Inversion-based Latent Bayesian Optimization
Authors: Jaewon Chu
Jinyoung Park
Seunghun Lee
Hyunwoo J. Kim
Venue: NeurIPS 2024 poster (2024)
Keywords: Latent Bayesian Optimization
Bayesian Optimization
Inversion
Primary area: probabilistic_methods

Abstract: Latent Bayesian optimization (LBO) approaches have successfully adopted Bayesian optimization over a continuous latent space by employing an encoder-decoder architecture to address the challenge of optimization in a high dimensional or discrete input space. LBO learns a surrogate model to approximate the black-box objective function in the latent space. However, we observed that most LBO methods suffer from the `misalignment problem', which is induced by the reconstruction error of the encoder-decoder architecture. It hinders learning an accurate surrogate model and generating high-quality solutions. In addition, several trust region-based LBO methods select the anchor, the center of the trust region, based solely on the objective function value without considering the trust region's potential to enhance the optimization process. To address these issues, we propose $\textbf{Inv}$ersion-based Latent $\textbf{B}$ayesian $\textbf{O}$ptimization (InvBO), a plug-and-play module for LBO. InvBO consists of two components: an inversion method and a potential-aware trust region anchor selection. The inversion method searches the latent code that completely reconstructs the given target data. The potential-aware trust region anchor selection considers the potential capability of the trust region for better local optimization. Experimental results demonstrate the effectiveness of InvBO on nine real-world benchmarks, such as molecule design and arithmetic expression fitting tasks. Code is available at https://github.com/mlvlab/InvBO.

Inversion-based Latent Bayesian Optimization
Jaewon Chu; Jinyoung Park; Seunghun Lee, Hyunwoo J. Kim
Computer Science Engineering Korea University
fallonsy07, 1pmn678, 1lsshh319, hyunsoojkim//korea. ac.kr
Abstract
Latent Bayesian optimization (LBO) approaches have successfully adopted Bayesian optimization over continuous latent space employing encoder- decoder architecture to address the challenge of optimization in high dimensional discrete input space. LBO learns surrogate model approximate the black-box objective function in the latent space. However, we observed that most LBO meth- ods suffer from the 'misalignment problem', which is induced by the reconstruction error of the encoder-decoder architecture. It hinders learning an accurate surrogate model and generating high-quality solutions. In addition, several trust region-based LBO methods select the anchor, the center of the trust region, based solely on the objective function value without considering the trust region's potential enhance the optimization process. To address these issues, we propose Inversion- -base Latent Bayesian Optimization (InvBO), a -ad--aa module for LBO. InvBO consists of two components: an inversion method and potential- -ware trust region anchor selection. The inversion method searches the latent code that completely reconstructs the given target data. The potential-aware trust region anchor selection considers the potential capability of the trust region for better local optimization. Experimental results demonstrate the effectiveness of InvBO nine real-world benchmarks, such as molecule design and arithmetic expression fitting tasks. Code is available at htttt://ggthubbbco.
1 Introduction
Bayesian optimization (BO) has been used in a wide range of applications such as material sci- ence (T] chemical design [ 3 ,and hyperparameter optimization 4 [5. The main idea BO is probabilistically estimating the expensive black-box objective function using surrogate model to find the optimal solution with minimum objective function evaluation. While BO has shown its success on continuous domains, applying BO over discrete input space is challenging [ 77. To address it, Latent Bayesian Optimization (LBO) has been proposed [8H14] LBO performs BO over latent space by mapping the discrete input space into the continual latent space with generative models such as Variational Auto Encoders (VAE) 15 consisting of an encoder ds and a decoder ps- Unlike the standard BO, the surrogate model in LBO associates a latent vector with objective function value by emulating the composition of the objective function and the decoder of VAE.
In LBO, however, the reconstruction eтror of the VAE often leads to one latent vector z being associated with two different objective function values as explained in Figure [] We observe that the discrepancy between y and y' (or and x') hinders learning accurate surrogate model and generating high-quality solutions. We name this the 'misalignment problem'. Most prior works [9] [2] use the surrogate model gt which is trained with the encoder triplet (x, y), and
equal contributions
CCorrsppondigg author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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g ene
dec
Encoder
Decoder
y f
X
Z
y
Po
(a) Encoder triplet
(b) Decoder triplet
Figure I: Misalignment problem. In LBO, a latent vector can be associated with two function
values y and y' due to the reconstruction error of the VAE, i.e. X & x'. (a) In Encoder triplet (x, 2, 4). latent vector z is associated with fxx,, where is the original input the encoder, i.e. (x).
(b) In Decoder triplet (x', 3/) z is associated with y' = fxx). which is the objective function value of reconstructed imput value x' using the decoder, i.e., x' po (z). The discrepancy between and y' hinders learning the accurate surrogate model g. We name this the "misalignment problem'.
generate solutions x' via decoder pe. Since gmte fails estimate the composite function of pe and f. this approach often results in suboptimal outcomes. Some works [3 [4] handle the misalignment problem by employing the surrogate model gde: trained with the decoder triplet (x', z, 3')"). However, they request a huge amount of additional oracle calls to obtain the decoder triplet, which leads to inefficient optimization. In addition, several existing LBO methods [3] 16 adopt the trust region method to restrict the search space and have shown performance gain. Most prior works select the anchor, the center of the trust region, as the current optimal point. This objective function value-based anchor selection overlooks the potential to benefit the optimization performance of the latent vectors within the trust region.
In this work, we propose an Inversion- Latent Bayesian Optimization (InvBO), plug aagg-an--pp and-play module for VAE-based LBO methods. InvBO consists of two components: the inversion method and a potential- aware trust region anchor selection. The inversion method addresses the misalignment problem by inverting decoder po to find the latent code that yields without any additional oracle call. We theoretically analyze that our inversion method decreases the upper bound of the error between the surrogate model and the objective function within the trust region. The potential-awan trust region anchor selection method selects the anchor considering not only the observed objective function value but also the potential to enhance the optimization process of the latent vectors that the trust region contains. We provide the experimental evaluation nine different tasks, Guacamol, DRD3, and arithmetic expression fitting task to show the general effectiveness of InvBO, Specifically, plug-and-play results of InvBO over diverse prior LBO works show a large performance gain and achieved state-of of-the-art performance.
The contributions of our paper are follows:
We propose the inversion method address the misalignment problem in LBO by generating the decoder triplet without using any additional oracle calls.
We propose the potential-awand trust region anchor selection, aiming to select the centers of trust regions considering the latent vectors expected to benefit the optimization process within the trust regions.
By combining the inversion method and potential-aware trust region anchor selection, we propose Inversion-based Latent Bayesian Optimization (InvBO), a novel plug-and-play module for LBO, and achieve state-of-the-art performance the nine different tasks.
Related Works
2.1 Latent Bayesian Optimization
The goal of Latent Bayesian Optimization (LBO) 8 9, 11111111 16 19 is to learn a latent space to enable optimization over a continuous space from discrete or structured input (e.g,, graph or image). LBO consists of Variational AutoEncoder (VAE) to generate data from the latent representation and a surrogate model (e.g., Gaussian process) to map the latent representation into the objective score. Some works the LBO have designed new decoder architectures [8] 20 23 to perform the

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reconstruction better, while other works have proposed learning mechanisms 19 17 to alleviate the discrepancy between the latent space and input space. LOL-BO [13] adapts the concept of trust-region to the latent space and jointly learns VAF and a surrogate model to search the data point in the local region. CoBO [4] designs new loss to encourage the correlation between the distances in the latent space and objective function.
2.2 Inversion in Generative Models
Inversion has widely been applied to a variety of generative models such as Generative Adversarial Networks (GANs) [24 25 and Diffusion models (26-29). Inversion is the process of finding the latent code Zin of a given image to manipulate images with generative models. Formally, given an image and the well-trained generator G, the inversion can be written as:
arg min dx(G((z,xx, ZE
Zinv
where -x)::) denotes the distance metric in the image space A' and E is the latent space. To solve Eq. (T). most inversion-based works can be generally classified as two approaches: optimization- based and learning-based methods. The optimization-baser inversion [30 iteratively finds latent vector to reconstruct the target image x through the fixed generator. The learning- basee inversion [25] 34 335 trains the encoder for mapping the image x to the latent code z while fixing the decoder. In this work, we introduce the concept of inversion to find the latent vector that can generate desired sample for constructing aligned triplet and we use the optimization-br based inversion.
3 Preliminaries
Bayesian optimization. Bayesian optimization (BO) is powerful and sample-efficiet optimization algorithm that aims at searching the input x with a maximum objective value f(x), which is formulated as:
arg max f(x), XEX
where the black-box objective function f : X + y is assumed expensive to evaluate, and X' is feasible set. Since the objective function f is unknown or cost-expensive, BO methods probabilistically emulate the objective function by a surrogate model g with observed dataset fxx") 1-1 With the surrogate model g. the acquisition funetion selects the most promising point x" "+I as the next evaluation point while balancing exploration and exploita- tion. BO repeats this process until the oracle budget is exhausted.
D ((x',''\l'
Trust region- based local Bayesian optimization. Classical Bayesian optimization methods often suffer from the difficulty of the optimization in high dimensional space [36]. To address this problem, TuRBO (36) adopts trust regions to limit the search space to small regions. The anchor (center) of trust region T is selected scheduled during the optimization process. At the beginning of the optimization, the side length of all trust regions is set to Linit. When the trust region T updates the best score Tsuce times in row, the side length becomes twice until it reaches Lmma- Similarly, when it fails to update the best score Thai times in row, the side length becomes half. When L falls below Limin- the side length of the trust region is set to Linit and restart the scheduling. Recently, LOL-BO [3] adapted trust region- based local optimization to LBO, and has shown performance gain.
current optimal point, and the size of the trust region is
4 Method
In this section, we present an Inversion- based Latent Bayesian Optimization (InvBO) consisting of an inversion and a novel trust region anchor selection method for effective and efficient optimization. We first describe latent Bayesian optimization and the misalignment problem of it (Section 4.1). Then, we introduce the inversion method to address the misalignment problem without using any additional oracle budgets (Section 4.2). Lastly, we present a potential-aware trust region anchor selection for better local search space (Section
3

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y
Unevaluated
Evaluated
Evaluated
Aligned triplet
X
X
X
A'
9 Encoder
Po: Decoder
Input space
Pe
qab
qá
Po
f Objective function
Z
Z
Zinv
Inversion process
Inversion
Z
Z
Latent space
(a) Recentering (Previous)
(b) Inversion (Ours)
Figure 2: Comparison of solutions to the misalignment problem. (a) Some works 13 14 solve the misalignment problem by the recentering technique that generates the aligned triplet (x' z, y²). However, it requests additional oracle calls as y' = f(x' is unevaluated, and does not fully evaluated function value = (x). (b) The inversion method (ours) aims to find Zime that generates the evaluated data get the aligned triplet (x, Zun, y) without any additional oracle calls.
the
4.1 Misalignment in Latent Bayesian Optimization
BO has proven its effectiveness in various areas where input space X' is continuous, however, BO over the discrete domain, such as chemical design, a challenging problem. To handle this problem, VAE-based latent Bayesian optimization (LBO) has been proposed [8] 11 [4] that leverages BO over continuous space by mapping the discrete input space X' to continuous latent space Z. Variational autoencoder (VAE) is composed of encoder 9o X + representation z of the input data x and decoder Z + X to generate the data x from the latent z.
compute the latent
Given the objective function f, latent Bayesian optimization can be formulated as:
arg max (p)(
where (z) is generated data with the decoder p& and Z is a latent space. Unlike the standard BO, the surrogate model g aims to emulate the function f Pa : Z + y. To the end, the surrogate model is trained with aligned dataset D [(x' z', v') where x po(z') is generated by the decoder po 2 + X' and y² fxx) is the objective value of x' evaluated via the black box objective function f: X y. In the rest of our paper, define that the dataset is aligned when all triplets satisfy the above conditions (i.e., all triplets are the decoder triplets explained in Figure [ and the dataset is misaligned otherwise. We define the 'misalignment problem' the misaligned dataset hinders the accurate learning of the surrogate model g-
Most existing LBO works [1+44] overlook the misalignment problem [ which originates from two processes: (i) construction of initial dataset DO and (ii) update of VAE.
Best scare upolato
Oraule
Construction of initial dataset DD, Since initial dataset D is composed of pairs of input data and its corresponding objective value [(x', y² ly" [(x*)] LBO requires latent vectors {z in Ji=1 to train the surrogate model. Most works com- pute a latent vector z' as z' = (x') under the assumption that VAE completely reconstructs ev- cry data points (i.e., pe (x)() = x'), which difficult to be satisfied in every case. This results in the data misalignment (x* * (zz))) during the construction of initial dataset D
Anguisition
Recentering
Anguistiio
Recentering
Figure 3: (Left) The number of oracle calls to evaluate the queries selected by the acquisition function (blue) and during the recentering (Red). (Right) The number of objective function evalua- tion that updates the best score.
Update of VAF. In LBO works, updating VAE during the optimization plays crucial role in adapting well to newly generated samples. However, due to the update VAE, the previously generated triplet (x, v, y) cannot ensure the alignment since was computed by the VAE before the update. Previous L.OO works 13 14] solve the misalignment problem originating from the VAE update with a recentering technique that requests additional oracle calls to generate the aligned dataset

