ((x) and (gg = k, = <(x) ((x), Vx, x' X. We can introduce the feature map
p = I
we then get the kernel matrix K, = D kx(x) 1,)(2) for all x E X and ho: = P.h.
Note that when the Hilbert space Hes is iinte-dimensioonl Euclidean space, +, is interpreted as the normal finieeddmennional matrix. In the more general setting where Hi, can be an infinite- dimensional space, 4, is the evaluation operator Ha, + R 2 defined as h = [b(2+)]Teqg> wh e H. with T as its adjoint operator.
Since the matrices (L, 4, + rf) Hk, + Hu and (2,47 + rf) : R + g! are strictly positive definite and
(0) 4, + r. 4. ( + +rl),
have
Φ ( +rl)-1 (0) 0. + rI) 1
(33)
Also from the definitions above (0. o, rl)((z) 86 k()) +r(()), and thus from Eq. 33 deduce that
o(x) =
(())= (0,00 +rl) -x())) + -
(34)
which gives
(x)) =(()))))))) (0.0.
T< +r (<(x)) +
(35)
This implies
(10, d, + +1) k(zx))) kx)x (Ke + +rl) +r)- kx))),
(36)
24

---

which is by definition the posterior variance (0 (x)) Now we can observe that
IØ(x) - kx (K,
+ +r) 90g
=1o(x) 9 - ((x) (, 0 + + - P.g
-16(x) 9 ((x) -0x) + +r) $ 4:9g
=lo(x)) 0, (1, +r|) (r)) $. r/]g ( t. + +rrr++-0g 10, Pigl
(rr((x)"(())) +r
$ ||r(0, t1 + +r()) li 1
= =I19|k, rr((z) (0,
( 2 +
r1 +r 2 + +
<B / rō(x) (o) or + (2, (2, d, + +r))
= =B 00c41 (x);
where the second equality uses Eq.
33 the first inequality is by Cauchy-Schwarde Schwartz and the final
equality is from Eq. (36). We define eo get lor, where I. = g(x,). We have.
(xx( (K, +r) sp
= =b((a) 8 ( + r1) '
= =((x)) (0, t, + +r) 1 co
$ ||c0) 2, +rl) 1/2 )((z) (2)) db +rl) +r1)-1/24/ E
=V T ( (0)0)++1)))())))
+r|)
(0. <0: T(D T. (
r -rr) -1 c0
=r 1--1 0 (z)) 30 0,0, (0.00 rI) c
1/2 g (1)
C KK parr-l'co co
ST 1/2 0/20 (x) 66.
-1/2 a, 1/2 //200//////00/ (x);
where the second equality is from Eq. 33 the first inequality is by Cauchy-Schwartz and the last
inequality follows by Eq. (22 and az = aie, 12,,, t).
1Ø(x) (((x)-9() [(4)
< (k((z)) (K, r1) " (0z)) (g(z) - kx(z) (Kr + r1) (0;)) (g(z) ke(x) [K, + +rl))''e)
<]ke(z) T (K, +rrl-1(aa)) g0t)1 + 19(x)) kx(x) (K, + + (r)r ( + (ī(x) ke(x) (Kr + +r)) +r(l) "oo
Sag (x) 2B, + +1/2/
B.3 Efficient Computations of Confidence Range for the Latent Expert function g
Leveraging the representer theorem (77 107 thanks to the RKHS property, the MLE problem and confidence range computation problem can be reduced to an O ((( )dimensioonal, tractable optimisation problem (37 problem (38 and problem (39).
L. MLE)
min Zo E off subject
Σ 2,1, Σ
log (1+ eZz)
TEQU
T
(37)
Zo K 1ark-1 Zo2 < B2
where Koj = (kg(In 2m))
25

---

(x)
max
2
of EIR eg
zER, SE
T
7
(38)
subject
K-1
K
S By
2/
e(
Di) 2 le
- 0. (c, (.,111
where K (kg
()),t, and (Z
D2)
Ereor 1,1,
Ererg log (1+e2)) is the LL value when the function value at is ZT.
9 (x)
min
2
Z o0 e
aER, sEX
Zo
Zo:
(39)
subject to 2
K K-1
$ B2.
0 2
E(Zog
Dt) > Le
an 10]1.t),
where Kog = (kig(t, (o),,,)))) (a.,,xe and eZ
Df) = Ereoy 2,1,
Ereof log (1+e2r) is the LL value when the function value at F, is Zy
B.4 Bound Cumulative Standard Deviation Sample Trajectory
Lemma B.3 (Lemma 4, 12
on (It) S V |110f1+ +2) =0
Σ
(40)
EE
Similarly, we have,
Σ (at) S 1 + +2h =0 -v
(41)
EE O4
B.5 Bound Cumulative Regret
We can then analyze the regret of our algorithm. We use Cr to denote the set E [T] = x7}-
ROf - -
1E2/
= Σ (s(x)) - [(x))}
Σ (fxx)) - f(x*)]
teR/nnr
IER//C
Σ ()(²)) - f(x))+
Σ (s(x)) f(₂)))
1E0//Cr
AAppears in the arXiv version: https///axx...rr/////////......0
26

---

For the first part, we have,
Σ [5(z5) f(xt)]
E (5(zg) +,0)) f(x))]
nERtnCT
Σ (s(x]) - +(00 409) - 1(x')
<
Σ (s(x]) - -6 + 4(4) - ((())) - [((x')
1ERGNCT
Σ (1))+ +
Σ -1(4))-1117))) 1,77)
S
1ER/NCT
228 f
Σ (x)+))
(f((5)) [1,0) - f(,)))(((77))
LE 1f
teOfner
where 0,, is given in Eq. (2b). the first inequality follows by Lem. 3.1 the second inequality follows by Lem. 3.1 and the line of Alg. [
Furthermore, we have,
Σ
(42)
1ER/ncr
Σ
[(((())- - L(
(43)
Σ
(44)
Σ 23, (2()
(45)
IER
Σ
(46)
EERtnCT
where the first incquality follows by the condition in line the Alg. the second inequality follows by the Lem. 3.1 and the third inequality follows by the condition in of the Alg. T
For the second part, we have,
Σ
((at))
- f(x))
(47)
1E0//0 VCT
= Σ (s(x)) - 1,(0%) + -,00))+ -
- f(x*))
(48)
122//CT
Σ ((xx)) -
f,('']
(49)
Σ 281
(50)
1E
228
Σ
o ) (2x)
(51)
1E0/ACT
where the first inequality follows by that f(x*) Z f (x*), the second inequality follows by the
optimality of at for the problem in line fine5lam and the Lem. 3.1 and the third inequality follows by the monotonicity of 8y, in t.
27

---

Hence,
of 52 + n1 Σ oz((2))
1E0/
52(2 + mm
4110/1 + 2)
-0 (1e4
B.6 Bound Cumulative Queries to Labeler
We can then analyze the cumulative queries to the expert. We notice that, Vi E 2g.
(x(x)) 9 (x1) 2 Jhn
(52)
Meanwhile, by Lem. B.2
(x(e))) 2, (2t) <4(2B, + r r/2 vas) 0g.
(x) (z).
(53)
Hence,
9hhr 4 (2B, +r 1/2 V age (x).
(54)
Therefore,
07 -10%
(55)
ΣV
(56)
BE Oz
Athr
(57)
9thr
DE 1E
4 (2By + +r- 0 (11e)
(58)
Ath
2BB + -
Σ
Tg. (1))
(59)
Ath
1E
(v Varniay 22 129
(60)
Dividing by 1 we obtain,
1 = 01y Vovangee
(61)
Hence,
Q = 2 = (
(62)
By setting 7. we have
(VIeA) log TW(B,, I/T, 1
oT =
(63)
8
Hence, dividing by / O on Eq. 62 again, we obtain,
Q = = 0 ((1Pet)) logg TN I/T, 11,o)
<0 ((78)211)) log TN I/T, |l 11110))
(64)
28

---

C Detailed Discussions on The Significance of Thm 4.1
Order-wise improvement can not be attained under current mild assumption. may contain no information (e.g.. = 0) or even adversarial. Even if human expertise is helpful, we can not guarantee an order-wise improvement either. For example, consider the following 9.
g(x) = f(x*) + if f(x) f(x)) S (f(x)
otherwise,
where e >> 0 is positive constant. In practice, such scenario means the human expert has some rough idea in a near-optimal region but not exactly sure where the exact optimum is. This is common in practice. In this case, human expert helpful in identifying the region with f(x) f(x*) + c but no longer helpful for further optimization inside the region (a E (() S =(xx) + ej. However, convergence rate is defined in the asymptotic sense. Hence, an order-wise improvement can not be guaranteed.
Assumption becomes unrealistic if we really want it. Some papers that show theoretical superiority [2. 6]. yet the assumptions are unrealistic. For example, [6] assumed that the human knows the true kernel hyperparameters while GP is misspecified, and [2] assumed the human belief function g has better and tighter confidence intervals over the entire domain. We can derive the better convergence rate of our algorithm than Al-only ones if we use [2] assumption, but this is unlikely to be true in reality. In fact, our method outperforms these method empirically (see Figure 5). This supports the superiority based on unrealístic conditions is not meaningful in practice.
Empirical success can be achieved without order-wise improvement on worst-case convergence. Our assumption is more natural; following [37]. we posit humans have better prior knowledge than GP and are only useful at the beginning as a warm starter. This assumption is widely accepted by the community and practitioners, which leads to real-world impact (e.g. Nature [42]). The warm- starting- based papers [36, 37, 44] have been published in reputable venues without such a theory. In manuscript, real-world applications also empirically demonstrate that our method not only improves the convergence of BO, but also maintains robustness despite varying labelling accuracy.
Worst-case convergence and hand-over guarantees matter. We believe that the value of theory the worst-case guarantee. To clear, starting point of human-Al collaborative BO that the experts not currently using BO. The scientific experts do very expensive tasks, which often cost millions of dollars and weeks to months to test one design (c.g. battery design). They are reluctant to employ BO due to its opaque and untrustworthy nature. The experts want to be involved in the AI decision-making process, otherwise they are forced to work as a robot feeding experimental results to the AL. But, they are also in the middle of trial and error, their advice is not always reliable. Our worst-case guarantee assures that at least their involvement does not harm the Al-only results, and also assures the automation in the later round. Thus, we believe approach applicable range of BO to high-stakes optimisation tasks. Furthermore, our handover guarantee assures that only limited human labeling effort is needed, which is also meaningful because the motivation to use BO is to alleviate the tedious human effort in the first place.
extend the
D Proof of the Kernel-Specific Bounds in Tab. 1
For the cumulative regret part, we have,
If the kernel function is linear, 7 (log 14DD, and thus Ri =
o (V ( log 1041).
If the kernel function is squared exponential, of jo41
O(loo (f))++1), Rof
o(ef) f log 2 a (1041) ((V> $
.
If the kernel function is Mátern, Mofi
Roj = (1of
30
loaa (12f
29

---

To bound the cumulative queries, we have,
1. Kg is linear kernel, then log (B, 1, ,1, 1000 O (log O (log T). By Thm. 5 in 84,
7 = =0(og]]
Hence,
09- = O (l) log T) = O (llg T) 2. kg is squared exponential kernel, then log, N (B _-1,111, ll O (log 2
O ((oggTyat)) (Example 4, [. By Thm. 4 in [49], have,
2 = ((log d)
Hence,
()) 2()+ (log T7)+1 (log 3. kg is a Matern kernel, then log. (B, T-, 1l00) = 0 (( log
O = (11g
= (logg))(a+t)
O YT T) (by Thm. 5.1 and Thm. 5.3 in [105).. By Thm. 4 in [49] have,
7 (1 ( log
Hence,
07 =0 -0(T (To++le+1) (log 2) logt) = 0(1 (1) T (log 2)
20(++1
where d(d+3tvv++10417
E Theoretical improvement of convergence rate
Algorithm COllaborative Bayesian Optimization with Helpful Labelling Experts (COBOHL).
Input and Initialization: function space ball Bg. and uncertainty threshold Jür-
2: Set B3 By. of 0, and O = 0.
3: for E [T] do
4: Generate Tt by solving the constrained auxiliary
optimization problem Expert-constraincd LCB Handover guarantee
minzex fz) subject to 9, (z) O.
5:
if (era))) 2, (x+)) > Ihr then
6:
Query the expert's label to get the feedback 1,,
7:
Update O = 0 U {1} and the posterior confidence set B Set Of o
8:
else
Evaluate the black-box function at the point It, and set of of_, U (t). Set O
9:
10:
Update the posterior mean/variance of the objective f.
Here, we give the analysis the regret of COBOHL,
Σ
= Σ (f,x))),((z)) + - ff,x)) + - -fx))))())))
-
(65)
16
te0f
Σ (f(x)) - (f((x))
(66)
1E0/
Σ 28 o (mx)
(67)
1E0/
$ 2 Σ (ar)
(68)
tE2%
0 (
f,89
V10%1
(69)
30

---

Table 2: Comparisons between our algorithm with the existing baseline methods,
blackhox human model?
no-rankahility assumption?
continunus guarantee?
data-driven trust?
handpver guarantee?
baselines
guarantee?
AV al. (2022) 1
X
Hvarfner et al. (2022)
X
Gupta et (2023)
Khoshvishkaic al. (2025)
50
X
X
X
Cisse et ul. (2023) 2
X
X
X
X
Adachi et (2023) t
Rodemann al. (2024) 70
X
X
X
AV al. (2024) 12
X
X
X
Hvarfner et (2024)
Ours
where the first incquality follows by the feasibility of z; in the expert-constrained LCB problem and (x()) f(="), the maximum information gain is defined over the set X := {xr Xlg(() S Jun).
(g fxx 10
Notably, the regret bound for vanilla LCB has maximum information gain defined over the region X. For commonly used kernel functions, the maximum information gain is proportional to the volume the set. Since X X, vol(X*)) I vol(2') and the maximum information gain gets reduced by ratio Therefore, the regret bound gets improved by a ratio of xol( vol(X'
Meanwhile, the regret bound vanilla LCB has a similar form of
|
of vol(X")
xol(X")
F Estimating norm bound online
By Assumption 2.4, there exists large enough constant B, that upper bounds the norm of the ground-truth latent black-box function g. However, tight estimate of this upper bound may be unknown to us in practice, while the execution of our algorithm explicitly relies on knowing a bound By (in Prob. (6) Bg is a key parameter).
So necessary to estimate the norm bound Bg using the online data. Suppose our guess is B. is possible that is even smaller than the ground-truth function norm || To detect this underestimate, observe that, with the correct setting of Bg such that Bg Z ||g|], we have that by Lemma 3.2 and the definition of maximum likelihood estimate,
(0, MLEE 2 (g) 2 (O (LME)) an (e, 6, 1))..B).
where 0/1 is the maximum likelihood estimate function with function norm bound B and on (e, 6, 12 , 4 (B) is the corresponding parameter as defined in Lemma 3.2 with norm bound ß. We also have 2B is a valid upper bound on |g and thus,
> (g) 2 (G an 8,,
2B
Therefore,
MLE > 4()) > ls (0 MLL)) an (c, 1,,1,
That is to say, ML. needs to be greater than equal to ( (c, (()1, when (
ß is valid upper bound on [|g]]-
Therefore, we can use the heuristic: every time we find that
(G ( (c,
we double the upper bound guess B.
G
Related Work
We summarized the baseline comparison in terms of five factors in Table Our algorithm is the first to offer data-driven trust level no-harm guarantee and a handover guarantee under no rankability assumption.
31

---

We briefly introduce the baseline methods used in the real-world experiments:
1. AV. et al., NeurlPS 2022 [I:
This algorithm initially proposed the human-AI collaborative setting. The approach is straightforward: human experts can intervene in the optimization process if they find the next query location suggested by the vanilla LCB BO to be un- promising. This method can be described as a 'human as constraint' approach, where the BO must adhere to the experts' recommendations regardless of the quality of their advice. This approach assumes that human experts are at least better than the vanilla LCB, thus requiring high level of trust in the experts. As shown in Figure 5 experts' input is not always reliable.
2. Khoshvishkaie al., ECML 2023 [50]:
This setting assumes that the querying budget is equally divided between human experts and the vanilla LCB BO. This means that once a point is selected by human experts, the BO will alternately select the next query. This method can select the vanilla LCB regardless of what the human expert selected, making it likely to achieve a no-harm guarantee, although no theoretical proof is provided. The trust level in experts in this method is low, as all expert inputs are treated equally regardless of their quality. Therefore, while this method performs well in unreliable settings, it is not as effective when experts are good advisors. To be fair, their work focuses more on imperfect cases and does not consider scenarios with effective experts.
3. Adachi al., AISTATS 2024 77 This setting assumes that the BO provides two possible candidates, from which the human selects one. Both candidates have convergence guarantees, thus ensuring a no-harm guarantee, although their proof is limited to discrete settings. However, the human must ultimately choose one of the candidates, maintaining a high level of trust in human experts. They introduced discounting function that hand-tunes the decaying rate of trust, gradually generating the same candidates. Although their work initiated the no-harm guarantee concept, the trust level adjustment is not data-driven and the proof is limited to discrete cases. To be fair, their main focus is on the explainability of black-box optimizers, which we did not consider in this work. Their method can be integrated into the GP surrogate model as plug-ant and -lay feature, making it easy to extend our work.
We did not compare against the following papers due to difficulty in aligning assumptions and similarity.
1.
43 42 24 : These works assume that humans explicitly express their beliefs
probability distribution, such as a Gaussian distribution centered at the most promising location. This assumption is too strong and incompatible with our black-box assumption of
human belief.
2. (39 12: : These methods are nearly identical to 50 Therefore, we selected 50 as a
representative work for this pessimistic approach.
3. [70]: This method is almost identical to [.. Thus, we selected as representative work for this pessimistic approach.
H
Comparison and Generalization to Other Feedback Forms.
H.1 Other feedback forms
(a) Pinpoint form: [T
39 50 adopt this form that the algorithm asks the humans to directly
pinpoint the next query location.
(b) Pairwise comparison: 7 adopts this form that the algorithm presents paired candidates, and the human selects the preferred one.
(c) Ranking:
12 adopts this form that the algorithm proposes a list of candidates, and the human provides a preferential ranking.
(d) Belief function: 4 42
adopt Gaussian distribution as expert input. Unlike the others, this form assumes an offline setting where the input defined at the beginning and remains unchanged during the optimization. Human experts must specify the mean and variance of the Gaussian which represent their belief the location of the global optimum and their confidence in this estimation, respectively.
32