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D D= ((x", :o f(x))) where x" Pa (x)((x))) as shown in Figure 2 (Left). But, it has limitation to consuming a huge amount of additional oracle calls while they do not update the best score (Figure 3) Note that prior works do not explicitly mention the additional oracle calls during the recentering technique, but can be verified by the official GitHub code. Further details about oracle consumption of the recentering are provided in the supplement Section
4.2 Inversion-based Latent Bayesian Optimization
Our primary goal is training the surrogate model g to correctly emulate the composite function f Z + Y/ via constructing an aligned dataset without consuming additional oracle calls. To the end, we propose an inversion method that inverts the target discrete data x into the latent vector that satisfies x = pe (z) for dataset alignment as shown in Figure 2 (Right), With a pre-trained frozen decoder po. the latent vector can be optimized by:
= arg min dx(x, dx(x,po(z)),
(4)
2EZ
where x is a target data and dx is a distance function in the input space X. We use the normalized Levenshtein distance (37) as distance function, dx. which can be applied to any string- form data. Our inversion method, however, flexible and can utilize any task-specific distance functions, such as Tanimoto similarity (38) for molecule design tasks. To solve the Eq. 4 we iteratively update latent vector z to find Ziav that reconstructs the target data x. We provide the overall pseudocode of the inversion method in Algorithm
The initialization strategy of latent vector z plays key role in the optimization-based in- version process. We set the initialization point of latent vector an output of a pre-trained encoder (x( given target discrete data x as in line 1. We iteratively update the latent vector with the cross-entropy loss used in VAE training in line until it reaches the maximum number
Algorithm 1 Inversion
Input: Encoder ge. decoder pe, target data x, max iteration T. distance function dx, learning rate 17. reconstruction loss L
I: Initialize (0) + 4x(x)
2: for = 0, .... 1 do
3: ( t z 1/ si L (po(z(")),,
of iterations T. Before the iteration budget is exhausted, finish the inversion process when
4: if (Edx(x,Poll (z(z())1)) < e then > Eq.
5:
return 2 (t+1) 6: end
the distance between the generated data po (z) and target data is less than finding the latent vector Zun that satisfies Eq. (4). which is denoted in line 4. The inversion method generates the aligned dataset during the construction of the initial dataset D and the update of VAE handle the misalignment problem.
our goal is
7: end for
8: return (2))
We theoretically show that optimizing the latent vector z to satisfy (x (z)) zo with inversion plays a crucial role in minimizing the upper bound of the error between the posterior mean of the surrogate model and the objective function value within the trust region centered at z.
Proposition 1. Let f be a black-box objective function and m be posterior mean of Gaussian process, pe be decoder of the variational autoencoder. c be an arbitrarily small constant, dx and d2 be the distance function on input X and latent Z spaces, respectively. The distance function dx is bounded between 0 and 1, inclusive. We assume that J. m and the composite function of I and po are Ia, Tiz, and Lipshiis continuous functions, respectively. Suppose the following assumptions are satisfied:
[f(x - m(z)l Sc (x (o(x)) S.
(S)
Then the difference between the posterior mean of the arbitrary point in the trust region centered at with trust radius ē and the black box objective value is upper bounded as:
(f(p)zz)) - m(z') c 6. (L2 Lz),
(6)
where dz 2z) S
The proof is available Section A
We assume that the black box function f. the posterior mean of Gaussian process m, and the objective function f o pe are Lipschitz continuous functions, which is
5

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common assumption in Bayesian optimization 39 40] or global optimization 41 Proposition | implies that the upper bound of the Gaussian process prediction error within the trust region can be minimized by reducing the distance between x and po (z), denoted as y. Our inversion method reduces y by generating an aligned dataset without additional oracle calls. In Eq. (6). the constant e represents the accuracy of the surrogate model, which can be improved during training. Some LBO works [12 [3] introduce regularization terms to learn a smooth latent space, implicitly reducing L3. the Lipschitz constant of the composite function f pa. CoBO [4] further proposes regularization losses that explicitly reduce L3 with theoretical grounding. Since the surrogate model m emulates the composite function f pe, its Lipschitz constant L2 can also be reduced along with Lg.
4.3 Potential-aware Trust Region Anchor Selection
Here, we propose a potential-aware trust region anchor selection to consider both the objective function value y and the potential ability of the trust region to henefit the optimization. Previous trust region-based BO works [13 36 42 select the anchor based the corresponding objective function value only. However, this objective score-based anchor selection does not consider that the trust region contains latent points expected enhance the optimization performance.
We design a potential score to measure the potential ability to enhance the optimization process of the trust region. To compute it, we use the acquisition function value, which is generally employed to measure the potential ability to improve the optimization process of a given data point. Specifically, we employ Thompson Sampling, well-established acquisition function used in previous trust region- based methods 13 14, 36 Formally, the potential score of each trust region Ti is computed as follows:
Ppot max f(z) where f ~ G' (f(z), k22,z2)),
where is an index of the candidate anchor point, Zind is the candidate set sampled from the trust region Ti and f is a sampled function from the surrogate model (e.g., GP) posterior.
As the scale of objective function values Y= (v', .2² 22,..._.... ... y*) and that of the potential ability of each trust region A = { Opat:... .... p } is changed dynamically during the optimization process,
calculate scaled potential score apoi by adjusting the scale of of a according to the Y:
Opot Amin
scaled Amax Amnn
x (Ymax Ymin),
(8)
where Amax = max; [afou] and Amia = min, [appa]] denote the maximum and minimum value of A, respectively. Ymax max; [y*] and min = min; [v']] indicate the maximum and minimum value of Y, respectively. Based the scaled potential score d'cated the final score s' of each anchor is calculated as:
y² + alaleed
Our final score takes into account the objective function value of the anchor (observed value) and the potential score (model's 's prediction) for the better local search space. Finally, we select the anchors with the highest final score s². We summarize our potential-aware trust region anchor selection schema in Algorithm of Section N
Experiments
5.1 Tasks
We measure the performance of the proposed method named InvBO on nine different tasks with three Bayesian optimization benchmarks: Guacamol [4). DRD3, and arithmetic expression fitting tasks
44 Guacamol and the DRD3 benchmark tasks aim find molecules with the most necessary properties. For Guacamol benchmarks, we use seven challenging tasks, Median molecules 2 (med2), Zaleplon MPO (zale), Perindopril MPO (pdop). Amlodipine MPO (adip), Osimertinib MPO (osmb), Ranolazine MPO (rano), and Valsartan SMARTS (valt). To show the effectiveness of our InvBO various settings, we also conducted experiments in a large budget setting, which is used in previous works [13 14] The goal of the arithmetic expression fitting task to generate single-variable expressions that minimize the distance from a target expression (e.g., 1/3 + x + sin (x æ)). More details of each benchmark are provided in Section K

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Table 1: Optimization results of applying InvBO to several trust region-based LBOs Guacamol
benchmark tasks. A higher score is a better one.
Task
Median molecules 2 (med2)
Valsartan SMARTS (valt)
Num Oracle
100
300
500
100
300
500
TuRBO-L
0.186=#0.000
0.186=0.000
0.186:10.000
0.000=100000
0.000=0.000
0.000=100000
TuRBO-L InvBO
0.186.10.000
0.002
0.202.10.001
0.000.10.000
0.024_ 0.017
0.212.10.092
LOL-BO
0.186:10.000
0.186=0.000
0.190:10.001
0.000:10.000
0.000=0.000
0.000:10.000
LOL-BO IO
0.189:10.002
0.204-0.005
0.227:1.0000
0.000
0.007-0.000
0.171:10.039
CoBO
08:::0000000 0.187:0.001
0. 0.0002
0.191++0.000
0.000++000000
0.000=0.000
0.000++00000
CoBO InvBO
0.203=0.004
0.004
0.000:10.000
0.042=0.013
0.348:10107
Zaleplon MPO (cale)
Osimertinib MPO (osmb)
Median molecules (med22
Valsartan SMAKTS (valt)
2:23
0.4
spoore
Best score
2.23
Best
Best
3.30
400
206
300
400
206
660
630
300
20
Number of Oracle
Number of Oracle
Number Oracle
Number Oracle
InvBO
CodO
C.BO
PG-LBO
* LOL-BO
W:LBO
TuRBO-L
LS.BO
Grapb-GIA
Figure 4: Optimization results on Guacamol benchmark tasks, The lines and ranges indicate the average and standard error of ten runs under the same settings. A higher score is better score.
5.2 Baselines
We compare six latent Bayesian Optimization methods: LS-BO, TuRBO 36
W-LBO IT. LOL-
widely used genetic algorithm for graph structure data. LOL-BO and CoBO employ the decoder triplet generated by the recentering technique, and the other baselines employ the encoder triplet during the optimization process. In the case of LOL-BO and CoBO, we substitute the recentering technique with the inversion
BO [13], CoBO [14,, and PG-LBB [17]. We also compare with GB-GA 45
method. We provide the details of each baseline in Section
5.3 Implementation Details
For the arithmetic expression fitting task, we follow other works 8 11 14 to employ Grammar- VAE model
(8. For the de novo molecule design tasks such as Guacamol benchmark and DRD3 tasks, we use SELFIES VAE [13 following recent works [13 [4]. In all tasks, we adopt sparse variational GP [46] with deep kernel 47 our surrogate model. In the inversion method, the learning rate is set to 0.1 in all experiments. The further implementation details are provided in Section M
5.4 Experimental Results
We apply our InvBO to several trust region-basee LBOs, TuRBO-L, LOL-BO 13 and CoBO [4], and provide optimization results on two Guacamol benchmark tasks, med2 and valt. All results are average scores of ten runs under the identical settings. Table demonstrates that our InvBO consistently improves all LBO models in two tasks by a large margin. In particular, in the valt task, all baseline models with InvBO demonstrate significant performance improvements and CoBO with InvBO achieves a 0.348 score gain while the baseline models without InvBO fail in optimization. More results of other baselines with InvBO on other tasks are provided in Section
Figure 4 provides the optimization results four Guacamol benchmark tasks including med2, zale, osmb, and valt. Each subfigure shows the number of evaluations of the black box objective function (Number of Oracle) and the corresponding average and standard error of the objective score (Best Score). Our InvBO built on CoBO achieves the best performance in all four tasks. Further results of other tasks are provided in Section F
We also conduct experiments to demonstrate the effectiveness of our InvBO on DRD3 and arithmetic expression fitting tasks and large-budget settings. The experimental results are illustrated in Fig-
7

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DRD3
Arithnetic Expeession
Valsartan SMARTS (valt)
Median molecules 2 (med2)
Dest soor
Best SUU00e
Beest score
Dest
Namteer of Oracle
Number of Oracle
Number of Oracle
Number of Oracle
(a) DRD3 and Arithmetic expression task
(b) Guacamol task under the large budget setting
PG-1BO
W-LBO
ToRtBO-
Grspb-Ci
(Duns)
Figure 5: Optimization results various tasks and settings. Note that: (a) A lower score is better
score. (b) A higher score is a better score.
Zaleplon MPO
Median molecules (med2)
Osimertinib MPO (osmb)
Valsartun SMARTS (valt)
023
0200
Best
Best
6.00
0131
200
30
Nomber of Oracle
Number of Orncle
Number Oracle
Number of Oracle
CaBO + Inv80 (INV, PAS)
* CoBO + INV
CoBD + PAS
CoBO
Figure 6: Optimization results of component ablation on zale, med2, osmb, and valt.
ure 5(a) and Figure 5(b] respectively. Please note that both DRD3 and arithmetic expression tasks aim to minimize the score while the goal of Guacamol tasks is increasing the score. Figure shows that our InvBO applied to CoBO achieves the best performance. We provide further optimization results other tasks under the large budget settings in Section G These results demonstrate that InvBO is effective in diverse tasks and settings.
Analysis
6.1
Ablation Study
We conduct additional experiments to verify the contribution each component in our InvBO: the inversion method (INV), and the potential-aware trust region anchor selection method (PAS). Figure shows the optimization results of the ablation study on med2, zale, osmb, and valt tasks. From the figure, models with the inversion method (i.e., CoBO with INV and InvBO) replace the recentering technique as the inversion method, while models without the inversion method (i.e., vanilla CoBO and CoBO with PAS) employ the recentering technique. Notably, both components of method contribute to the optimization process, and the combination work, InvBO, consistently obtains better objective scores compared other models. Specifically, in osmb task, the average best score achieved by the methods with the PAS, INV and both (i.e., InvBO) shows 0.784, 0.792, and 0.804 score gains compared to vanilla CoBO, respectively.
6.2 Analysis on Misalignment Problem and Inversion
To further prove that the misaligned dataset hinders the accurate learning of the surrogate model (i.e., misalignment problem), compare the performance of the surrogate model trained with aligned and misaligned datasets on the med2 task. Figure 7 shows the fitting results the surrogate model trained with encoder triplets "g left) and decoder triplets (gee right), respectively. Figure demonstrates that the surrogate model gdee approximates the objective function accurately, while got fails to emulate the objective function. Further details of an experiment are provided in Section
In Figure g we compare the optimization results of CoBO using decoder triplets and encoder triplets on the valt and med2 tasks. Both models use the potential-aware anchor selection, and the decoder triplets are made by our inversion method. CoBO using the decoder triplets shows superior optimization performance over the CoBO using the encoder triplets both tasks. This implies that
8

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GP trained w/ encoder triplet
GP trained w/ decoder triplet
Valsartan SMARTS (valt)
Median molecules 2 (med2)
e.e
Predlited value
0215
135
0.2
Best accrrs
0.3
Best acore
Prodicted
03
0.1
0300
2.7
001
00
0/4
25
400
600
22
462
Troe value
True value
CoBO wi decoder triplet (Ours)
C-CBO w/ encoder triplt
Figure 7: Gaussian process fitting results trained with encoder triplets and decoder triplets.
Figure 8: Optimization results using encoder triplet and decoder triplet on valt and med2 tasks.
Valsartan SMARTS (valt)
Median moleculos 2 (med2)
Mediza molecules (med2)
Median melecules 2 (med2)
0225
1/x0s mnte1
2.0
0.12
Best scure
Best scare
0335
Best
0308
300
305
300
400
50c
600
130
200
Number Oracle
Number Oracle
Number
Oracle
Cetlu Imx3K)
V-RIO--- Im00
Cotlu
THRBOOG4 PAS
Figure 9: (Left) Gaussian process prediction error Figure 10: Optimization results with diverse trust within the trust region. (Right) Optimization re- region anchor selections valt and med2 tasks. sults on med2 tasks.
the misaligned dataset hinders the accurate learning of the surrogate model, which leads suboptimal optimization results, and the inversion method handles the problem by generating the decoder triplet.
6.3 Effects of Inversion on Proposition
1
In Section 4.2 we provide the upper bound of the error between the predicted and ground-truth objective value. In Eq. (6). the Lipschitz constant of the objective function Lı and the trust region radius 5 is fixed or the hyper-parameter and the constant can be improved by learning the surrogate model. In the end, we can reduce the upper bound of the objective value prediction error by reducing the three components: 7, L2, and Lg. which implies the distance between x and pe (z). the Lipschitz constant of function m and f f pe, respectively. Our inversion method reduce by searching the latent Po (in Previously, CoBO [14] proposed regularization losses that
vector Zun that satisfies reduce La- Since the surrogate model emulates the composite function f o po, these regularization losses can reduce L2 along with Lz.
(left shows the regularization losses of CoBO and our inversion method reduces the objective value prediction error. CoBO-based models (i.e., CoBO+PAS and CoBO+InvBO) employ the CoBO regularization losses, and models with InvBO (i.e., TuRBO-L+InvBO and CoBO+InvBO) employ our inversion method. TuRBO-L does not use the CoBO regularization losses nor our inversion method, and models with PAS (i.e., TuRBO-L+PAS and CoBO+PAS) employ encoder triplets. Applying the regularization losses and our inversion method reduces the objective value prediction error, respectively, but our inversion method shows a larger error reduction. The combination of regularization losses and our inversion shows the smallest prediction error, which implies our inversion method robustly complements existing methods. We provide the optimization results of each model on the med2 task in Figure (right). These results demonstrate that reducing the objective function prediction error plays a crucial role in optimization performance.
6.4 Comparing Diverse Anchor Selection Methods
To further prove the importance of our potential-awar anchor selection method, we perform BO with the diverse anchor selection methods: random, acquisition a, and objective score y. Random indicates randomly selected anchors, and acquisition and objective indicate anchors are selected based on the max acquisition function value and objective score, respectively. All models use the inversion method, and the optimization results on valt and med2 tasks in Figure 10 Ours and objective

---

score-based anchor selection rapidly find high siigh-quaity data compared to the random and acquisition- based selections. However, objective score-based anchor selection shows inferior performance than our potential-aware anchor selection. This indicates that both the uncertainty of the surrogate model and objective function value need to be considered for exploration and exploitation.
Conclusion
We propose Inversion-based Latent Bayesian Optimization (InvBO), plug-and-play module for LBO. We introduce the inversion method that inverts the decoder to find the latent vector for generating the aligned dataset. Additionally, we present the potential-aware trust region anchor selection that considers not only the corresponding objective function value of the anchor but also the potential ability of the trust region. From our experimental results, InvBO achieves totte--ofthh-ar performance on nine LBO benchmark tasks. We also theoretically demonstrate the effectiveness of our inversion method and provide comprehensive analysis to show the effectiveness of our InvBO.
Broader Impacts
One of the contributions of this paper is molecular design optimization, which requires careful consideration due to its unintentional applications such as the generation of toxic. We believe that our work has lot of positive aspects accelerate the development of chemical and drug discovery with an inversion- teeesion-bbsed latent Bayesian optimization method.
Limitations
The performance the methods proposed in the paper depends on the quality the generative model. For example, if the objective function is related to the molecule property, the generated model such as the VAE should have the ability to generate the proper molecule to have good performance of optimization.
Acknowledgement
This work was partly supported by ICT Creative Consilience Program through the Institute of Information Communications Technology Planning & Evaluation (ITTP) (ITP--224-RS--22 20-2000- I1201819, 10%) and the National Research Foundation of Korea (NRF) TRRF-2033RIA22200 45%) grant funded by the Korea government (MSIT), and the Virtual Engineering Platform Project (Grant No. PO022336, 45%), funded by the Ministry of Trade, Industry & Energy (MoTIE, South Korea).
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A A Proof of Proposition 1
In this section, we provide the proof of Proposition I
Proposition 1. Let f be a black-box objective function and be posterior mean of Gaussian process, Pe be decoder of the variational autoencoder, e be an arbitrarily small constant, dx and d2 be the distance function on input X and latent Z spaces, respectively. The distance function dx is bounded between 0 and 1, inclusive. We assume that f. m and the composite function of f and are L1, L2, and Li-Lichchitz continuous functions, respectively. Suppose the following assumptions are satisfied:
If(x) -m(=)== S
(10)
dx po(z)) n.
Then the difference between the posterior mean of the arbitrary point v' in the trust region centered at z with trust radius â and the black box objective value upper bounded as: (s(pazz)) - m(z')l S c 7 L1 + (L2 + L3),
(11)
where dz (2, 2 S 5.
Proof. With L-Lipschitz continuity, we have:
If(x) - f(po (2)) s L1 (x po(2)) S L1. [m(2) m(2') s L2 (2 x') s 5. I.2,
(12)
(p)) - ((o((2))) S Ta dz(z, x S 5- 8.3.
Thus, the difference between the posterior mean of point within the trust region of length 8 centered and the black-box objective value is upper bounded follows:
(po(2z)) - mmzz))
= 1(f(x) mmz)) + (f(po(z)) - f(x)) + (f(po(z)) - ((po(2))) + (m(()) mm(2))))
(13)
$ =f((x) m(z)l + [f(p(zz) - f(x) + (((po(z))) f(vo(z)) + [m(z) mm(x)l
cc-7- L1 - 8. (L2 + L3)
B Exploration of Potential-aware Trust Region Anchor Selection
Median molecules 2 (med2)
Valsartan SMARTS (valt)
500
500
Num Oracle
Num Oracle
400
400
300
300
200
200
100
100
0-
Step
Step
CoBO+INV,PAS
CoBO+INV
Figure 11: Exploration effectiveness ablation study of the potential aware anchor selection (PAS) on CoBO+InyBO (INV, PAS).
We measure the number of searched unique data samples for each iteration to validate the exploration ability of our potential- aware trust region anchor selection in 4.3 We compare with the objective score-based anchor selection, which selects the best anchors based on their objective score, used in most prior works [3 14 36). From the figure, our potential-awar anchor selection searches more diverse data compared to the objective score-based anchor selection on both tasks. These results demonstrate that our anchor selection method improves the exploration ability without the loss of the exploitation ability.
14

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C Dissimilarity in Latent Bayesian Optimization
Median molecules 2 (med2)
Valsartan SMARTS (valt)
044
MMmamyy
0.3
dx
dx
0.0
0.0
120
Step
Step
COBO+INV,PAS
CoBO+PAS
Figure 12: Dissimilarity between x² and po(2') with and without inversion on med2 and valt tasks. The measurement of dissimilarity is the normalized Levenshtein distance between the SELFIES token. (x-axis: number of iterations, y-axis: normalized Levenshtein distance.)
We here provide the comparison of with and without inversion empirically prove that the inversion generates aligned data. Given a pair of data sample x' and its corresponding latent vector z we mea- sure the dissimilarity between the input xf and the data pz(z') generated from the decoder. The latent vector generated with the inversion method z is defined as x arg mingez dx(x',, po)2)) (Sec- tion 4.2 The latent vector without inversion method z go (x+) constructed by feeding the input x² into the encoder 4o similar to prior LBO works 11 17].
Figure T shows the dissimilarity comparison results without and with the inversion on med2 and valt tasks. We measure the dissimilarity the normalized Levenshtein distance between two SELFIES tokens. The x-axis indicates the iteration, and the y-axis indicates dissimilarity between x and pz()"). From the figure, the inversion achieves zero dissimilarity for every iteration on both tasks, which indicates that the inversion always generates aligned data. On the other hand, BO without the inversion mostly generates misaligned data. These results demonstrate the necessity of the inversion in LBO.
D Applying PAS to TuRBO on standard BO benchmark
40D Ackley
Best score
-100
-150
-175
-253
1000
Number of Oracle
MIRBO-PAS (Ours)
TuRBO
Optimization results of TuRBO and applying PAS to TuRBO the synthetic Ackley function with 40 dimensions. The lines and ranges indicate the mean and a standard deviation of ten runs with different seeds.
Figure 13:
We analyze the effectiveness of PAS along with standard BO approaches. We provide the opti- mization performance of TuRBO and TuRBO with PAS in the Ackley benchmark function with 40 dimension, with input ranges from 1-32.768, 32.768]. Our implementation is based the codebase TuRBO (36 provided in the BoTorch 4 tutorial, and we use the RBF kernel with Automatic Relevance Determination (ARD) lengthscale
49 The number of initial data is 80 and the number of
15