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Primal-dual x
pinpoint x
MVN Delief'mode
Simple regret SR,
vanilla LCB
(ndverstrial)
Simple ragret SR,
LCB strong weak: wrong
(random)
2
(informative)
0
0
10
20
30
40
50
0
20
30
50
100
200
Function evaluation 12,1
Function evaluation 10,
(a) (a) Primal-dual vs pinpointing
(b) (b) MVN human belief model
Figure 6: Different forms of human feedback
H.2 Adaptation
Slight modification can adapt these forms to our method.
(a) Pinpoint form:
We can simply replace the expert-augmented LCB line 4 Algorithm
with the pinpointed candidate.
(b) Pairwise comparison:
By adopting the Bradley-Terry-Luee (BTL) model [17], we can extend our likelihood ratio model to incorporate preferential feedback. This allows us to obtain the surrogate, while the other parts of our algorithm remain unchanged.
(c) Ranking: Ranking feedback can be decomposed into multiple pairwise comparisons. Therefore, we can apply the same method as the pairwise comparison.
(d) Belief function:
We can use this Gaussian distribution model as the surrogate.
H.3
Comparison
We demonstrate the adaptation of (a) pinpoint and (d) belief function forms in Fig. 6 The pinpoint strategy employs a sample from the expert belief function as Ie on line in Algorithm [ while keeping the remaining lines the approaches, particularly in later iterations. This is because expert sampling does not incorporate GP information. Generally, humans excel exploration in the beginning, while GP excels at finding precise locations in the later stages. This finding is supported by other literature, such as [48]. involving human expert studies.
as the original. performs worse than the original primal-dual
In Fig. 6 ()) we employed the multivariate normal distribution (MVN) belief model proposed by 43 This model represents the human belief function أ NNxx; E), where μ is the mean vector representing the estimated location of the global optimum xx,, and the covariance matrix, representing the confidence of the estimation. We use Σ I, the identity matrix I, as suggested
by 43) We transform: [0. | 1/2] + [0, 1). and we use this normalised belief function as the acceptance probability of a Bernoulli distribution
p given location (note that p 0 is acceptance). Following [43, we set three levels of beliefs: strong, weak, and wrong. These levels are established by adjusting the mean vector to be offset from x². 'Strong' aligns with x², 'wrong' is the furthest possible location from x", and 'weak' is an intermediate location. Our algorithm robustly converges for any level of trust.
As such, the primary reason we adopted binary labelling is due to its empirical success, as demon- strated in Fig. and Fig. [6] None of the other formats, including (a) pinpoint form [1I] 50 and (b) pairwise comparison 77. outperforms our method. the experiments by 77. the authors showed that (a) pairwise comparison outperforms both (d) belief form (43). Therefore, it logically follows that our binary labeling format yields the best performance.
The main reasons why the binary format works better are as follows:
(a) Pinpoint form: The accuracy pinpointing is generally lower than that of kernel-based models. Humans excel at qualitative comparison rather than estimating absolute quantities
Numerous studies [1 48 50 70 have confirmed that manual search (pinpointing) by
human experts only outperforms in the initial stages, with standard BO with GP performing
33

---

Synthetic human learning rate
simple regret
Cumulative queries
(informative))
2
noneatinary dotted sationury
200
Cumulative queries !Q
Feedback
regret SR
accuracy
9,-0.1
Simple
2
100
0.00
(random)=+0
0
10
20
30
40
50
0
10
20
30
40
50
%
100 iteration
200
Function evaluation (
Function evaluation
12
Figure 7: Non-stationary human accuracy.
shows that this type of feedback only outperforms when the expert's manual sampling is consistently superior the standard BO. However, such cases are rare in our examples (e.g., Rosenbrock), and [I] 70 corroborate this conclusion.
better in later rounds. 39
(b) Pairwise comparison: This format relies two critical assumptions: transitivity and completeness. Transitivity assumes no inconsistencies, which are often referred to as a pock--aper-scissooss" relationship. However, real-world human preferences frequently exhibit this issue [20). Completeness assumes that humans can always rank their preferences at any given points. In practice, when user is unsure which option is better, this assumption does not hold. Our imprecise probability approach avoids these issues by not relying on an absolute ranking structure [10
(c) Ranking: Ranking is an extension of pairwise comparison and has been classically re- searched as the Borda count, which is known not satisfy all rational axioms. Theoretically, the Condorcet winner in pairwise comparison is the only method that is known to identify the global maximum of ordinal utility.
(d) Belief function:
This is another form of absolute quantity, which humans are generally not proficient estimating. Additionally, the offline nature of this method does not allow for knowledge updates.
I Potential Extensions for Future Work
L.1 Extension to Time-varying Human Feedback Model
In practice, human's belief in the black-box function may be influenced by the online evaluation results of the ground-truth black-box function. To further incorporate such online influence, need to model the change of human feedback model.
Simple extension, yet not promising performance gain. The most naïve approach for non-stationary model windowing, i.e., forgetting the previous queried dataset. This can be very easy to apply to our setting, as it simply removes the old data outside the predefined iteration window.
Fig. 7 shows the scenario where the accuracy of human experts' labelling improves over time, rep- resented by a 2(1 cxp(=our/1Q) )). where air controls the learning rate. The non- non-stationary model employs windowing, retaining only the most recent w-th data points, with = 5. The stationary model does not use windowing, thereby retaining all labelled datasets. The plots represent the average of 10 runs without standard error for improved visibility. While simple regret showed slight improvement initially, the performance gain varied depending on atr- In contrast, the cumula- tive number of queries Q significantly increased due to the increased uncertainty introduced by windowing.
More sophisticated extension. Another more sophisticated approach is modelling the dynamics of behavioural change. A potential idea is modelling the human behaviour change as an implicit online learning process of the latent function That is, It+1 = F(g.. 2t, 2t,3e), )²), where g, is the human latent function step l. The forward dynamics F captures the update of human latent function when observing the new data point. One potential F is gradient ascent of log- liglliiklihood as shown in
It+1 = 9: AV, log (xt,, )t), where Py (at,, ) is the probability of observing ye at the input x given the black-box objective function is g. We can then combine this dynamic with our likelihood
34

---

Table 3: The complete list of hyperparameters and their settings.
hyperparameters
initial value
data-driven optimisation?
tuning method
kernel hyperparamters
Torch default Borch default
maximising the marginal likelihood copying kemel values
kernel hyperparamters
in Eq
le-4
fixed
[
in Eq
algorithm using 40
By in Lemma 3.1
standardised (=1)
fixed
in Lemmaa
0 =
fixed
in Lemma 3.T
0.01
fixed
5y. in L.mma
using the equation in Lemma 3.1
Xe in 6
using dual update in Eq.
in Eq.
0.02
fixed
Bg in Eq
the method in Appendix the method in Appendix
0q in Eq. o
0.01
in line. i Alg.
fixed
in line, in Alg.
le-5
fixed
ratio model. Since this part requires significantly different analysis and experiments, we leave it as future work.
I.2 Extension to Adaptive Trust Weight !7
In line 6 of Alg. 2 the weight is fixed. An adaptive y could offer better resilience to adver- sity. However, even without such a scheme, our no-harm guarantee holds, both thcoretically and empirically.
Adaptation through the posterior standard deviation. Although rj is set be constant in our current design of the algorithm, there is still adaptation on trusting human or the vanilla BO algorithm through the time-varying posterior standard deviation. Intuitively, if originally the expert-augmented human-preferred
solution of is trusted more, more samples are allocated to human- preferred region and a, (2,)) drops
quickly. Intuitively, if we keep sampling of and xt + xf, o (a²)) would finally be larger than vjo, (xr{) and we switch to sampling x/.
Choice does not need to be very large in practice. Intuitively. 7 captures the belief on the expertise level of the human. The more trust we have on the expertise of the human, the larger y can choose. But larger 7) increases the risk of higher regret due potential over-trust in adversarial human labeler. In our experience, does not need to be very large. Indeed, 3 already achieves superior performance in our experiment (see Fig.
1.3 Extension to Different Acquisition Function
Our algorithm can be easily extended to other acquisition functions. For example, we can indeed use similar idea to extend expected improvement (EI) acquisition function to human constrained expected improvement (HCEI) to generate zz
of E arg max PP is accepted by human)El(aa ZEX
(70)
J Experiments
J.1 Hyperparameters
summarized the comprehensive list hyperparameters used in this work and their settings in Table 3 Most of these are standard in typical GP-UCB approaches. The newly introduced hyperparameters are primarily tunable data-driven manner, and we provided a sensitivity analysis in the experiment section for those that are not.
35

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J.2 Synthetic Function Details
J.2.1 Task Definitions
Ackely
Ackley funciton is defined as:
d
d
f(z) exp
24
exp
cos()
+a + exp(1)
(71)
where a = 20, c 2 d = We take the negative Ackley function as the objective of BO to make this optimisation problem maximisation. This is d-mmennsioaal function bounded by [-1, 1 The global optimum x² [0, 0, 0, 0] and f(x*) = O.
Hölder Table
Hölder Table funciton is defined as:
f(x) = sin(x1) (ost))oo(((2))))) (x2) exp
(72)
π
where T; is the i-th dimensional input. This is a 2-dimensional function bounded by x E [0, 10
The global optimum is xx = [8.05502, 9.66459) and [(x") 19.2085.
Rastringin Rastringin function is defined as:
d f(x) = 10d Σ
[ot 10 co0ms2mm
(73)
2=1
where 2; is the i-th dimensional input. This is 2-dimensional function bounded by x E
[-5.12, 5.12).. The global optimum is [0, 0] and f(*))
Michalewiez Michalewicz funciton is defined as:
ix?
f(x) = Σ sin( sin 2m
(74)
where x; is the i-th dimensional input and m 10. This is a 5-dimensional function bounded by E [0, Tíd. The global optimum is f(=") - 4.687658.
Rosenbrock Rosenbrock funciton is defined as:
d-1
f(x) = 1100(;441 - =2?) + (x; - 1
(75)
where 2; is the i-th dimensional input. This is 3-dimensional function bounded by x E [-5, 10)
The global optimum is x [1]d and f(x*) a 0.
J.2.2 Computational time and elicitation efficiency
the comprehensive experimental results, including overhead and cumulative queries. Overhead refers to the wall-clock time in seconds required to generate the next query location. While the time taken query the objective function is excluded, the time query human (or synthetic) experts is included. Our overhead is the largest among the simple baselines; however, an average of around 10 seconds per query is reasonable when compared more computationally expensive algorithms, such as information- -thooretic acquisition functions, which typically require several hours per query. In most experiments, we observe a plateau in cumulative queries, indicating a handover guarantee. In the case of the Michalewicz function, plateau has not yet been reached due to its high-dimensioonl nature. Nevertheless, we observe convergence acceleration in both simple and cumulative regrets.
36