---

total oracle calls is 1,000. Figure 13 shows that applying PAS on TuRBO consistently improves the optimization performance during the iteration. This result demonstrates that PAS is also effective standard BO benchmark tasks.
E
Experiments of Plug-and-Play
We provide further results of applying our InvBO to previous LBO works, LS-BO, TuRBO-L, W- LBO [M], LOL-BO [33, PG-LBO [[7] and CoBO [4] on Guacamol and arithmetic fitting tasks. All results are average scores ten runs under identical settings. Since LS-BO, W-LBO, and PG-LBO are not trust region-based LBO, we report the optimization results applying inversion (INV) only on these baselines. The experimental results are illustrated T The table shows that applying our InvBO to all previous LBO works consistently improve the optimization performance in all Guacamol and arithmetic expression fitting tasks.
F Experiments on Guacamol Benchmark with Small Budget
Amlodipine MPO (adip)
Ranolazine MPO (ramo)
Perindopril MPO (pdop)
0.58
0.5
6.70
0.56
Best score
0.56
Best score
2.54
0.32
0:52
Best
6:59
0.48
0.64
0:66
350
200
300
4000
500
Number of Oracle
Number of Oracle
Number of Oracle
CaBO InvBO (Ours)
CoBO
PG-LBO
TuRBO-L
LS-BO
Graph-CiA
Figure 14: Optimization results on Guacamol benchmark tasks, excluding the tasks in Figure 4 The lines and ranges indicate the average and standard error of ten runs under the same settings. A higher score is better score.
Figure 14 provides the optimization results three Guacamol benchmark tasks including pdop, adip, and rano. The y-axis of each subfigure denotes the best-found score, and the x-axis denotes the number of the objective function evaluation. Our InvBO built the CoBO shows the state-of-the-art performance on all three tasks. In particular, CoBO with our InvBO achieves 0.56 best score in the pdop task while the best scores of all other basclines are under 0.48.
G Experiments on Guacamol Benchmark with Large Budget
We provide further optimization results on five Guacamol benchmark tasks, adip, osmb, pdop, rano, and zale under the large budget setting demonstrate the effectiveness of our InvBO in diverse budget settings, which are used in previous works [13] [4]. The experimental results are illustrated in Figure [15] In the figure, our InvBO CoBO achieves superior performance on large-budget settings. Specifically, InvBO built on the CoBO showed large margin from the baseline and achieved more than 0.9 best score in adip tasks. These results imply that our InvBO is effective in diverse settings and tasks.
H Analysis on Recentering Technique
here provide the details about additional oracle calls the recentering technique 13 14]
Firstly, CoBO explicitly mentioned the oracle call for recentering in L8-9 of Algorithm 1 in the paper. Secondly, LOL--O's oracle calls for recentering can be verified by its official GitHub code. See, L.188-
228 of hoooloo/olboo///oo 1olbo. py and L54 of lolbo/1olbo. /atettt/llaett..mmmmm satcc__pace_oooe joeacc_eeeeetivee.. py.
16

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Table 2: Optimization results of applying InvBO or inversion method (INV) to several LBOs on Guacamol benchmark tasks and arithmetic expression task, including the task in Table A higher score is better for all tasks except the arithmetic expression task.
Task
Aripiprazole similarity (adip)
Median molecules 2 (med22
Num Oracle
100
300
500
100
300
500
LBO LBO INV
0.493=0.003
0.497=20.005
0.499=0.006
0.186:100000
0.186:1:0.000
0.186:+:0.00
0.495=0.002
0.501
0.509=0.008
0.000
0.188:10.002
0.190:10.002
W-LBO
0.497=0.003
0.499=000006
0.502=0.006
0.1 18:+000000
0.186:t0.000
0.186
W-LBO + INV
0.50
0.523=0.002
0.526=0.002
0.1 8.00000000
0.206:1:.001
0.207=100001
PG-LBO
0.497=0.003
0,518:0.0008
0.528=0.008
0. 186::0.000
0.1 186::00000
0.186:10.00 000
PG-LBO INY
0.003
0.523 0.006
0.558 0.007
0.186 +0,000
0.186++0.00
0.191
TuRBO-L
0.491=0.000
000000
0.491=0.000
0.186+00000
0.186:10.000
0.186:+0.000
TuRBO-L + InvBO
0.499=0.006
0.518=0.008
0,541=0.008
0.186::0.000
0.194110.022
0.202:10.001
LOL-BO
0.401=0.000
0,501 =000044
0,512=0.005
0.186+:0.000
0.186:t:0.000
0.190:00.001
LOL-BO + InvBO
0.500=0.002
0.545=0.0099
0.578
0.1891100002
0.204:10.005
0.227:0.010
CoBO CoBO InvBO
0.491=0.000
0,501 0.0004
0.509=0.005
0.186::00000
0.188:100002
0.191
0,502 0.004
0.542 0.006
0,581
0.008
0.187
0.203 0.004
0.214+0.006
Task
Osimertinib MPO (osmb)
Valsartan SMARTS (valt)
Num Oracle
100
300
500
100
300
500
LBO LBO INV
0.762=0.000
0.763=00001
0.769=0.005
0.000+100000
0.000:10.000
0.000:10.000
0.767=0.002
0.773=00004
0.782=0.004
0,000+000000
0.0001100000
0.000:100000
W-LBO
0.762-0.001
0.766+40.003
0.771 -..014
0.000++00000
0.000+00000
0.001-+0.001
W-LBO + INV
0.770
0.786
0.789=0.001
0.000:000000
0.002:10.001
0.116:10.010
PG-LBO
0.763-0.002
0.764=20.02
-.771-0.003
0.000+00000
0.000:E0.000
0.000+-0.000
PG-LBO INV
0.765=0.003
0.784
0.804 7.804:0.004
0.000++0.000
0.000:10.000
0.159::0.079
TuRBO-L
0.762=0.000
0.762=00.00
0.762=0.000
0.000000000
0.000110.000
0.000:100000
TuRBO-L + InvBO
0.765=0.002
0.785 0.003
0,799-0,004
0. 0.000++00000
0.024:0.017
0.212:10.092
LOL-BO
0.762=0.000
0.769==0.002
0.777=0.003
0.000+00.000
0,000++00000
0,000+10,000
LOL-BO + InvBO
0.775=0.002
0,807 0.005
0.000+000000
0.007:10.005
0.171
CoBO
0.762=0.000
0.763=20.001
0.772 0.003
0.000+10.000
0.000-100000
0.000 000
CoBO InvBO
0.769 0.002
0.795 # 0.004
0.804 0.004
0.000+000000
0.042
Task
Perindopril MPO (pdop)
Ranolazine MPO (rano)
Num Orucle
100
300
500
100
300
500
LBO LBO INV
0.458=0.002
0.458=0.002
0.458=0.002
0.650:100006
0.655:10.007
0.6641:0012
0.466
0.469-0.00
0.658-10.007
0.669:10.08
0.682:40.009
W-LBO
0.460-0.004
0.462+00.04
0.464-0.005
0.649+0.006
0.674++0.007
0.681++0.005
W-LBO + INV
0.468
0.408
0.483=0.001
0.683-10.002
0.700:10.003
0.70710.003
PG-LBO
0.458-0.002
0.458:0.002
0.465=0.004
0.650110.003
0.681110.009
0.720
PG-LBO INV
0.458=0.002
0.005
0.490=0.000
0.674:10.010
0.713-10.013
0.729:10001
TuRBO-L
0.456=0.000
0.45620.000
0,456=0.000
0.6421:0000
0.642-10.000
0.650110.007
TuRBO-L + InvBO
0.470
0.506:0000K
0,534=0,MM 0.008
0.688100.004
0.743-10.011
0.791
LOI
0.456-0.008
0.462+0.004
0..70-0.004
0.642+00000
0.671+H0000
0.688
LOL-BO + InvBO
0.512=0.017
0.546
0.565=0.014
0.699:10.004
0.762:10.012
0.787:10014
CoBO CoBO InvBO
0.456=0.000
0.460=0.004
0.461=0.004
0.642-10.000
0.654110.006
0.705
0.001
0.014
0.561 0.015
0.686:10.004
0.708:10.012
0.756:10.015
Task
Zaleplon MPO (zale)
Arithmetic expression
Num Oracle
100
300
500
100
300
500
LBO LBO INV
0.344=0.009
0.357=0.010
0.368=0.009
1.27410.147
0.512:10.046
0.425:+0.020
0.370
0.387=0.010
0.394_0.011
1.306:0.112
0.506:1.1114
0.31710.048
W-LBO
0.369=0.014
0.390=00013
0.407=0.014
1,280:10.169
0.686:10. 103
0.538:10.068
W-LBO + INV
0.383=0.003
0.416=0.002
0.423=0.002
1.52410.008
0,466:10.006
0.44510.006
PG-LBO
0.369=0.009
0.410=0.002
0.415=0.009
1.290-10.146
0.913:10.1 140
0.83240.. 146
PG-LBBO INV
0.372 0.016
0.420 0.007
0.452 0.006
1.515++0.000
0.872++0.177
0.595
TuRBO-L
0.327=0.000
0.332:00.005
0.332=0.005
1.424+00.12
0.885:10.083
0.605:10.094
TuRBO-L + InvBO
0.411
0.461
0.475
0.898110.161
0.692:10.139
0.374-10.00
LOL-BO
0.327=0.000
0.002
0.00100.006
0.953++-..109
0.578:1:0003
0.412:+0.039
LOL-BO + InvBO
0.400=0.011
0.456=0.001
0.477=0.011
0.739110.142
0.510:10.031
0.392-10.023
CoBO
0.327=0.000
0.393=0.014
0.421=0.010
0.752-10112
0. 0.526-10.057
0.391 +.00
CoBO InvBO
0.435-0.014
0.495-0.023
0.513 0.012
0.840++0.173
0.513+-0.075
0.252
17

---

Amlodipine MPO (adip)
Osimertinib MPO (osmb)
Perindopril MPO (pdop)
0.51
090.92
Best score
0.80
0.33
0.75
0.70
Best
0.00
0.00
0.89
0.60
0.55
0.44
0.50
10C
20K
40K
S0K
80K
10K
20K
40K
10K
23K
40K
60k
Number Oracle
Number of Oracle
Number Oracle
Ranolazine MPO (rano)
Zaleplon MPO (zale)
0.93
0.75
00
score
0.70
0.65
020.80
0.60
6.75
0.5
10K
20C
40K
GCK
Number Oracle
Number of Oracle
CoBO InvBO (Ours)
CoBO
PG-LBO
LOL-BO
W-LBO
TURBO-L
LS-BO
Graph-GA
Figure 15: Optimization results Guacamol task under the large budget setting, excluding the tasks in Figure 5(b] The lines and ranges indicate the average and standard eтror of five runs under the same settings. A higher score is better score.
We present the experiments on additional oracle calls in recentering in Figure I6 the optimization results of CoBO that use recentering with additional oracle calls and recentering without additional oracle calls on med2 and valt tasks. In both tasks, CoBO that uses the recentering technique without additional oracle calls fails in optimization while consuming additional oracle calls shows progress. These results show that the additional oracle calls are essential in the recentering technique, while the previous works do not explicitly mention it.
The figure shows
I Analysis on Misalignment Problem
We here provide the experimental details about the Gaussian process fitting with the encoder triplets and the decoder triplets. We train the surrogate model 300 train points and predict 100 test points, all points are randomly sampled. The encoder triplets {xx' fxxx)
and the decoder triplets [(x² z'm f(x')) } _ share the same discrete data x, where x Po (2) The dots in the figure denote the
predictions of the surrogate model and bars denote 95% confidence intervals. Figure 17 provides the Gaussian process fitting results trained with the encoder triplets (Left) and decoder triplets (Right) that predict the training points on the med2 task. The figure implies that the Gaussian process trained with encoder triplet fails to fit the composite function f while trained with decoder triplet emulates the function accurately.
Efficiency Analysis
Although Bayesian optimization assumes the objective function is cost-expensive, it is still important to consider the efficiency of the algorithm. We present an efficiency analysis comparing with baseline methods [3] 17], and TuRBO-L, BOOO. We conducted experiments under the same condition for fair comparison: a single NVIDIA RTX 2080 with the CPU of AMD EPYC 7742. Wall-clock time comparison with the baseline model is in Table 3 The table shows that applying our InvBO to CoBO achieves state-of-the-art performance not only under the same oracle calls but also under the same wall-clock time.
18

---

Median molecules (med2)
Valsartan SMARTS (valt)
0.194
Best score
Best score
1.5
0.186
-9-
200
300
Num Oracle call
Num Oracle call
Recentering with oracle calls
Recentering without oracle calls
Figure 16: Optimizaiton results on Guacamol tasks, med2 and valt. We compare the CoBO applying InvBO that uses the recentering technique with and without additional oracle calls. The lines and ranges indicate the average and standard error runs under the same settings. A higher score is better score.
Train with encoder triplet
Train with decoder triplet
Predicted value
Predicted value
0.16
0.13
0.10
0.07
0.04
True value
True value
Figure 17: Gaussian process fitting results trained with encoder triplets and decoder triplets. Note that the datapoints are in the training
K Details of Benchmark Tasks
We demonstrate the effectiveness of InvBO through three optimization benchmarks: Guacamol 3
DRD3, and arithmetic expression fitting task [8 [T [3 [4]. Guacamol benchmark tasks aim to design molecules with the desired properties, which are scored with a range between to 1. The number of initial points is 100 and the number of oracle calls is 500 in small-budget setting, and the number initial points is 10,000 and the number of oracle calls is 70,000 in a large-budget setting, In the DRD3 benchmark task, we aim to design ligands (molecules that specifically bind to protein) that bind to dopamine receptor D3 (DRD3). We evaluate the docking score ligands with the Dockstring library [50]. The number of the observed data points is 100 and the number of max oracle calls is 3,000. Lastly, the arithmetic expression fitting task is to generate single-variable expressions that close to target expression (e.g., 1/3 + x sin (xr 2)). The number of initial access oracle is 100 and the number of max oracle calls is 500. All tasks are assumed to be noiseless settings,
and we select the initial points randomly from the dataset provided in [14] and the same initial data was used in all tasks.
Details of Baselines
We compare our InvBO with six latent Bayesian Optimization methods: LS-BO, TuRBO (3),, W-
LBO M. LOL-BO [3]. CoBO 14, and PG-LBO [17), and one graph-based genetic algorithm:
19

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Table 3: Wall-clock time and corresponding best score for each model.
Task
Model
CoBO+InvBO
CoBO
PG-LBO
LOL-BO
TuRBO
LS-BO
W-LBO
Found Best Score
0.2143
0.1907
0.1856
0.1904
0.1856
0.1856
0.1856
Oracle call
500
500
500
500
500
500
500
Wall-clock time (s)
420.12
285.02
1582.23
308.45
842.25
152.31
573.21
med2
Found Best Score
0.1938
0.1883
0.1856
0.1856
0.1856
0.1856
0.1856
Oracle calls
284
314
286
182
236
Wall-clock time (s)
152.31
152.31
152.31
152.31
152.31
152.31
152.31
Found Best Score
0.7325
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Oracle calls
500
500
500
500
500
500
500
Wall-cluck time (s)
246.12
148.23
443.65
163.67
101.58
152.96
133.48
valt
Found Best Score
0.1173
0.0000
0.0000
0.0000
0.0000
0.0000
0,0000
Oracle calls
262
184
403
426
500
439
Wall-clock time (s)
101.58
101.58
101.58
101.58
101.58
101.58
101.58
We adopt the standard LBO for LS-BO approach following other work [1] with VAE update during the optimization process. TuRBO leverages trust regions to alleviate the over- exploration. To adapt TuRBO to the latent space, employ TuRBO-L, LS-BO with trust region, following other latent Bayesian optimization baselines [4]. W-LBO weights the data based on importance in order to focus samples with high objective values. LOL-BO adapts the concept of the trust region to the latent space and proposes VAE learning methods to inject the prior of the sparse GP models for better optimization. CoBO designs novel regularization losses based on the Lipschitz condition to boost the correlation between the distance in the latent space and the distance within the objective function. PG-LBO utilizes unobserved data with pseudo-labeling technique and integrates Gaussian Process guidance into VAE training to learn a latent space for better optimization. In PG-LBO, we sample the pseudo data by adding the Gaussian noise, and dynamic thresholding.
GB-GA [45].
M Implementation Details
use p BoTorel GPy- T [52,, and Guacamo software packages. In the Guacamol tasks and DRD3 task (50], we use the SELFIES VAE [3] which is pretrained in an unsupervised manner with 1.27M molecules from Guacamol benchmark. The Grammar VAE [ is pre-trained with 40K expression data in an unsupervised manner. The dimension of latent space is 256 the SELFIES VAE and 25 in the Grammar VAE. The size of the data obtained from the acquisition function and k are hyperparameters, which are presented in Table
Our implementation based on the codebase of [3] @
M.1 Hyperparameters Setting
The learning rate used in the inversion method is 0.1 in all tasks, as we empirically observe that always finds the latent vector that generates the target discrete data. The maximum iteration number of the inversion method 1,000 all tasks. The rest of the hyperparameters follow the setting used in [14]. Since arithmetic fitting tasks and Guacamol with the small budget tasks use different initial data numbers used in [14], we set the number top-k data and the number query points N, same as DRD3 task, as they use the same number of initial data points.
N Pseudocode of Potential-aware Trust Region Anchor Selection
Here, we provide the pseudocode of the potential-aware trust region anchor selection method. We get the max acquisition function value, which is a Thompson Sampling, of each trust region 7 given a
Copyright (c) 2016 G-Faccbook, Inc (Adam Paszke) 51 Licensed under BSD-style license
'Copyright (c) Meta Platforms, Inc. and affiliates. Licensed under MIT License
'Coyyright (c) 2017 Jake Gardner. Licensed under License
'Coyyright (c) 2020 The Apache Software Foundation, Licensed under the Apache License, Version 2.0.
20

---

Table 4: Other hyperparameters for various benchmarks.
Hyperparameters
Guacamol-small
Guacamol-large
DRD3
Arithmetic
Number of initial data points 1
100
10000
100
100
Number of query points Ng
5
10
5
Number of top-k data
50
1000
10
VAE update interval Naal
10
10
10
10
Max oracle calls
500
70000
3000
500
Algorithm 2 Potential-aware Anchor Selection
Input: Data history {x', z', ,' Ji= surrogate model GP
1: for 1,2,..., do
2: Get a candidate set Zand with random points in the trust region T around z²
3: Opo max f(z) where f f~rf-: GP (u(z), k(z, xE
Eq.
4: end for
5: Y {y" )=
6: A {aput i=
7: for = 1,2, 2,
n do a'calld f Calculate(a Y, A)
8:
Eq.
9:
+ waaod
Eq.
10: end for
S in
12: return z' where is index of max (S)
candidate set Zend in lines 2-3. After scaling the max acquisition function values, we calculate the final score of each trust region of in lines 8-9. Finally, we sort the anchors in the descending order of the final score and select the anchor.
Pseudocode of InvBO Applied to CoBO
We provide the pseudocode of CoBO applying our method, InvBO, which consists of the inversion method and the potential-aware anchor selection. The inversion method is used in lines and and the potential-aware anchor selection method is used in line 11. As in CoBO, we use the subset of data history, which composed of the top scored data and the most recently evaluated data during the optimization process in line 4. When we fail Nrai times to update the best score, we train the VAE with CoBO loss LCOBO and update the dataset to be aligned using the inversion method. After that, we train the surrogate model with 2' and negative log-likelihood loss Laur except for the first iteration and select the trust region anchor with the proposed potential-awars trust region anchor selection in line 1. We select N, next query points from the trust region in line 12. Then, evaluate the next query points with the objective function and update the data history in lines 17, and 18.
21

---

Algorithm 3 InvBO built on CoBO
Input: Pretrained encoder 4s- decoder pe- black-box function surrogate model g. acquisition
function a, oracle budget T", latent update interval Naa number of query point Ng. initial data
D = [(xx'>))) learning rate for inversion distance function dx
1: D [xxxx zi v²) (x', 8²) E DO, z Inversion (x', 9p. Pe, p, dx) }
D> Algorithmm 1
2: A'ail f +0
3: for t = 1,2,... I do
4:
D' t- CONCAT (Di-1(-Ng Ng :],
(opk(D
5: if Mial Naa then
6: Peail +
7: Train 2s and pe with LcoBo, D
{(x' z/m.. v') (x', z'yy') Dt. zin Inversion (x', qo. po. n, dx) DD'|
8:
D
Algorithm
9:
end if
10:
Train 9 with Learr D' if & else DO Zaunchor <- Trust Region Anchor Selection(D', ,g)
Algorithm
12:
Z +- Select N₂ query points in trust region centered on anchor 8" + (xx(xx2)))D= y
13:
14:
if (po(zz)) S y², Vz' E Zuext then
15:
Thfil Miail +1
16:
end
7real Daww + (po(z'), ( 'z' E Znexa i=1
17:
18:
D' + CONCAT (D', Dnew)
end for
20:
0²,2".10 <- arg (x. y
return x"
22