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Ackley
Hölder Tuble
Rostringin
Michalewicz
x10
Rosesbrock
Simple
regret SR.
20
10-
0
Camulative
regeet Rg:
400
600
120
x10
400
60
4
10
10
10
10
Overhead (8)
10
10
10
10"
10
10
20
30
40
10
20
30
40
Flunction evaluation (?)
Function evaluation [0]
Function evaluatioo (2)
Flanction evaltaation
(2)
Function evaluation (Q)
120
120
Qumdative
queries Q?
80
30
80-
80-
80
40
40
40
50
100 Meration
130
200
30
100
150
200
9
50
100
150
200
50
100
150
200
0
100
150
200
iteration
Itentiion
iteration
Vanilla I.C
random
expect sampling
Ours
Figure 8: Simple and cumulative regrets, overhead, and cumulative queries for synthetic experiments.
feedback accuracy a
trust weight /
3
vanilla LCB (adversarial)
vanilla LCB
regret SR,
1-=
Simple
2
(random) (informative)
2
y
10 1=100
50
100
150
200
0
50
100
150
200
Function evaluation 12
Function evaluation 12
Figure 9: Confirming no-harm guarantee.
J.2.3 Comprehensive check for no-harm guarantee
We examine the no-harm guarantee by extending the iterations confirm that our algorithm can converge at rate comparable to the vanilla LCB. We tested with the two adversarial cases; (1) varying feedback accuracy
{-2, 1, 0, 1, 2} for the fixed trust weight 7 and (2) varying trust weights 1) E (3.5. 5. 10, 100} for the fixed accuracy = -2. Our algorithm converges to the same regret as the vanilla LCB over multiple iterations in both cases. We observed saturation behavior, where the convergence drop starts at similar locations among larger n, indicating that the no-harm guarantee assured regardless of how large 7) becomes. Particularly, the convergence curves of 10 and 100 are almost identical, supporting the saturation perspective.
J.3
Human experiment details
J.3.1
Task definitions
The task involves identifying the optimal electrolyte material combination to maximize ionic con- ductivity lithium-ion batteries. Ionic conductivity crucial for reducing internal resistance, which is essential for fast charging. Slow charging remains one of the biggest challenges for the widespread adoption of electric vehicles. Therefore, finding the best electrolyte combination is crucial to ad advancing electric vehicle development and realizing sustainable socicty.
37

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In our study, we considered four types of electrolyte materials. For demonstration purposes, we did not conduct physical experiments. Instead, we utilized an open dataset and fitted functions to interpolate between data points, creating continuous search space. Experiments were then performed on this synthetic data using software and four human experts. In real-world development, researchers and engineers synthesize these materials, which is expensive, making the expert's 's labeling process significantly cheaper than objective queries.
Li standard design The first task involves the EC-DMC-EMC-LIPP, system 226 7 where EC, DMC, and EMC are ethylene carbonate, dimethyl carbonate, and ethyl methyl carbonate, respectively, and LiPF; is lithium bexafluorophosphate Ionic conductivity depends on both lithium salt molarity and cosolvent composition. Using the dataset from [26), we fitted the Casteel-Amis equation [19] and extended it to continuous space. The input features are (1) LiPFs molarity, (2) DMC vs. EMC cosolvent ratio, and (3) EC vs. carbonates cosolvent ratio, with inputs bounded as ø1 E [o, 2),
22 [0. 1]. and 23 [0, 1]. The output is generated by adding i.i.d zero-mean Gaussian noise with a variance of to the noiseless function. We take the negative of the ionic conductivity in log mS/cm as the minimization objective.
Li+ methyl-acetate The second task involves the MA-DMC-EMC-LiPP, system being methyl acetate. Using the dataset from [56], we fitted the Casteel-Amis equation and extended it to continuous space. The input features are (1) LiPFs molarity, (2) DMC vs. EMC cosolvent ratio, and (3) MA vs. carbonates cosolvent ratio, with inputs bounded as æι [0, 2]. x2 [0, 1], and I3 E [0, IJ. The output is generated by adding i.i.d. zero-mean Gaussian noise with a variance of 1 to the noiseless function. We take the negative of the ionic conductivity in log mS/em as the minimization objective.
56 7 with MA
Lit polymer-nanocomposit The third task involves the PEO-LLZTO nanocomposite elec- trolyte system (108]. where PEO is polyethylene oxide, and LLZTO is lithium garnet ALa (0,,) nanoparticles. Using the dataset from [1o]] we fitted GP model and extended it to continuous space. The input features are (1) PEO volume %, (2) LLZTO volume %, and (3) LLZTO particle size in micrometers, with inputs bounded as I1 E [700 95), 22 E [5, 30,, and It E [0.04, 10). The output is generated by adding i.i.d. zero-mean Gaussian noise with a variance of 1 to the noiseless function. We take the negative of the ionic conductivity in log mS/cm as the minimization objective.
Li Ionic liquid The fourth task involves the bmimSCN-LiCIO, -LITFSI LLTT ionic liquid [72), where bmimSCN is lithium bestriftoroonehhnnsullfonnn)nn
thiocyanate, LiCIO, is lithium perchlorate, and LiTFSI is Using the dataset from (72), we fitted a GP model and extended it to continuous space. The input features are (1) LiCIO, molarity, (2) LiTFSI molarity, and (3) bmimSCN molarity, with inputs bounded as æ E [0, 4], x2 E [o, 1.5], and x3 E [3, 5). The output is generated by adding i.i.d. zero-mean Gaussian noise with variance of I to the noiseless function. We take the negative of the ionic conductivity in log mS/cm as the minimization objective.
J.4 How Do Human Experts Reason?
We explore how experts reason through these optimization tasks. lonic conductivity is roughly estimated by the product of movable ion density and diffusivity, as described by the Nernst-Einstein equation. Experts base their evaluations on this relationship.
Lit standard design In this system, EC plays a crucial role in both factors. LiPFa provides movable ions (Li+ and PF ). but these ions are not mobile in their raw state due strong electrostatic forces. EC, a highly polarized but non-charged solvent, dissolves LiPFs through solvation. Increasing EC concentration can raise movable ion density, but EC's high viscosity slows diffusivity, creating a convex curve. Experts generally agree that the global maximum is around 30% EC and 1 M LiPFs. but the optimal EMC/DMC ratio remains uncertain. EMC and DMC are similar, with EMC being larger and asymmetric, and DMC being smaller and symmetric. Smaller molecules tend to be more diffusive, so higher DMC ratio is expected to be better, although the asymmetric structure of EMC could disrupt higher-order solvation networks, contributing to diffusivity.
38

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In summary, experts vaguely know the whole function shape and possible global optimum location for two variables, yet others are unknown.
Li methyl-acetate his task involves replacing EC with MA from the first task, making the overall system similar. However, MA is an unusual material, and none of the participants are familiar with it. We will explain how experts reasoned this change in the optimization task.
EC plays a central role in dissolving LiPFs, increasing movable ion density, although is viscous. While one knows methyl acetate, it can be inferred that it also dissolves LiPF,, The challenge lies in determining its polarization ability and viscosity. EC is a planar molecule with a five-membered ring, resembling 'small sheet magnet" with strong magnetic power but easy stacking, Conversely, MA a small, con-ring-structured asymmetric molecule. This asymmetry prevents MA molecules from stacking, enhancing diffusivity. However, the asymmetry also reduces polarization, leading to weaker solvation effect and lower movable ion density.
Thus, MA has mix of positive and negative effects, making it difficult for experts to predict the exact shape of the convex curve. Nonetheless, in most "less viscous" solvent systems, the peak typically occurs around 1.5 M of LiPFg- Experts can roughly estimate this position, and this estimation is fairly accurate, as the true position is at 1.35 M.
Li pancomstnanoc This task is completely different from the previous two tasks. Our electrolyte is now solid-state rather than liquid, so the NemsstEinnttiin equation may not be applicable. However, the core idea remains the same. PEO is a framework material without ionic conductivity, whereas LLZTO has ionic conductivity. Generally, a higher LLZTO content should result in greater conductivity. Other factors are less certain.
We can anticipate the effects in both directions. Smaller particle sizes might be better because they distribute more evenly within the PEO, increasing ionic conductive paths. However, smaller particles might also be worse due to increased grain boundaries and aggregation caused by electric forces. Thus, most experts expected convex relationship with particle size and monotonic increase with LLZTO ratio.
In reality, experimental results showed that conductivity improved monotonically with smaller particle sizes and displayed a convex relationship with LLZTO volume. Therefore, the experts' advice was somewhat inaccurate.
Looking back, experts were partially correct. Aggregation did create the convex shape in LLZTO volume ratio, indicating their understanding of the phenomenon. However, they did not identify the correct input dimension where aggregation mattered. For particle size, the thorough mixing procedure with ball milling used in the dataset prevented aggregation, leading to misconceptions about the function shape.
Na Tonic liquid This task is completely different from the previous tasks. Although electrolyte is liquid, all materials are ionically conductive. As the name suggests, ionic liquids are special materials that can dissolve themselves without the need for cosolvent. Consequently, the movable ion density factor remains almost unchanged, as all components are conductive regardless of composition. Therefore, diffusivity becomes the dominant factor. Diffusivity primarily depends two factors: molecule size and electric interaction. Smaller molecules are generally more mobile, but they also have stronger electric interactions when the charge the same (all ions in this system are monovalent).
This dual dependence leads to different expectations: if size the dominant factor, smaller molecules (like LiCI,) are expected to perform best. Conversely, if electric interaction is dominant, the results will differ.
Most experts anticipated monotonic change in all dimensions, expecting both LiCl, and LiTFSI to show increased performance due to their smaller size compared to bmimSCN. However, experimental results showed double peak shape for LiTFSI vs. bmimSCN and convex shape for LiCl, and bmimSCN. Thus, the experts' advice was inaccurate. The real physical phenomena were more complex than initially thought, with electric interactions playing more dominant role.
39

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Effective advise
Unreliable advise
Li standard design
methyl-ncetate
Li pulymer-nunneoompesiee
Li lonic liquid
Best Observed
0.6
0.3.
Overhcad
10"1
10
10
10
10a
10
10
10
20
30
40
0
10
20
30
40
10
20
30
40
10
20
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Function evaluation
10
Function evaluation
10
Function evaluation
10/
Function evaluation
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Figure 10: Simple and cumulative regrets, overhead, and cumulative queries for real-world experi- ments.
JA.11
Computational time and elicitation efficiency
Figure TO illustrates the full experimental results, including the best observed values max Y overhead, and cumulative queries O The overhead definition remains consistent with that in the synthetic experiments. Note that these experiments include only four human trials, resulting in noisier data compared to the synthetic experiments, which used 10 random seeds. The overhead for our method and the baselines is approximately the same, around 10 seconds per query. This is manageable compared to the significantly slower methods, such as information-theoredt acquisition functions, which take several hours per query.
241"
Regarding cumulative queries, only our method demonstrates a handover guarantee. While baseline methods continue to request human intervention even the experiments conclude, our method stops requesting input midway through the experiments, thereby freeing the human expert from the task. Our approach allows for more effective input from experts in cases where their advice is beneficial and reduces input in unreliable cases. In contrast, the baselines request input regardless of the quality of the advice. Notably, the method described in increases the frequency of requests when experts provide incorrect information. This occurs because disagreements between the surrogate f and human beliefs prompt human experts to intervene, aiming to prevent the BO from proceeding in wrong direction. Unfortunately, this intervention can act as adversarial response. In contrast, our algorithm avoids such scenarios through active learning constraints (as highlighted line 6). thus achieving no-harm guarantee in unreliable cases.
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Reviews:
[{"review_id": "yhZHtOxu63", "reviewer": "Reviewer_72YW", "summary": "The paper studies the integration of human expert knowledge into Bayesian Optimization (BO) processes through binary accept/reject recommendations. The authors introduce an approach that ensures two key guarantees: Handover Guarantee: The approach proves a sublinear bound on the cumulative number of binary labels needed from experts. Over time, the need for expert labels decreases, thereby saving effort and computation time. No-Harm Guarantee: The algorithm adapts the trust level in expert advice based on data, ensuring that the convergence rate will not be worse than using BO without expert advice, even in cases where the expert's advice may be adversarial. The proposed method is empirically validated in real-world tasks, particularly in the design of lithium-ion batteries, demonstrating its robustness and superiority over existing methods.", "questions": "Given the potential computational overhead of the proposed method, have the authors compared the computational time of your proposed method with other baselines?\n\nMoreover, particularly in high-dimensional spaces, have the authors considered any strategies for reducing computational complexity? For example, could dimensionality reduction techniques be integrated into the process?", "limitations": "na", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Originality: The paper presents an innovative approach by integrating human expert knowledge into BO with theoretical guarantees. The introduction of the \"handover\" and \"no-harm\" guarantees is novel, addressing a significant gap in existing literature where human-in-the-loop methods often lack formal theoretical guarantees.\n\nQuality: The authors provide clear theoretical formulations with proofs, that establish the validity of their approach. The empirical evaluations uses both synthetic and real-world datasets, and the results are compelling, showing the practical effectiveness of the method.\n\nClarity: The paper is well-organized and clearly written. The concepts, while complex, are explained in a step-by-step manner, making the paper accessible to readers. Given this is 41-page long paper, I especially like the table of contents in Appendix. \n\nSignificance: The approach could have broad implications for how expert knowledge is integrated into automated systems, making it a valuable addition to the literature.", "weaknesses": "Limited Generalizability: While the paper provides strong theoretical guarantees, the method's effectiveness may be limited to specific types of expert interactions (binary labeling). The approach might not generalize well to scenarios where expert input is more complex or continuous.\n\nComputational Complexity: The algorithm's dependence on GP and the primal-dual method could be computationally expensive, particularly in high-dimensional settings. This could limit the practicality of the approach in real-time applications or for problems with very large datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "OcYIjNc6Jm", "reviewer": "Reviewer_Ct1E", "summary": "This paper introduces a robust Bayesian optimization algorithm that incorporates human expert knowledge to accelerate the optimization process. The key theoretic contributions include a handover guarantee, ensuring the number of expert labels required decreases over time, and a no-harm guarantee, ensuring the optimization performance is not adversely affected even with unreliable expert advice. The algorithm combines Gaussian processes for objective functions and likelihood ratio models for expert beliefs, using a primal-dual approach to mix these two surrogates, thereby enhancing the efficiency of handling expert inputs. Empirical validation on synthetic benchmarks and real-world tasks, such as lithium-ion battery design, shows the method's effectiveness and robustness, outperforming existing baselines.", "questions": "Some suggestions to improve clarity have been mentioned in the **Weakness** section. I have two additional questions:\n\n1. I would like to see a justification for the linear scaling in the synthetic agent response model in Example 2.2, as it is used throughout the synthetic experiment as well as the robustness and sensitivity analysis. Is this scaling general enough to represent typical synthetic agent responses?\n\n2. I am interested in learning about the actual human labor efforts involved in the battery design task to understand the applicability of this work in real-world applications. The experiment results include the cumulative queries, but I would like to know more about the actual time spent on labeling. This can be a rough estimate, like a comparison in magnitude to the computational time and the function evaluation time.", "limitations": "The authors do have adequately addressed the limitations, even though I still have concerns in how costly the human labors in a real-world task (e.g., the battery design) can be.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper introduces a first-of-its-kind principled approach to Bayesian optimization by integrating human expert knowledge. The authors provide solid theoretical guarantees, including the handover and no-harm guarantees, which are well-justified and supported by detailed proofs.\n\nThe empirical experiments are comprehensive, covering five synthetic benchmarks and one real-world application, as well as robustness and sensitivity analysis. The experiment results show the algorithm's improvement over existing baselines.\n\nThe paper is overall well-written and structured, with a clear presentation of its contributions, theoretical foundations, and experimental results. The inclusion of visual aids and detailed plots enhances understanding and reproducibility, though there is a lack of explanations on notations and terminologies, which I will elaborate in the **Weakness** section.\n\nThis paper's contribution has the potential to advance the state of the art in Bayesian optimization and inspire further research in human-AI collaboration.", "weaknesses": "The main weakness of this paper lies in its clarity.\n\n1. The term \"cost function\" is misleading as it typically refers to black-box objective function evaluation costs in cost-aware Bayesian optimization, whereas this paper uses it to denote expert belief.\n\n2. Despite having a notation paragraph at the end of Section 2 and an algorithm paragraph at the beginning of Section 4.2, it is still difficult to locate the definitions of the notations used in Algorithm 1. For instance, the notations in Line 1 are not clearly defined until Line 9, where it becomes apparent they represent the prior confidence set. Including a line of parameters at the beginning of the pseudocode would be helpful.\n\n3. This issue extends to Figure 2. The caption could be more detailed to clarify the notations used in the figure, as it currently requires effort to understand.\n\n4. Both LCB and UCB are defined, but only LCB is used in their algorithm, and the authors claim it is still based on the GP-UCB algorithm. It needs more clarification on when to use LCB and when to use UCB.\n\n5. The term \"LL values\" appears multiple times but lacks a clear explanation. It seems to be associated with the LL maximizer mentioned somewhere in the paper but needs more clarification.\n\n6. The x-axis of Figure 3 is confusing. For example, the number of function evaluations seems to go up to 50, yet the number \"50\" does not appear.\n\n7. There is a lack of ablation study on the kernel choice and the number of dimensions. While I understand these may not be the primary focus of this paper, I am interested in the authors' thoughts on these aspects.\n\n8. Additionally, it takes time to figure out how the synthetic agent response (including feedback accuracy and expert belief distribution) is modeled in the synthetic experiments. The authors should refer to Example 2.2 again or directly restate the expression in the **synthetic dataset** paragraph.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for the positive feedback and the helpful comments. The following are our detailed responses.\n\n# Point-by-Point Responses to Weaknesses \n1. To avoid any confusion, we have changed it to 'expert function' or just 'function' instead.  \n2. We have added the notations with explanations and some initializations at the beginning of the Algorithm 1.\n3. We have added more details to clarify the notations used in Figure 2.   \n4. In the GP-UCB paper [75], maximization formulation is adopted. However, we consider the minimization formulation (as common for many applications in engineering) in our paper. So GP-UCB is modified to LCB in our paper. Vanilla LCB in this paper is essentially the same as GP-UCB in [75]. To clarify this issue, we have inserted a footnote in line 321.\n5. We have added an explanation clearly stating that 'LL value' refers to the log likelihood value over the historical data up to step $t$ with a function $\\hat{g}\\in\\mathcal{B}_g$.\n6. We have updated Figure 3 to avoid any confusion. \n7. Kernel choice and scalability to high dimension are common challenges for BO. Theoretically, Table 1 shows the impact of the kernel function and dimension on both the cumulative regret and cumulative queries. This is similar to the GP-UCB algorithm. In practice, existing generic techniques, such as decomposed kernels, can be applied in our algorithm to choose kernel functions and achieve scalability in high-dimensional spaces. We have added similar discussions around Table 1 in the manuscript.  \n8. We have added some discussions to state it again at line 271.\n\n# Point-by-Point Responses to Questions\n> I would like to see a justification for the linear scaling in the synthetic agent response model in Example 2.2, as it is used throughout the synthetic experiment as well as the robustness and sensitivity analysis. Is this scaling general enough to represent typical synthetic agent responses? \n\nIt is challenging to encompass all possible agent responses in the experiments. In the synthetic response model $p_{x\\succ_g0} = S(a \\rho(f(x)))$, our linear scaling function, combined with the accuracy coefficient, can cover the three important and common scenarios, \n* **Helpful human expert**: $a$ is significantly positive (e.g., $a=1$). In this case, $x^\\star$ is mapped to a high probability of being accepted, and the point with large $f(x)$ value is mapped to a low probability of being accepted. \n* **Purely random response**: $a=0$, every point is accepted with probability of $0.5$.\n* **Adversarial human expert**: $a$ is significantly negative (e.g., $a=-1$). In this case, $x^\\star$ is mapped to a low probability of being accepted. \n\nWhile Theorem 4.1 assures the versatility of our algorithm for any response function, we also wanted to empirically confirm its reliability with another possible response function. Thus, we added experiments in Fig. R2(a) in the global response, selecting [37] as a representative example from the literature. [37] adopted human belief as a multivariate normal distribution (MVN), where the mean and variance of the MVN correspond to the most promising location of the global optimum and the confidence in the belief, respectively. We have tested the strong, weak, and wrong models by following [37]. Fig. R2(a) show that the outcomes do not differ significantly from those in Example 2.2. \nIn particular, the “wrong” case is centred at the point farthest from $x^*$ within the domain, representing another adversarial response and demonstrating the efficacy of the no-harm guarantee.\n\n> I am interested in learning about the actual human labor efforts involved in the battery design task to understand the applicability of this work in real-world applications. The experiment results include the cumulative queries, but I would like to know more about the actual time spent on labeling. This can be a rough estimate, like a comparison in magnitude to the computational time and the function evaluation time.