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Reviews:
[{"review_id": "gsIJdWQPOD", "reviewer": "Reviewer_k47A", "summary": "Latent Bayesian optimization has been tackled in this work. In order to solve an optimization problem on a continuous latent space, it utilizes auto-encoder-based neural networks. In particular, this work attempts to solve a misalignment problem in the latent Bayesian optimization. Some experimental results are demonstrated to validate the method proposed in this work.", "questions": "- Does the misalignment problem certainly degrade the performance of Bayesian optimization? Is there any particular evidence?\n- I think that the proposed method lets each decision focus on exploitation. What do you think about this issue?\n- Equation (4) doesn't seem to be inversion. It just finds the nearest output of the decoder.\n- How did you choose the dimensionality of the latent space?\n- The proposed method seems to require theoretical analysis.\n- Considering the nature of Bayesian optimization, which is to solve black-box optimization, the benchmarks used in this work are too domain-specific to show algorithm's performance. Under the assumption of optimizing a black-box function, the proposed method actively utilizes the information of objective functions. Do you expect that your algorithm works well in more general optimization problems?\n- I cannot find the details of the neural architectures used. How did you design such networks?\n- Could you elaborate on the description of Figure 3?", "limitations": "There are no specific societal limitations of this work.", "rating": 3, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "- Latent Bayesian optimization, which is solved in this paper, is a compelling topic in the Bayesian optimization field.", "weaknesses": "- Motivation of this work is weak.\n- Thorough analysis on the misalignment problem is not provided.\n- Experiments are domain-specific.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "- **[W1] Motivation of InvBO.**\n    \n    Other reviewers provided positive comments regarding the motivation of our paper as follows:\n    \n    > Reviewer 8vpX: This paper identifies and addresses an overlooked issue in several latent space Bayesian optimization methods.\n    > \n    \n    > Reviewer wVbQ: The way the authors motivated, defined, and applied InvBO is clear.\n    > \n    \n    > Reviewer nFyz:  The authors highlight a systematic problem present in all VAE-based Bayesian optimization architectures.\n    > \n    \n    > Reviewer tYNa: The inversion method is a novel and principled way to address the important problem of misalignment between the input and latent spaces in LBO.\n    > \n    \n    Here we clarify the motivation behind our proposed InvBO, which consists of Inversion and PAS.\n    \n    - **Motivation of Inversion.**\n    \n    LBO suffers from the misalignment problem caused by the reconstruction error of the VAE. Previous works handle this problem with a recentering technique; however, Figure 3 shows that this requires additional oracle calls. This motivated us to design an inversion method, a solution to the misalignment problem that does not require any additional oracle calls.\n    \n    - **Motivation of PAS.**\n    \n    Most prior trust region-based approaches select the anchor as the current optimal point without considering the potential of the latent vectors within the trust region to improve optimization performance. This prompted us to design a novel anchor selection method that considers the potential of the latent vectors within the trust region.\n    \n- **[W2, Q1] Evidence and analysis of misalignment problem.**\n    \n    We have already presented evidence that the misalignment problem certainly degrades optimization performance. Figure 12 in Section C shows the discrepancy between $\\mathbf{x}$ and $p_\\theta(\\mathbf{z})$. As shown in Figure 7 from the main paper, this discrepancy leads to the misalignment problem. Furthermore, Figure 8 in the main paper shows that optimization performance is significantly lower with a misalignment problem (yellow line) compared to when the problem is addressed by inversion (blue line).\n    \n- **[W3, Q6] Diversity of experimental domains and general optimization ability of InvBO.**\n    \n    We already measure the performance of our InvBO on diverse domains such as molecule domains (e.g., Guacamol and DRD3 tasks) and an arithmetic expression fitting task. Figures 4 and 5 in the main paper demonstrate the general optimization ability of our InvBO across various domains and settings. \n    \n- **[Q2] Exploration capability of InvBO.**\n    \n    InvBO does not make exploitation-centric decisions. In Figure 1 of the provided PDF, we conduct additional experiments measuring the number of unique data searched in each iteration of CoBO with and without InvBO. Figure 1 demonstrates that InvBO does not lose exploration capability compared to CoBO. On the other hand, Figure 4 in the main paper shows that applying InvBO enhances the exploitation ability. These results indicate that InvBO makes decisions while balancing exploitation and exploration.\n    \n- **[Q3] Clarification on the appropriateness of the term 'Inversion'.**\n    \n    ‘Inversion’ is broadly used to refer to the reverse process of generation and has been widely applied to generative models such as GANs and Diffusion models [1-5]. As mentioned in Section 2.2 of the paper, ‘Inversion’ is a process of finding a latent code that generates the original data $\\mathbf{x}$ through a generator $G$. Without loss of generality, we also use the term `Inversion' in the same sense with decoder $p_\\theta$ and formally define it in Equation (4).\n    \n    - Reference\n        1. Xia, Weihao, et al. \"Gan inversion: A survey.\" TPAMI, 2022.\n        2. Zhu, Jiapeng, et al. \"In-domain gan inversion for real image editing.\" *ECCV,* 2020.\n        3. Wang, Tengfei, et al. \"High-fidelity gan inversion for image attribute editing.\" CVPR, 2022.\n        4. Xu, Yiran, et al. \"In-N-Out: Faithful 3D GAN Inversion with Volumetric Decomposition for Face Editing.\" *CVPR,* 2024.\n        5. Gal, Rinon, et al. \"An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion.\" *ICLR,* 2023\n- **[Q4, Q7] Details of the dimensionality of latent space and the Variational Autoencoder (VAE) used in experiments.**\n    \n    For a fair comparison, we use the same dimensionality of latent space and VAE following baseline works. As we mentioned in the main paper, we use SELFIES VAE [1] for the de novo molecule design tasks (e.g., Guacamol and DRD3) and Grammar VAE [2] for the arithmetic expression fitting task.\n    \n    - Paper\n        1. Maus, Natalie, et al. \"Local latent space Bayesian optimization over structured inputs.\" NeurIPS, 2022.\n        2. Kusner, Matt J., et al. \"Grammar variational autoencoder.\" *ICML*, 2017.\n- **[Q5] Theoretical analysis of InvBO.**\n    \n    In Proposition 1, we theoretically show that inversion plays a crucial role in minimizing the upper bound of the GP prediction error within the trust region. Figure 9 in the main paper shows the experimental results of GP prediction error within the trust region (left) and the corresponding optimization results (right). These results demonstrate that the inversion method minimizes the GP prediction error and results in the improvement of the optimization process.\n    \n- **[Q8] Detailed description of Figure 3.**\n    \n    The left figure illustrates the number of oracle calls made by the acquisition function and recentering. The right figure displays the number of best score updates achieved by the acquisition function and recentering. From both figures, the acquisition function updates the best score 5 times within approximately 150 oracle calls, whereas the recentering fails to update the best score with about 350 oracle calls. These results demonstrate that the recentering wastes a huge amount of oracle calls.", "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": "iMsKsPcqUX", "reviewer": "Reviewer_8vpX", "summary": "This paper identifies and addresses an overlooked issue in several latent space Bayesian optimization methods and proposes a new trust region anchor selection method (PAS) that incorporates the \"potential\" of a trust region to improve optimization. Specifically, the paper proposes an inversion method to correct the misalignment between the encoding that generated a sample and the resultant encoding of the generated sample. Aligning these representations allows for a better estimate by the surrogate and improves optimization. Further, the trust region anchor selection incorporates both the observed objective value at a given point and the potential for improvement within that trust region (evaluated through Thompson sampling), which similarly improves the optimization.", "questions": "PAS is not specific to LSBO and can be employed wherever TuRBO-derived methods are used. Was this tried on one of the standard BO benchmarks (Robot, Lunar, etc)?", "limitations": "The noted limitation of InvBO being sensitive to the quality of the generative model is fair and to be expected.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- They identify and address the misalignment problem using an inversion method. Logically, this is a more direct and sample efficient way of doing it compared to previous methods which perform oracle evaluations to score unevaluted points. Additionally, this allows for the GP to produce a better fit of the objective which improves optimization.\n- The inversion method is plug-and-play with LOLBO and CoBO, and could possibly be used by other LSBO methods that perform fine-tuning of the VAE.\n- The potential aware trust region selection is more flexible than TuRBOs and allows the optimization to revisit previous regions in the optimization trajectory. Their ablations show the effectiveness of this.\n- The methods are intuitive and straightforward to implement.\n- InvBO performs particularly well over baselines in the low-budget regime.", "weaknesses": "- Methodologically the contribution is a bit weak due the fact the proposals here are largely extensions of LOLBO and CoBO.\n- The Lipschitz assumption for the VAE decoder and objective function doesn't seem well motivated. This isn't a significant issue with respect to the method, but I do question the relevance of Proposition 1.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "- **[W1] Methodological contribution of InvBO beyond LOLBO and CoBO.**\n    \n    Our InvBo can be applied beyond trust region-based LBO methods (e.g., LOLBO and CoBO). We show that each component of our InvBO, such as Inversion and Potential-aware trust-region anchor selection (PAS), can be implemented with diverse BO approaches, as specified below:\n    \n    - **Inversion.**\n        \n        Our Inversion can be applied to any LBO method, including trust region-based LBO methods (e.g., LOLBO and CoBO). In Table 2 of Section D, we have already shown that the inversion has successfully been adopted to diverse LBO works with and without using trust region. \n        \n    - **PAS.**\n        \n        Our PAS can be extended to any trust region-based standard BO method (e.g., TurBO). To validate it, we conduct additional experiments applying PAS to TuRBO on standard BO tasks in Rover and Lunar and report the optimization results in Figure 2 of the provided PDF. These experimental results demonstrate that InvBO can be applied not only to TR-based LBO but also to standard TuRBO.\n        \n- **[W2] Appropriateness of Lipschitz assumptions in Proposition 1.**\n    \n    Thank you for your valuable question. In Proposition 1, we assumed the black box function $f$ and the composite function $f\\circ p_\\theta$ of the black box function and the decoder of VAE as a Lipschitz continuity function. Assuming the black box function or the objective function as the Lipschitz continuous function is common in Bayesian optimization [1-5] or global optimization [6].  \n    \n    - Reference\n        \n        [1] González, Javier, et al. \"Batch Bayesian optimization via local penalization.\" *Artificial intelligence and statistics*. PMLR, 2016.\n        \n        [2] Scarlett, Jonathan. \"Tight regret bounds for Bayesian optimization in one dimension.\" *International Conference on Machine Learning*. PMLR, 2018.\n        \n        [3] Hoang, Trong Nghia, et al. \"Decentralized high-dimensional Bayesian optimization with factor graphs.\" *Proceedings of the AAAI Conference on Artificial Intelligence*. Vol. 32. No. 1. 2018.\n        \n        [4] Kim, Jungtaek, and Seungjin Choi. \"On local optimizers of acquisition functions in bayesian optimization.\" *Machine Learning and Knowledge Discovery in Databases: European Conference, ECML PKDD 2020, Ghent, Belgium, September 14–18, 2020, Proceedings, Part II*. Springer International Publishing, 2021.\n        \n        [5] Lee, Seunghun, et al. \"Advancing Bayesian optimization via learning correlated latent space.\" *Advances in Neural Information Processing Systems* 37 (2023).\n        \n        [6] Christodoulos A. Floudas and Panos M. Pardalos, editors. Encyclopedia of Optimization, Second Edition. Springer, 2009.\n\n- **[Q1] Applying PAS to TuRBO on standard BO benchmarks.**\n    \n    Thank you for the suggestion. We provide the optimization performance of TuRBO and TuRBO with PAS in two standard BO benchmarks, Rover and Lunar. Our implementation is based on the codebase of TuRBO provided in the BoTorch tutorial and uses the same hyperparameters ($e.g.$, batch size, and number of initial data points) with TuRBO. The experimental results are reported in Figure 2 of the provided PDF. The figure shows that applying PAS on TuRBO consistently improves the optimization performance. These results demonstrate that PAS is also effective in standard BO benchmark tasks.", "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": "p73gwg3O8G", "reviewer": "Reviewer_wVbQ", "summary": "The authors propose Inversion-based Latent Bayesian Optimization (InvBO), a novel approach to improve latent space Bayesian optimization (LBO) by introducing two novel components. First, to fix the misalignment problem that typically plagues LBO methods that rely on encoder-decoder models, InvBO introduces an inversion method that can be used to recover the latent code that decodes to a given data point. This allows the misalignment problem to be largely circumvented without the need for additional black-box function evaluations. Second, InvBO proposes a new strategy for the selection of the center of the trust region for trust-region based LBO. In existing LBO methods that use trust regions, the trust region centers are usually chosen to be the latent data point associated with the best objective value observed thus far. InvBO proposes a new method for trust region center selection that encourages selection of trust region centers that give the trust region the highest potential to improve local optimization performance. The authors refer to this method as “potential-aware trust region anchor selection”. By combining these two components (the inversion method and the potential-aware trust region anchor selection method), the authors show that InvBO can be applied to substantially outperform current state-of-the-art LBO methods across nine high-dimensional, discrete optimization tasks. These include some of the most difficult tasks from the GaucaMol benchmark suite of molecule optimization tasks.", "questions": "I am interested in your ideas for future work. Do you have plans for how you might build upon this work and continue to improve methods for LBO?", "limitations": "Yes.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "Originality: InvBO proposes two novel ideas that greatly improve performance of LBO. The inversion method is a very straight-forward and effective means of dealing with the misalignment problem that avoids using extra function evaluations. The potential-aware trust region anchor selection method represents a novel means of selecting trust region centers that improves upon the fairly ad-hoc strategy people have been using of just centering the trust region on the best data point observed so far. \nQuality: The paper is well-written and concise. Additionally, the figures and tables are all of good quality - they are both easy to parse and do a nice job of displaying results. Figure 2 does a nice job of illustrating the author’s inversion method. \nClarity: The paper is clear and easy to follow from start to finish. The figures and tables are clear and easy to read. The way the authors motivated, defined, and applied InvBO is clear. \nSignificance: LBO has emerged as one of the most promising ways to optimize black-box functions defined over discrete, high-dimensional spaces. These discrete, high-dimensional optimization problems are of particular interest to the community because relevant real-world problems, particularly in biological design, are defined over discrete high-dimensional spaces (i.e. the discrete search space of molecules or proteins). The results in this paper show a very substantial performance improvement over state-of-the-art methods in LBO, including on some of the most difficult molecular design benchmark optimization tasks in the popular GuacaMol benchmark suite. This paper therefore represents a significant improvement in our ability to optimize discrete high-dimensional black-box functions and will be of interest to the community.", "weaknesses": "Figure 1 colors should be changed to be more friendly to red-green color blind folks. \n\nIt would be interesting to see an additional comparison to the Genetic Expert-Guided Learning (GEGL) method for the molecular design tasks in the results section. GEGL is an reinforcement learning (RL) method that obtains state-of-the-art performance across tasks in the GuacaMol benchmark suite (see GuacaMol results in Table 2 of their paper here https://arxiv.org/pdf/2007.04897). While I do think that this additional comparison to GEGL would strengthen the paper, I do not think that it is strictly necessary for this paper to be accepted because RL is an orthogonal method and the results currently in the paper compare to all relevant LBO baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "- **[W1] Modifying the colors of Figure 1 for red-green color blind folks.**\n    \n    Thank you for the suggestion. We will modify the colors of Figure 1 for red-green color-blind folks in our camera-ready version if the paper gets accepted.\n    \n- **[W2] Additional comparison to GEGL.**\n    \n    Thank you for the suggestion. We provide the experimental results of GEGL optimization on adip and osmb tasks in Figure 3 in the provided PDF. We include only a subset of baselines in Figure 3 to enhance the clarity of additional experimental results. While GEGL demonstrates superior optimization performance compared to Graph-GA, CoBO with InvBO still achieves higher optimization performance.\n    \n- **[Q1] Future work of InvBO.**\n    \n    LBO has become a promising approach for optimizing structured data such as molecules or proteins. However, research on multi-objective Bayesian optimization over latent space has not been fully explored [1, 2]. Here, we propose InvBO for single-objective Bayesian optimization over latent space, but we believe that the misalignment problem also occurs in multi-objective Bayesian optimization over latent space. We will explore the adaptation of InvBO to multi-objective Bayesian optimization over latent space.\n    \n    - Reference\n        1. Stanton, Samuel, et al. \"Accelerating bayesian optimization for biological sequence design with denoising autoencoders.\" *International Conference on Machine Learning*. PMLR, 2022.\n        2. Gruver, Nate, et al. \"Protein design with guided discrete diffusion.\" *Advances in neural information processing systems* 37 (2023).", "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": "ATjIVKepaa", "reviewer": "Reviewer_nFyz", "summary": "&nbsp;\n\nThe authors propose two empirical improvements to VAE-based Bayesian optimization methodology. First, the authors propose a correction for the eponymous misalignment problem which they characterize in the paper, the idea being that the latent z corresponded to an encoded x may not correspond to the x' decoded from the same latent z. The correction proposed by the authors is based on inversion. The authors demonstrate that inversion systematically improves a suite of VAE-based Bayesian optimization methods and furthermore, include diagnostic experiments that demonstrate how the Gaussian process surrogate model fit is improved following inversion. Secondly, the authors propose an improvement to trust region-based optimization in the VAE latent space based on using \"potential-aware\" scoring. Given that the method is broadly applicable and the experimental validation of the approach is rigorous, I recommend acceptance with the following points for the authors to consider. I am ready to increase my score if these points are addressed.\n\n&nbsp;", "questions": "&nbsp;\n\n1. In Section L, the authors state that they pretrained the SELFIES VAE with 1.27M molecules from the Guacamol benchmark. Do the authors assess whether the generated molecules are contained within the pre-training dataset?\n\n2. What do the authors think the interplay between metric learning and inversion is?\n\n3. On line 248 what is the particular type of approximation that the authors use for their sparse GP implementation? I note that this is also not mentioned in Section L of the appendix. From the code it would appear to be the Sparse Variational Gaussian Process (SVGP) model.\n\n4. Why are Figures 7 and 16 different?\n\n\n&nbsp;", "limitations": "&nbsp;\n\n1. The use of the Levenshtein distance for diagnosing the effect of inversion on the misalignment problem may not be the best criteria for the molecule experiments. Tanimoto similarity would be more indicative of chemically meaningful differences since it is a distance metric between molecules directly and not their string representation. \n\n2. The aforementioned lack of justification for the potential-aware scoring criterion for the trust regions is an additional limitation for the work in its current form.\n\n3. It would be great if instructions could be included in the codebase to reproduce the results.\n\n\n&nbsp;", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "&nbsp;\n\n1. The principle strength of this work is the generality of the approach. Concretely, the authors highlight a systematic problem present in all VAE-based Bayesian optimization architectures and solve it with their inversion approach.\n\n2. The diagnostic experiments on the GP fit provided by the authors provides excellent evidence as to why the inversion approach improves performance, shedding light on the underlying mechanism of the inversion approach.