\n\nHuman labelling costs vary among experts but typically range from a few seconds to several minutes. As shown in Figures 6 and 7, the BO overhead ranges from a few seconds to tens of seconds, which is comparable. However, we assume that the function evaluation time is very expensive. The battery design tasks in Figure 5 are demonstration purposes using open datasets. In the real-world development, creating a prototype battery requires at least three days for manufacturing and one week for testing. This experiment costs at least \\$10,000 when outsourced, making the labelling cost negligible by comparison. This high expense is why experts are reluctant to rely solely on opaque AI algorithms for decision-making. They prefer to be involved in the design process for trustworthiness, despite being uncertain of their effectiveness of aiding BO in advance. \nFrom this human perspective, our approach can be seen as augmenting human ability through BO with a provable convergence guarantee. Thus, human involvement is more of a prerequisite in our setting rather than an additional cost. In fact, the handover guarantee can reduce their effort compared to a manual search (see Figure 7, where the cumulative queries of ours are smaller than those of a manual search).\n\nHowever, even though the above cases are our main target and perspective, one may wish to extend to other scenarios. For example, users might want to be involved in scenarios with cheap feedback, such as hyperparameter tuning of lightweight ML models. In this setting, we agree that the labelling cost would matter.\n\nWe have added this limitation and similar discussion to the manuscript.\n\n# Response to Limitations\n>  The authors do have adequately addressed the limitations, even though I still have concerns in how costly the human labors in a real-world task (e.g., the battery design) can be.\n\nPlease see our response to the last comment.\n\nThank you again for valuable feedback. We hope our rebuttal clarifies our points.", "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": "KKhqV9Xpon", "reviewer": "Reviewer_dzPz", "summary": "The paper proposes to use human rejection as a feedback in human-AI collaborated Bayesian optimization. The proposed solution uses constrained optimization to explore the regions humans think might be beneficial. The authors offered guarantee that 1) the proposed algorithm has the same regret as the vanilla BO algorithm and 2) the number of queries needed from humans converges to 0. Empirically, the authors show that the proposed method outperforms existing method and has a human study with real human experts.", "questions": "See weakness.", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The human-AI collaboration in bayesian optimization is important. The authors show an interesting human-AI collaboration system that humans provide acceptance-rejection feedback to help the BO process. The empirical results are good.", "weaknesses": "Insufficient theoretical results - what is human's value in the proposed Human-AI system: The authors showed two theoretical results, 1) the human-AI system is no worse than the AI system and 2) humans will be out of the system in the long run. But the key theoretical results are missing. The most important results should be how humans improve the proposed system compared to AI-only systems. Moreover, the authors should show how humans' expertise affect the convergence rate. In my opinion, the current theoretical results are not meaningful. This is the most important point the authors should address. \n\nFeedback form: the authors use 'acceptance-rejection' feedback, which seems limited. Intuitively, humans can directly provide $x_t^c$ in this problem setting. Why the authors choose this form of feedback instead of a richer form of feedback. \n\nHuman choice model: humans are only required to choose from $x_t^c$ and reject in the paper. The vanilla LCB is never presented to the humans. Can authors explain why it is the case, maybe humans can also reject bad vanilla LCB suggestions. \n\nChanging human behaviors: in practice, humans' behavior is often changing, especially in such interactive systems that humans observe new feedback at each time step. How can the proposed method can adapted into such settings?\n\nHuman study - No IRB approval: my understanding is for any human study. Investigators are required to obtain either an exempt determination or IRB approval before these activities can be initiated. While authors seem don't have either based on the checklist.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the valuable feedback. The following are our responses.\n# Insufficient theoretical results\nWe agree that acceleration through expert knowledge is central to our contribution. However, we do not believe that \"theoretical\" acceleration is essential for the completeness of our work. Our main contribution is the **algorithm**, which outperforms 6 baselines on 5 synthetic and 4 real-world tasks, which should be sufficient for empirical validation. While theoretical validation is nice to have for showing its effectiveness, it is not the only form of validation. Notably, our setting has the **gap** between what theory can offer and what users want.\n\nIn the expert-guided BO, there is a near consensus that the expert is a warm starter [34, 36, 37, 42, 44]. It is natural that experts can guide AI to more promising regions in the beginning. However, in the later stages, with a larger dataset, GP is more accurate in finding the global optimum, as confirmed in various literature [34, 36, 37, 42, 44]. This means that convergence in the later stage will not be very different from an AI-only system. Still, early-stage acceleration is of paramount importance for high-stakes optimization, such as battery design, where engineers strive to launch better products than competitors as early as possible. \n\nHowever, theoretical analysis focuses on worst-case asymptotic convergence rates. This means that the warm-starter assumption does not significantly affect the theoretical convergence rate. This is why theoretical acceleration is not crucial in our setting. We believe the value of \"conservative\" theory lies more in providing safeguard guarantees, assuring we can safely and effectively combine experts and AI suggestions for early-stage acceleration.\nIndeed, ours is the *first* algorithm to enjoy a full guarantee (see Table 2).\n\nWe are constrained by the space to explain theoretical details, but please read the P5 for sMdb. In short, order-wise improvement is unattainable under current mild assumption and assumption can become unrealistic if we really want it. The baselines [2, 6] have theoretical acceleration under unrealistic assumptions. Figure 5 shows they do not outperform ours, highlighting such theoretical acceleration do not always translate to empirical success.\n\n# Feedback form\nWe choose `acceptance-rejection' feedback for two reasons:\n\n**Pinpoint form feedback does not necessarily perform better empirically.**\nWe added an experiment (Fig. R3(a) in the global response) comparing pinpoint and primal-dual approaches. The results show that our primal-dual approach performs better across varying feedback accuracy, at least within our framework. Existing works [6, 44] employ pinpoint feedback in different frameworks. However, Figure 5 clearly shows that neither [6] nor [44] performs better than ours. This fact is confirmed by several studies from different institutions, such as [2] and [63], both of which demonstrate that [6] performs even worse than vanilla LCB. Similarly, [33] shows that this type of feedback only works better if the expert's manual sampling is consistently superior to vanilla LCB. Such cases are rare in our examples (e.g., Rosenbrock), and [2, 63] confirm the same conclusion.\n\n**Acceptance-Rejection feedback is efficient and accurate in extracting human knowledge for BO.**\nFrom the BO perspective, there are two reasons why we adopt the acceptance-rejection feedback. (1) Functional estimation approaches like ours can exploit more information. They can query multiple locations to experts to elicit their belief function, and reflects on the next query. (2) The precision of human pinpointing can be low. Humans excels at qualitative comparison than absolute quantities in feedback [41]. This is a widely accepted concept in human preference learning and economics, which has a long history of moving from cardinal utility (absolute) to ordinal utility (comparison) for robust estimation. Furthermore, acceptance-rejection feedback is easier for human to give. \n\n# Human choice model\nWe do not for two reasons.\n\n**The value of presenting the vanilla LCB suggestion is low.**\nRejecting LCB suggestions may be helpful to elicit human knowledge, but using $g_t$ distributional information to select the most informative points would be more query-efficient. More specifically, as the logic in line 5 of Algorithm 1 indicates, the vanilla LCB solution $x_t^u$ is used when either human augmented solution $x_t^c$ can not be the optimal solution for sure or the uncertainty of $x_t^c$ is already very low as compared to $x_t^u$. In both cases, we would sample the vanilla LCB solution $x_t^u$ regardless of the acceptance/reject feedback from human. So it is not necessary to query human for this point.\n\n**Not presenting the vanilla LCB suggestion reduces the human labeling effort significantly.**\nIf we presented the vanilla LCB solution to the human, we would have suffered from a linear growth of cumulative queries to the human labeler. Instead, by not presenting it, we show a small sublinear bound on the cumulative human label queries. This is meaningful since it reduces the human labeling effort in a provable order-wise way.\n\n# Changing human behaviors\nThis is the same with P4 of sMdb, please read them as our response. In short, yes, our algorithm can do and we ran experiments in Fig. 1 in global response.\n\n# No IRB approval\nAlthough we do not have IRB approval, our institution has reviewed and approved our study as low-risk since all experiments involve running software on open-source datasets. According to the NeurIPS 2024 ethics guidelines, adhering to existing protocols at authors' institution is required, and IRB approval is just one form of such protocols. Due to anonymity, we cannot provide evidence now, but we promise to attach it if accepted.\n\nThank you again for your valuable comments and suggestions. We have incorporated all of our discussions into the relevant sections of the manuscript.", "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": "pF3kJ5BQa8", "reviewer": "Reviewer_sMdb", "summary": "The paper introduces a Bayesian optimization algorithm designed to include human expert knowledge.  Expert input comes in the form of simple accept/reject feedback (as in \"this experiment is / is not worth doing\").  To incorporate feedback two models are maintained, (1) a standard GP model of the objective, and (2) a likelihood ratio model of human expert feedback.  Theoretical performance guarantees are proven, precisely a so-called handover guarantee demonstrating sub-linearity in the number of expert queries, and a convergence-rate guarantee to ensure sub-linear performance.  Experiments include a practical example on li-ion battery design.", "questions": "- are you assuming that the expert labeling model is time-invariant here?  I am curious what might happen as the expert learns over the course of the optimization, particularly if the BO turns up new and unexpected results (which, in my experience, is not uncommon).\n- if I'm reading the regret-bound (6a) correctly this simply shows that, in the worst-case, the algorithm converges at the usual rate for BO.  In the optimistic case - that is, assuming an expert providing non-adversarial feedback - do you have any thoughts on how/if this might be used to improve the regret bound?  This would be non-trivial I suspect, but it would be good to better understand/bound the potential acceleration from human expert feedback.", "limitations": "n/a", "rating": 9, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "I am very impressed by this paper!  The problem it considers is important but under-studied.  The algorithm and its justification is novel and well-motivated, and the experimental results are impressive.\n\n- The core idea of using expert feedback in simple accept/reject form is certainly attractive compared to the more usual ranking approach and avoids many of the problems therein.\n- While I'm not entirely familiar with the model used for human feedback (likelihood ratio model) it seems to be a sensible choice.\n- The paper is reasonably easy to follow and has a clear flow (for the most part).\n- The theoretical analysis is interesting - I particularly appreciate the handover guarantee, which (as far as I know) is unique in the literature.\n- The results clearly demonstrate the potential of the algorithm.", "weaknesses": "Small points:\n- does assumption 2.3 really need a justification?  This is a standard assumption.\n- line 114: is it usual to call $r$ a regularization term?  Would it be easier to simply use $\\xi$ as per assumption 2.6?\n- equation (4): use of $\\xi$ here is a potential notational clash with assumption 3.7.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your positive feedback and the helpful suggestions. The following are our detailed responses. \n\n## P1\n> does assumption 2.3 really need a justification? This is a standard assumption. \n\nWe do some justification because it is also an important assumption and we want to differentiate from gradient-based optimization (where compact set assumption is often not needed).\n\n## P2\n> line 114: is it usual to call $r$ \n a regularization term? Would it be easier to simply use $\\xi$ \n as per assumption 2.6?\n\nYes, this is common in theoretical studies. We are aware that $r$ is typically referred to as the Gaussian noise variance. Nevertheless, we follow the definition in [16], because even in the noiseless case, we need the regularisation term $rI$ for the invertibility of $K_{\\mathcal{Q}_t^f}$ to avoid numerical instability.\n\n## P3\n> equation (4): use of $\\xi$ \n here is a potential notational clash with assumption 3.7. \n\nWe have replaced $\\xi$ with $\\zeta$ to represent the dual update step size in the manuscript. \n\n## P4\n>  are you assuming that the expert labeling model is time-invariant here? I am curious what might happen as the expert learns over the course of the optimization, particularly if the BO turns up new and unexpected results (which, in my experience, is not uncommon).\n\nHere are our thoughts on extending our results to time-varying human behavior model:\n\n- **Simple extension, yet not promising performance gain.**\nWe have added the experiments (Fig. R1 in global response).\nThe most naïve approach for non-stationary model is windowing, i.e., forgetting the previous queried dataset. This can be very easy to apply to our setting, as it simply removes the old data from $g_t$ using the predefined iteration window. This increases the predictive uncertainty of $g_t$, and therefore requires more queries to the human per iteration, yet our framework still works.\nAs shown in the Fig. R1, this non-stationary approach can offer a slight performance gain when the human learning rate is very quick ($\\alpha_{lr} = 1$). However, we do not know if this is always the case, and it also requires more labeling effort from humans.\n\n- **More sophisticated extension.** Another more sophisticated approach is modelling the dynamics of behavioural change. A potential idea is modelling the\nhuman behaviour change as an implicit online learning process of the latent function $g$. That is, \n$g_{t+1}=F(g_t, x_t, y_t),$\nwhere $g_t$ is the human latent function at step $t$. The forward dynamics $F$ captures the update of human latent function $g$ when observing the new data point $(x_t, y_t)$. One potential $F$ is gradient ascent of log-likelihood as shown in $g_{t+1}=g_t+\\lambda\\nabla_g\\log p_g(x_t, y_t)$, where $p_g(x_t, y_t)$ is the probability of observing $y_t$ at the input $x_t$ given the black-box objective function is $g$. We can then combine this dynamic with our likelihood ratio model. Since this part requires significantly different analysis and experiments, we leave it as future work.\n\nWe have added similar discussions to the appendix of the manuscript.\n\n## P5\n> if I'm reading the regret-bound (6a) correctly this simply shows that, in the worst-case, the algorithm converges at the usual rate for BO. In the optimistic case - that is, assuming an expert providing non-adversarial feedback - do you have any thoughts on how/if this might be used to improve the regret bound? This would be non-trivial I suspect, but it would be good to better understand/bound the potential acceleration from human expert feedback. \n\nThanks for the suggestion. Under our current mild assumption on the latent expert function $g$, an *order-wise* convergence improvement is not attainable. Indeed, assumption becomes unrealistic if we really want an order-wise improvement. Therefore, we only demonstrate the empirical superiority of expert-assisted BO in the experiments. \nThe following are the more specific discussions.\n- **Order-wise improvement can not be attained under current mild assumption**. $g$ may contain no information (e.g., $g=0$) or even adversarial. Even if human expertise is helpful, we can not guarantee an *order-wise* improvement either. For example, consider the following $g$,\n$g(x)= f(x^\\star)+c,  \\text{if } f(x)-f(x^\\star)\\leq c, \\text{ and } f(x)  \\text{ otherwise},$ \nwhere $c>0$ is a positive constant. In practice, such a scenario means the human expert has some rough idea in a near-optimal region but not exactly sure where the exact optimum is. This is common in practice. In this case, human expert is helpful in identifying the region with $f(x)\\leq f(x^\\star)+c$ but no longer helpful for further optimization inside the region {$x\\in\\mathcal{X}|f(x)\\leq f(x^\\star)+c$}. However, convergence rate is defined in the asymptotic sense. Hence, an order-wise improvement can not be guaranteed.\n\n- **Assumption becomes unrealistic if we really want it**. \nSome papers that show theoretical superiority [2, 6], yet the assumptions are unrealistic. For example, [6] assumed that the human knows the true kernel hyperparameters while GP is misspecified, and [2] assumed the human belief function $g$ has better and tighter confidence intervals over the entire domain. We can derive the better convergence rate of our algorithm than AI-only ones if we use [2] assumption, but this is unlikely to be true in reality. In fact, our method outperforms these method empirically (see Figure 5). This supports the superiority based on unrealistic conditions is not meaningful in practice. \n\nWe have updated the manuscript by adding some discussion near Theorem 4.1 and in the Appendix.", "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": "bI9ce3pawR", "reviewer": "Reviewer_iJqg", "summary": "This submission discusses the application of human's expertise knowledge into Bayesian Optimization algorithm. A principled approach, COBOL, was proposed which provides two novel guarantees: handover guarantee that ensures queries for expert's label diminishes to zero with iterations; and no-harm guarantee that preserves comparable convergence rate to vanilla BO using a data-driven trust level adjustment. Numerical results show the robustness of COBOL and real world application proves COBOL outperforms state-of-the-art methods.", "questions": "1. The estimation on the hyperparameter $B_g$ in lines 182-189 seems heuristic to me. Is there any reasons to initialize the norm bound at a small value and double it during the optimization?\n\n2. Could the authors discuss briefly the possibility of applying such human collaborated BO formulation to different acquisition functions (e.g. Expected Improvement)?\n\n3.  In the plotting of real world Li+ methyl-acetate experiment in Figure 5, how can the curve for [44] KDD 2023 go up after 45th evaluation? Best observed value should be non-increasing with iterations, could the authors explain how they achieve these results? \n\n4. Could the authors be more specific when referring to proofs and sections? e.g. not using phrases like \"will be seen\" (line 148), \"detailed later\" (line 171), \"later experimental section\" (line 176)", "limitations": "The authors mention limited capability of the algorithm on high-dimensional problems due to its GP-UCB based nature. The experiments are only conducted up to 4D. See also above for limitations.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.  This paper presents comprehensive theoretical proofs to support the proposed method. The math seems strong to me. Notations and assumptions are clearly stated at the beginning. Follow-up theorems are easy to follow.\n\n2. I quite like the trust-level adjustment method. The authors present novelty of their work by differentiating themselves from those who requires hand-tuned trust-level functions. I like the idea of treating expert belief as the regularization of the acquisition function and the corresponding primal-dual optimization problem formulation seems reasonable to me.", "weaknesses": "My only concern regards the no-harm guarantee. From observation in Figure 3, the no-harm guarantee indeed presents comparable convergence rate as vanilla LCB, but never arrives at the same optimum as vanilla LCB within the given iterations under adversarial cases. By looking at line 5 in Algorithm 1, it seems adversarial expert's label could harm the quality of query point continuously, especially when the constant value of $\\eta$ is set to be large so the confidence condition can be easily satisfied. To my understanding, $\\lambda_0$ improves the quality of solution from expert-augmented LCB by reducing its value when the data tells expert's feedback is not trustful. Resilience to adversarial label is achieved through updating the primal-dual weight $\\lambda_0$, but I feel like $\\eta$ should also vary with iterations since it is a critical hyperparameter in the algorithm that actually controls whether to accept expert-augmented query point $x_t^c$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable feedback and helpful suggestions. The following are our detailed responses.\n\n# Response to the concern regarding the no-harm guarantee and $\\eta$. \n\nThe suggestion for an adaptive trust weight $\\eta$ is interesting and could offer better resilience to adversity. Given the limited time, we leave the design of a principled scheme to adapt the weight $\\eta$ as future work. However, we want to emphasize that, even without such a scheme, our no-harm guarantee holds, both theoretically and empirically. We have prepared several experiments for further clarification.\n- **Supporting experimental results.**\nTo clarify the robustness in the concerned cases, we added experiments (Fig. R2(b) in the global response) with longer iterations ($T = 200$) to confirm the convergence on par with vanilla LCB. We confirmed the no-harm guarantee for both cases of adversarial feedback accuracy and large trust weight $\\eta=100$ on adversarial feedback. For the large trust weight $\\eta=100$, we observed saturation behavior, where larger $\\eta$ does not significantly change the convergence rate with given increase in $\\eta$.\n- **Adaptation through the posterior standard deviation.**\nAlthough $\\eta$ is set to be a constant in our current design of the algorithm, there is still adaptation on trusting human or the vanilla BO algorithm through the time-varying posterior standard deviation. Intuitively, if originally the expert-augmented solution $x_t^c$ is trusted more, more samples are allocated to human-preferred region and $\\sigma_t(x_t^c)$ drops quickly. Intuitively, if we keep sampling $x_t^c$ and $x_t^u\\neq x_t^c$, $\\sigma_t(x_t^u)$ would finally be larger than $\\eta\\sigma_t(x_t^c)$ and we switch to sampling $x_t^u$.      \n- **Choice of $\\eta$ does not need to be very large in practice.**\nIntuitively, $\\eta$ captures the belief on the expertise level of the human. The more trust we have on the expertise of the human, the larger $\\eta$ we can choose. But larger $\\eta$ increases the risk of higher regret due to potential over-trust in adversarial human labeler. In our experience, $\\eta$ does not need to be very large. Indeed, $\\eta=3$ already achieves superior performance in our experiment. And in our updated Figure R2(b), the experimental results become insensitive to $\\eta$ when $\\eta>10$. \n\nWe have added more discussions in the manuscript around Algorithm 1 to discuss the choice of $\\eta$.\n\n# Responses to other comments.\n> The estimation on the hyperparameter $B_g$ in lines 182-189 seems heuristic to me. Is there any reasons to initialize the norm bound at a small value and double it during the optimization? \n\nYes, this is a rough estimation. Our algorithm requires knowledge of a valid upper bound on the norm of $g$, which is unknown a priori. Thus, we empirically check the LL value as a criterion. If we find the current bound $B_g$ to be invalid (that is, smaller than the norm of $g$), we double it. We do not use a too big (big enough so that $\\geq||g||$) initial $B_g$ to avoid over-exploration in $g$. \n\nWe would like to note that the MLE in GP also does not provide a principled method to estimate its hyperparameters (Karvonen et al.). Regret theory typically assumes the true hyperparameters are given. Thus, this is a shared challenge, and we have found heuristics that work in practice, similar to GP. Developing a more principled tuning method is our future work, and we believe a statistical test is a promising line of research.\n\n- T. Karvonen et al., Maximum Likelihood Estimation in Gaussian Process is Ill-Posed. JMLR 2023.\n\n> Could the authors discuss briefly the possibility of applying such human collaborated BO formulation to different acquisition functions (e.g. Expected Improvement)?\n\nOur algorithm can be easily extended to other acquisition functions. For expected improvement, we can indeed use similar idea to constrained expected improvement to generate $x_t^c$, \n$$x_t^c \\in \\arg\\max_{x\\in\\mathcal{X}} \\mathbb{P}(x\\text{ is accepted by human})\\mathrm{EI}(x). $$\nHowever, our convergence rate depends on the GP-UCB algorithm, so the convergence rate will change to that of the selected acquisition function. \n\n> In the plotting of real world Li+ methyl-acetate experiment in Figure 5, how can the curve for [44] KDD 2023 go up after 45th evaluation? Best observed value should be non-increasing with iterations, could the authors explain how they achieve these results?\n\nUpon checking the raw data, we found that a few results failed at the point of ascent. We were averaging after performing individual minimum operations. We have reviewed all results and rerun the experiments. Please review the new Figure for the confirmation (Fig. R3(b) in global response).\n\n> Could the authors be more specific when referring to proofs and sections? e.g. not using phrases like \"will be seen\" (line 148), \"detailed later\" (line 171), \"later experimental section\" (line 176)\n\nYes, we updated the manuscript to be more specific. Both lines 148 and 171 refers to Theorem 4.1 and Appendix B, and line 176 refers to Figure 3.\n\n> The authors mention limited capability of the algorithm on high-dimensional problems due to its GP-UCB based nature. The experiments are only conducted up to 4D. See also above for limitations.\n\nScalability to high dimensions is a common challenge for BO. In practice, existing generic techniques, such as decomposed kernels, can be applied in our algorithm to choose kernel functions and achieve scalability in high-dimensional spaces. We realized one minor mistake in the explanation of the Michalewicz function in Appendix F.2.1, which was written as 2D but is actually a 5D function. Thus, our results are up to 5D. However, we are not claiming our results are scalable to high dimensions without the above-mentioned techniques.\n\nWe have added similar discussions around Table 1 in the manuscript.", "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:
# Global response and added experiments