\n\n&nbsp;", "weaknesses": "&nbsp;\n\nI highlight below some points of concern with the paper. Of particular note is the justification for the potential-aware method. In terms of the sample-efficiency matters benchmark, I believe this would greatly increase the impact of the paper by showcasing the potential for VAE-based BO methods against other molecule generation approaches in the ML literature. If the authors can address these points I will increase my score.\n\n&nbsp;\n\n__MAJOR POINTS__\n\n&nbsp;\n\n1. In Proposition 1, how important is the assumption that the distance function d_\\Chi is bounded on [0, 1]? \n\n2. The Levenshtein distance may not be the most appropriate distance for objects such as molecules which are defined using the SMILES and/or SELFIES syntax. Other distance metrics such as Tanimoto similarity [8, 9] would be more chemically meaningful.\n\n3. I think it would be a great idea for the authors to assess their method on the sample efficiency matters benchmark [13]. While this is not a weakness of the current work per se, I list this here as seeing the performance of this method on the benchmark in comparison to other molecule generation approaches would be very interesting to see and would highly encourage me to increase my score.\n\n4. It would be great if the authors could include the T-LBO method from [14] as a baseline as it would be interesting to understand the effect of metric learning on performance. In particular, metric learning is hypothesized to smoothen the latent function and hence make it easier to fit the GP on the latent points. It would be very interesting to understand the interplay between inversion and metric learning. In other words, does inversion directly solve the problem that metric learning is trying to address? For the arithmetic task, the results from this paper should be directly comparable if the experiment was run under the same settings.\n\n5. For the experiments reported in Section C of the appendix, it would be great if the authors could report the Tanimoto similarity between the molecules that correspond to the SELFIES strings as this would be a much more chemically meaningful measure of similarity relative to Levenshtein distance.\n\n6. For the \"potential-aware\" method of anchor selection the authors use the acquisition function value is scaled because of the effect of using multiple local GPs to model each trust region?. Do the authors have a motivation for using the sum of the objective function value of the anchor together with the scaled maximum of the Thomson sample as the scoring criterion? Presumably the \\alpha term accounts for the local region and the objective function accounts for the known quality of the anchor. Why should this objective be substantially different than taking the maximum of a Thompson sample directly i.e. just keeping the \\alpha term? \n\n7. Could the authors provide clear instructions for reproducing the experimental results in the supplied code by means of a README? Currently it is not clear how to reproduce the results.\n\n&nbsp;\n\n__MINOR POINTS__\n\n&nbsp;\n\n1. There are some missing capitalizations in the references section e.g. \"Bayesian\" and \"Gaussian\".\n\n2. When mentioning variational autoencoders it would be worth citing the original paper [1].\n\n3. On line 68, it may be worth mentioning that the goal of LBO is to learn a latent space to enable optimization over a continuous space from a discrete or structured input space where \"structured\" refers to objects such as graphs and images.\n\n4. In the related work section it would be worth mentioning the following works on VAE-based Bayesian optimization [2-7].\n\n5. In Proposition 1, f, m, and the composition function of f and p_\\theta are assumed to be L_1, L_2, and L_3 Lipschitz continuous respectively. It may be beneficial to clarify that these functions are not 1-Lipschitz, 2-Lipschitz continuous etc. but rather the Lipschitz constants can be arbitrary.\n\n6. The PyTorch, BoTorch, and GPyTorch papers [10-12] should be cited given that the packages are used.\n\n7. In Section K of the appendix the citation for LS-BO is not given (presumably Tripp et al., NeurIPS 2020).\n\n8. In Figures 5 and 6 the x-axis label, \"Number of Oracle\" is somewhat confusing. Perhaps \"queries\" would be more appropriate? Additionally, it would be great if the authors could give the number of random trials for which the error bars are reported?\n\n9. Although it is mentioned in the main text, it might be worth adding the number of trials and the fact that the uncertainty bars are standard errors in the captions for Figures 4, 5, and 6. \n\n10. For Table 1, it would be great if the caption appeared above the table rather than below it.\n\n11. The diagnostic experiment on the misalignment problem in Section H of the appendix is interesting. It would be great to report a quantitative R^2 value in addition to the plots.\n\n12. Line 323, \"This indicates that both the uncertainty of the surrogate model and objective function value need to be considered for exploration and exploitation\". The same can be said about the acquisition function itself?\n\n13. In Algorithm 2 of Section M of the appendix it would be great to explicitly provide the definition of the Calculate subroutine.\n\n14. In Table 3 of Section I of the Appendix it would be worth stating that the rows are fixed wall clock time and fixed oracle calls respectively.\n\n&nbsp;\n\n__REFERENCES__\n\n&nbsp;\n\n[1] Kingma and Welling, Auto-Encoding Variational Bayes, ICLR, 2014.\n\n[2] Stanton, S., Maddox, W., Gruver, N., Maffettone, P., Delaney, E., Greenside, P. and Wilson, A.G., 2022, June. Accelerating Bayesian optimization for biological sequence design with denoising autoencoders. In International Conference on Machine Learning (pp. 20459-20478). PMLR.\n\n[3] Notin, P., Hernández-Lobato, J.M. and Gal, Y., 2021. Improving black-box optimization in VAE latent space using decoder uncertainty. Advances in Neural Information Processing Systems, 34, pp.802-814.\n\n[4] Lu, X., Gonzalez, J., Dai, Z. and Lawrence, N.D., 2018, Structured variationally auto-encoded optimization. In International conference on machine learning (pp. 3267-3275). PMLR.\n\n[5] Siivola, E., Paleyes, A., González, J., & Vehtari, A. (2021). Good practices for Bayesian optimization of high dimensional structured spaces. Applied AI Letters, 2(2), e24.\n\n[6] Maus, N., Wu, K., Eriksson, D. and Gardner, J., 2023, Discovering Many Diverse Solutions with Bayesian Optimization. In International Conference on Artificial Intelligence and Statistics (pp. 1779-1798). PMLR.\n\n[7] Verma, E., Chakraborty, S. and Griffiths, R.R., 2022. High-Dimensional Bayesian optimization with invariance. In ICML Workshop on Adaptive Experimental Design and Active Learning.\n\n[8] Tanimoto TT (17 Nov 1958). \"An Elementary Mathematical theory of Classification and Prediction\". Internal IBM Technical Report. 1957 (8?).\n\n[9] Bajusz, D., Rácz, A. and Héberger, K., 2015. Why is Tanimoto index an appropriate choice for fingerprint-based similarity calculations?. Journal of cheminformatics, 7, pp.1-13.\n\n[10] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L. and Desmaison, A., 2019. PyTorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32.\n\n[11] Balandat, M., Karrer, B., Jiang, D., Daulton, S., Letham, B., Wilson, A.G. and Bakshy, E., 2020. BoTorch: A framework for efficient Monte-Carlo Bayesian optimization. Advances in neural information processing systems, 33, pp.21524-21538.\n\n[12] Gardner, J., Pleiss, G., Weinberger, K.Q., Bindel, D. and Wilson, A.G., 2018. Gpytorch: Blackbox matrix-matrix Gaussian process inference with GPU acceleration. Advances in neural information processing systems, 31.\n\n[13] Gao, W., Fu, T., Sun, J. and Coley, C., 2022. Sample efficiency matters: a benchmark for practical molecular optimization. Advances in Neural Information Processing Systems, 35, pp.21342-21357.\n\n[14] Grosnit, A., Tutunov, R., Maraval, A.M., Griffiths, R.R., Cowen-Rivers, A.I., Yang, L., Zhu, L., Lyu, W., Chen, Z., Wang, J. and Peters, J., 2021. High-dimensional Bayesian optimisation with variational autoencoders and deep metric learning. arXiv preprint arXiv:2106.03609.\n\n&nbsp;", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "- **[W1] Importance of $d_{\\cal X}$ bounding assumption in Proposition 1.**\n    \n    For analytical convenience, we use the bounding assumption of the distance function.  This is similar to the concept of image normalization, where pixel values are scaled to a specific range (often 0 to 1) to facilitate more effective processing and analysis. We would like to note that this scaling does not affect the generality or applicability of our findings, since the underlying principles remain unchanged regardless of the bound.\n    \n- **[W2, L1] Appropriateness of Levenshtein distance for molecules.**\n    \n    Thank you for the good suggestion. Since we evaluate the optimization problem in diverse domains, such as molecule and arithmetic expression tasks, we use a more general distance metric, Levenshtein distance, compared to Tanimoto similarity. Our InvBO can utilize any distance function, including Tanimoto similarity. We will include it in the final version.\n    \n- **[W3] Assessment of InvBO on the Sample Efficiency Matters Benchmark.**\n    \n    Thank you for the suggestion. We conduct additional experiments applying InvBO to LBO in the sample efficiency matters benchmark. We apply InvBO to the SELFIES VAE provided by the benchmark on 7 tasks. The experimental results are in the table below:\n    \n    |  | REINVENT SMILES | Graph GA Fragment | SELFEIS VAE - InvBO | SELFIES VAE - LBO |\n    | --- | --- | --- | --- | --- |\n    | amlodipine_mpo | 0.635 $\\pm$ 0.035 | 0.661 $\\pm$ 0.020 | 0.594 $\\pm$ 0.011 | 0.516 $\\pm$ 0.005  |\n    | median2 | 0.276 $\\pm$ 0.008  | 0.273 $\\pm$ 0.009 | 0.236 $\\pm$ 0.018 | 0.185 $\\pm$ 0.001 |\n    | osimertinib_mpo | 0.837 $\\pm$ 0.009 | 0.831 $\\pm$ 0.005 | 0.823 $\\pm$ 0.007 | 0.765 $\\pm$ 0.002 |\n    | perindopril_mpo | 0.537 $\\pm$ 0.016 | 0.538 $\\pm$ 0.009 | 0.538 $\\pm $ 0.004 | 0.429 $\\pm$ 0.003 |\n    | ranolazine_mpo | 0.760 $\\pm$ 0.009 | 0.728 $\\pm$ 0.012 | 0.762 $\\pm$ 0.011 | 0.452 $\\pm$ 0.025 |\n    | valsartan_smarts | 0.179 $\\pm$ 0.358 | 0.000 $\\pm$ 0.000 | 0.003 $\\pm $ 0.005 | 0.002 $\\pm$ 0.003 |\n    | zaleplon_mpo | 0.358 $\\pm$ 0.062 | 0.346 $\\pm$ 0.032 | 0.384 $\\pm$ 0.008 | 0.206$\\pm$ 0.015 |\n    | SUM | 3.582 | 3.377 | 3.34 | 2.103 |\n    | Rank | 1 | 2 | 3 | 23 |\n    \n    We re-rank the method provided in the sample efficiency matters benchmark on 7 tasks by AUC Top-10 from 5 independent runs. In the table, applying InvBO to SELFIES VAE achieves rank 3 on 7 tasks while vanilla LBO with SELFIES VAE achieves rank 23. These results demonstrate that LBO with InvBO is highly competitive with other non-BO baselines.  \n    \n- **[W4, Q2] Additional comparison to T-LBO and analysis of the interplay between Inversion and metric learning.**\n    \n    We appreciate for valuable suggestion. We provide the optimization results of T-LBO with and without inversion on the arithmetic expression task in Figure 5 in the provided PDF. The metric learning proposed in T-LBO adjusts the latent space to be smooth, making it easier to fit the GP. Meanwhile, inversion constructs an aligned dataset without additional oracle calls, enabling the GP to correctly emulate the objective function. Although these are orthogonal approaches, metric learning helps reduce the $L_3$ constant in Proposition 1, and it is expected to produce a synergistic effect with InvBO. We will include it in the final version.\n    \n- **[W5] Tanimoto similarity between $\\mathbf x$ and $p_\\theta(\\mathbf z)$.**\n    \n    Thank you for the suggestion. We additionally conducted experiments measuring the Tanimoto similarity between $\\mathbf{x}$ and $p_\\theta(\\mathbf{z})$ in Figure 4 in the provided PDF. Figure 4 shows the Tanimoto similarity comparison results with and without inversion on the med2 and valt tasks. We used rdkit library to measure the Tanimoto similarity. From the figure, the inversion achieves a Tanimoto similarity of 1.0 for every iteration, while CoBO without inversion achieves about 0.8 Tanimoto similarities in both tasks.\n    \n- **[W6, L2] Motivation for $\\alpha$ normalization in PAS.**\n    \n    We normalize the $\\alpha$ value to ensure that it has the same influence relative to the anchor point’s objective score $y$. Without scaling, regions with high uncertainty may be overly influenced by the $\\alpha$ value compared to the objective score $y$.\n    \n- **[W7, L3] Providing clear reproduction instructions.**\n    \n    We will publish our code, including clear reproduction instructions in the camera-ready version if the paper gets accepted.\n    \n- **[Q1] Assessment of generated molecules against the pre-training dataset.**\n    \n    \n    |  | med2 | valt |\n    | --- | --- | --- |\n    | Ratio of novel data (%) | 100 $\\pm$ 0.00 | 100 $\\pm$ 0.00 |\n    \n    We additionally conducted experiments to assess whether the generated data are contained within the pre-training Guacamol benchmark dataset. The table above provides the ratio of generated data that is not contained in the pre-training dataset on the med2 and valt tasks across five runs. The table demonstrates that all generated data during the optimization process are not contained in the pre-training dataset.\n    \n- **[Q3] Sparse GP approximation method used in InvBO.**\n    \n    As shown in our codebase, we use Sparse Variational Gaussian Process (SVGP) for the sparse Gaussian process implementation. We will add this detail in our camera-ready version if the paper gets accepted.\n    \n- **[Q4] Discrepancies between Figures 7 and 16.**\n    \n    Good observation. Figures 7 and 16 are different because they are conducted on different splits. Figure 7 illustrates the Gaussian process fitting results on the test set, while Figure 16 shows the results on the training set. \n    \n- **[Minor points in Weakness]**\n  \n    We appreciate your detailed review. We will update the final version considering these minor 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": "mvWhGNpDmJ", "reviewer": "Reviewer_tYNa", "summary": "This paper proposes Inversion-based Latent Bayesian Optimization (InvBO), a plug-and-play module for latent Bayesian optimization (LBO) methods. The key components of InvBO are: 1) An inversion method to find latent codes z that exactly reconstruct input samples x, addressing misalignment between the latent space and input space. 2) A potential-aware trust region anchor selection method that considers both observed function values and the potential of the trust region to improve optimization. Empirically, InvBO boosts the performance of several existing LBO methods on nine tasks spanning molecule design and arithmetic expression fitting. Theoretically, the authors prove InvBO reduces an upper bound on the surrogate model's prediction error within trust regions.", "questions": "Can you comment more on the limitations of LBO and InvBO for practical applications? For example, what size dataset is needed to train an effective VAE model? What happens if the dataset is small?\n\nProposition 1 assumes the inversion mapping exists and has a small error. Can you discuss what happens if these do not hold, for example, if the inverse mapping is ill-posed?\n\nCan you provide any theoretical insight into why considering the acquisition function in anchor selection (PAS) helps? For example, can it be related to the quality of local optima or regret bounds?\n\nWhy is the improvement on DRD3 and the symbolic regression task smaller than on Guacamol? What properties of these tasks make them challenging?\n\nCan you discuss Validating INV and PAS, how about implementing them on LBOs, TuRBO-L, LOL-BO.", "limitations": "Limitations of needing a large dataset to train the VAE\n\nLimitations of the theoretical assumptions, and what happens if they are violated in practice\n\nIncluding this discussion, even if speculative, would help contextualize the contributions for practitioners. The paper already acknowledges several mathematical assumptions. Expanding this to real-world considerations would be a positive addition.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The inversion method is a novel and principled way to address the important problem of misalignment between the input and latent spaces in LBO. The method finds decoder triplets (x, z, y) without additional function evaluations.\n\nPotential-aware trust region anchor selection considers both observed values and the acquisition function when selecting local search regions. This expands on prior methods that only use observed values.\n\nThe theoretical result in Proposition 1 provides justification for the inversion method. It shows that minimizing the reconstruction error d(x, p(z)) with inversion reduces an upper bound on the surrogate model's error.\n\nExtensive experiments demonstrate the effectiveness of InvBO. It improves over several strong LBO baselines on molecule design (Guacamol, DRD3) and symbolic regression tasks. Ablations confirm both INV and PAS contribute to performance.\n\nThe writing is generally clear and easy to follow. Figures, tables and algorithms complement the main text well. The Related Works section provides helpful context.", "weaknesses": "The paper could include more discussion on the limitations and practical considerations of LBO in general. For example, what are the trade-offs compared to standard BO? How much data is needed to train an effective VAE? Some discussion is given but more would help ground the work. What are the real applications of LBO?\n\nProposition 1 relies on several key assumptions, like Lipschitz continuity of f and small reconstruction error d(x, p(z)). While these enable a clean result, it would help to discuss how realistic the assumptions are and what can go wrong if they are violated.\n\nTo validate INV and PAS, it is also important to implement them on LBOs, TuRBO-L, LOL-BO.\n\nThe potential-aware anchor selection mainly seems motivated by intuition. A theoretical grounding, even if loose, could strengthen this contribution. For example, can we say anything about the quality of the local optima or the regret?\n\nThe improvement on DRD3 and the arithmetic expression tasks (Figure 5a) appears smaller than on Guacamol. It would be good to comment on what makes certain tasks more or less challenging for the method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "- **[W1, Q1, L1] Discussion on the limitations and practical considerations of LBO.**\n    \n    Good suggestion. While standard BO struggles with discrete data, LBO addresses this by mapping discrete data to continuous space. To bridge the gap between the discrete and continuous space, LBO utilizes a Variational AutoEncoder (VAE), the quality of which depends on the number of unlabeled data available. Thus, LBO requires a large amount of high-quality unlabeled data to train an effective VAE.\n    \n- **[W2, Q2, L2] Realism and violated situation of assumptions in Proposition 1.**\n    \n    Thank you for your valuable question. The Lipschitz continuity assumption for the objective function $f$ is common in Bayesian optimization [1-5] or global optimization [6]. Furthermore, we have shown that the distance $d_{\\cal X}(\\mathbf x, p_\\theta(\\mathbf z))$ can be reached to zero through our inversion method in Figure 12. These findings imply that our assumptions in Proposition 1 are highly realistic. Notably, the non-zero values of the distance are the main motivation of InvBO. If the assumptions are violated, GP prediction error increases, resulting in suboptimal optimization performance as we have already shown in Figure 8. \n    \n    - Reference\n        \n        [1] González, Javier, et al. \"Batch Bayesian optimization via local penalization.\" *Artificial intelligence and statistics*. PMLR, 2016.\n        \n        [2] Scarlett, Jonathan. \"Tight regret bounds for Bayesian optimization in one dimension.\" *International Conference on Machine Learning*. PMLR, 2018.\n        \n        [3] Hoang, Trong Nghia, et al. \"Decentralized high-dimensional Bayesian optimization with factor graphs.\" *Proceedings of the AAAI Conference on Artificial Intelligence*. Vol. 32. No. 1. 2018.\n        \n        [4] Kim, Jungtaek, and Seungjin Choi. \"On local optimizers of acquisition functions in bayesian optimization.\" *Machine Learning and Knowledge Discovery in Databases: European Conference, ECML PKDD 2020, Ghent, Belgium, September 14–18, 2020, Proceedings, Part II*. Springer International Publishing, 2021.\n        \n        [5] Lee, Seunghun, et al. \"Advancing Bayesian optimization via learning correlated latent space.\" *Advances in Neural Information Processing Systems* 37 (2023).\n        \n        [6] Christodoulos A. Floudas and Panos M. Pardalos, editors. Encyclopedia of Optimization, Second Edition. Springer, 2009.\n\n- **[W3, Q5] Validation of INV and PAS on various LBO methods.**\n    \n    We have already provided the experimental results of applying InvBO over mentioned LBOs, TuRBO-L, and LOL-BO in Table 2 of Section D. Table 2 shows that applying our InvBO to previous trust region-based LBOs (TuRBO-L, LOLBO, CoBO) and non-trust region-based LBOs (LBO, W-LBO, PG-LBO) consistently improves the optimization performance in 8 tasks.\n    \n- **[W4, Q3, L3] Theoretical grounding for PAS.**\n    \n    Thank you for the good suggestion. We also have a deep interest in the theoretical grounding for PAS and recognize its significance. To provide a comprehensive theoretical analysis of PAS, we are actively conducting ongoing research in this area. We anticipate sharing these findings in future work.\n    \n- **[W5, Q4] Analysis of performance improvement on DRD3 and Arithmetic expression tasks.**\n    \n    In the DRD3 and arithmetic expression tasks, the dissimilarity between $\\mathbf{x}$ and $p_\\theta(\\mathbf{z})$ in previous LBOs without InvBO is relatively small, resulting in less room for performance improvement from inversion compared to other tasks. We conducted additional experiments measuring normalized Levenshtein distance between $\\mathbf x$ and $p_\\theta(\\mathbf z)$ with and without inversion on DRD3 and arithmetic expression tasks in Figure 6. in the provided PDF. Compared to Figure 12 in Section C which has a mean of dissimilarity of 0.3, the mean of dissimilarity of these two tasks is lower than 0.1.", "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:
Thank you to the reviewers for the thorough feedback on our paper. Based on the reviews, we have organized the key strengths of our paper that reviewers identified:

### **1. Convincing motivation.**

Most of the reviewers (8vpX, wVbQ, nFyz, tYNa) provided positive feedback on our motivation. We address the misalignment problem that has been overlooked by prior LBO works, despite its presence in all VAE-based Bayesian optimization architectures.

### **2. Superior optimization performance.**

To validate the effectiveness of InvBO, we measured the optimization performance on nine different tasks. Reviewers (8vpX, wVbQ) responded positively to the superior optimization performance of InvBO, especially in the low-budget setting.

### **3. Novelty.**

Reviewers highlighted the novelty as a strength of InvBO. InvBO consists of two components: the inversion and the PAS method. (wVbQ, tYNa) The inversion method provides a novel and principled way to address the misalignment problem. (wVbQ) The PAS method improves the trust region anchor selection, which most previous works approached with ad-hoc strategies.

### **4. Generality.**

Reviewers (8vpX, nFyz) appreciated the generality of InvBO. As mentioned in the paper, InvBO is a plug-and-play algorithm compatible with previous LBOs. Furthermore, we provided experimental results showing that PAS can be applied to TuRBO on standard BO tasks (e.g., non-LBO tasks) in Figure 2 of the provided PDF.

### **5. Thorough analysis.**

Reviewers gave positive feedback on our thorough analysis of InvBO. Reviewers (8vpX, tYNa) noted that the ablation studies of InvBO demonstrated the effectiveness of each component. Additionally, the experiments on GP fitting provided strong evidence of the effectiveness of the inversion method (8vpX, nFyz).

---

We appreciate all the reviewers for their thoughtful feedback. We will address all issues raised by the reviewers below.

Meta-review: This paper overall is quite strong, and the reviewers are nearly unanimous about this. The authors have clearly identified a subtle but very important and, in hindsight obvious, problem with existing state of the art methods in latent space BO and their approach to solving it "feels" like the right one. This is clearly borne out in empirical evaluation, and the authors' method is clearly quite strong. It feels like essentially every software implementation of LOL-BO should probably adopt InvBO as an on-by-default feature. The fact that so many oracle calls can be saved by "in model" changes since they aren't being done directly in service of optimization is great.
Decision: Accept (poster)