The authors would like to thank all reviewers for their effort. As per the reviewers' requests, we have added five additional plots to the rebuttal PDF and incorporated these figures into our manuscript.

- **Fig. R1**: Nonstationary Human Accuracy (**dzPz, sMdb**)

We tested the scenario where the accuracy of human experts’ labeling improves over time. Our algorithm can extend to such scenarios, but the performance gain is limited to certain conditions. Further work is needed for the future work.

- **Fig. R2(a)**: Variants of Human Belief Model (**Ct1E**)

We tested another human belief model proposed by [37]. Our algorithm works for the strong, weak, and wrong belief models introduced in [37].

- **Fig. R2(b)**: Confirming No-Harm Guarantee (**iJqg**)

To confirm that our algorithm converges on par with vanilla LCB in adversarial scenarios, we extended the iterations to 200. Our algorithm converges to the same regret as vanilla LCB for both feedback accuracy adversity and strong trust weights on adversarial feedback.

- **Fig. R3(a)**: Pinpoint Form (**dzPz**)

We tested the efficacy of pinpoint feedback suggested by **dzPz**. The results do not show a promising performance gain over our primal-dual approach.

- **Fig. R3(b)**: Bug in Figure 5 (**iJqg**)

Thank you for pointing this out. We have rerun the experiments, and the bug has been resolved.

We would like to thank all reviewers again for their comments and suggestions, which have significantly improved the quality of our manuscript. We have already updated our manuscript to reflect our fruitful discussion. We hope this rebuttal clarifies any initial confusions and concerns. We are happy to discuss further if more clarification is needed.

Meta-review: This paper addresses a very interesting problem, has solid empirical and theoretical results, and was very well-received by the reviewers. I also commend the authors on how they engaged with the reviewers. Excellent work! But, as always, please take care to incorporate feedback on doing the final edits for camera-ready.
Decision: Accept (spotlight)