MITIGATING OVER-EXPLORATION IN LATENT SPACE OPTIMIZATION USING LES
Authors: Omer Ronen
Ahmed Imtiaz Humayun
Richard Baraniuk
Randall Balestriero
Bin Yu
Venue: ICML 2025 poster (2025)
Keywords: VAE
Drug-Design
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: We develop a method to mitigate over-exploration in Latent Space Optimization

Abstract: We develop Latent Exploration Score (LES) to mitigate over-exploration in Latent Space Optimization (LSO), a popular method for solving black-box discrete optimization problems. LSO utilizes continuous optimization within the latent space of a Variational Autoencoder (VAE) and is known to be susceptible to over-exploration, which manifests in unrealistic solutions that reduce its practicality. LES leverages the trained decoder’s approximation of the data distribution, and can be employed with any VAE decoder–including pretrained ones–without additional training, architectural changes or access to the training data. Our evaluation across five LSO benchmark tasks and twenty-two VAE models demonstrates that LES always enhances the quality of the solutions while maintaining high objective values, leading to improvements over existing solutions in most cases. We believe that new avenues to LSO will be opened by LES’ ability to identify out of distribution areas, differentiability, and computational tractability.

Mitigating over-exploration in latent space optimization using LES
Omer Ronen' Ahmed Imtiaz Humayun? Richard Baraniuk?
Randall Balestriero Bin Yu'
Abstract
We develop Latent Exploration Score (LES) to mitigate over-exploration in Latent Space Op- timization (LSO),, a popular method for solv- ing black-box discrete optimization problems. LSO utilizes continuous optimization within the latent space of Variational Autoencoder (VAE) and is known to be susceptible to over- exploration, which manifests in unrealistic solu- tions that reduce its practicality. LES leverages the trained decoder's approximation of the data distribution, and can be employed with any VAE decoder-includee pretrained ones wetthut addi- tional training, architectural changes access to the training data. Our evaluation across five LSO benchmark tasks and twenty-two VAE mod- els demonstrates that LES always enhances the quality of the solutions while maintaining high objective values, leading to improvements over ex- isting solutions in most cases. We believe that new avenues to LSO will be opened by LES' ability to identify out of distribution areas, differentiability, and computational tractability.
111+x+snn(1/3)
x2+sin(s++11n)
x2+xxp(3)+(+++
Figure I. Incorporating LES promotes valid solutions. We con- sider the task of approximating the expression 1/3 + + sin (x * x), using LSO. Optimization trajectories with (blue) and without (red) LES constraint in the latent space of VAE are projected onto a two-dimensional subspace that contains the starting point and the end-points obtained after 10 gradient ascent steps. In the left panel, we show the LES score for latent vectors on the pro-dimensional subspace, with darker shades correspond- ing to lower LES. In the right panel, we show the validity of the decoder oulputs for each latent vector, with orange denoting in- valid generations. High LES values correlate with valid areas, and incorporating LES LSO produce an expression that adheres to the grammatical rules of Example 1.1.
et al., 2018). LSO transfers the optimization problem to the domain of the latent space of VAF, which can be ef- ficiently explored using continuous optimization methods. However, ensuring that LSO solutions respect the structure of the original space remains a challenge. To illustrate this issue, we provide some examples.
1. Introduction
Many important tasks in scientific fields, such as small molecule discovery and protein engineering, can be framed as discrete black-box optimization problems. In con- trast to conditional sampling-based approaches. including GFlowNet (Bengio et al., 2023) and Diffusion (Corso al., 2022; Igashov et al., 2024), which are better suited for appli- cations like linker design (Du et al., 2024), optimization is particularly effective when the goal is to improve a specific property, such as enhancing a drug's safety.
Example 1.1 (Arithmetic expressions). An expression built up using numbers, arithmetic operators and parentheses is called arithmetic expression. However, not every se- quence of the above elements correspond to a valid expres- sion. For instance, the expression "sin (x) + xx is valid expression while "sin (xxx" is not.
Example 1.2 (Simplified molecular-input line-entry sys- tem (SMILES)). SMILES provides a syntax to describe molecules using short ASCII strings. Atoms are represented by letters (e.g.. water:"O") bonds are represented by sym- bols (e.g., triple: "#", double: "=", ...), ), branches are represented in parentheses and cyclic structures are repre- sented by inserting numbers at the beginning and the end. Like the arithmetic expressions case, not every combina- tion of the elements described above corresponds to valid molecule. For example, while "ciceccc1" " is valid, both "C1 CCCCC2 and " "1 CCCCC) are not.
LSO was recently developed to enhance the sample effi- ciency of discrete optimization algorithms, such genetic algorithms, in the black-box setting (Gómez-Bombarelli
'CC Berkeley RRice Univeristy 'Brown University. Correspon- dence to: Omer Ronen <omer ronen @ berkeley.cdu>>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Mitigating over-exploration in latent space optimization using LES
Example 1.3 (Quality filters for molecules). Chemists seek molecules that not only optimize desired chemical proper- ties but are also stable and easy to synthesize. This has led to the development of rules such as Lipinski's 's Rule of Five IRO5 Lipinski et al. (1997)), which helps determine if the bioavailability (i.e., the proportion of a drug or other sub- stance that enters the circulation when introduced into the body) of a given compound meets a certain threshold. For example, RO5 suggests that poor absorption is more likely when the octanol-water partition coefficient (logP) exceeds 5. Similarly, the Pan Assay Interference Compounds (Baell false & Holloway, 2010) filter helps in identifying false positives in assay screenings. The rd. fiiees (Walters, 2019) puckuge has curated many such rules and is considered a "high precision, low recall surrogate measure" " (Brown et al., 2019). Following (Notin al., 2021) we consider sample valid if it passes the rd.filters filters'
Our results demonstrate that LES is highly effective identifying regions that preserve the structure of the sequence space. Although LES' computation scales cu- bically with the latent dimension, it is up to 80% faster than the current state-of-the-an for identifying out-of- distribution data points in the latent space of generative models for discrete sequences (Tables and 22).
We develop a numerically stable optimization proce- dure to incorporate LES constraint in LSO.
We evaluate LES-coontra LSO across thirty opti- mization tasks, including twenty-two VAEs and five benchmark problems, demonstrating its robustness in generating valid solutions and achieving high objective values. Specifically, in 19 out of the 30 LSO exper- iments, our method either finds the best solution on average or achieves a solution within I standard devi- ation of the best solution across 10 independent runs. This outperforms the six alternative methods we con- sidered by 19% (Tables 20 and 21).
Numerous directions have been explored to overcome the challenge of providing valid solutions, including special- ized VAE architectures (Kusner al., 2017: Jin et al., 2018) or robust representations for discrete data (Krenn et al., 2020). Additionally, constrained objectives can be formu- lated under the assumption that one has access to a function which quantifies the validity of any point in the latent space (Griffiths& & HeenándezzLobbtoo 2020). However, in many realistic scenarios, such as Example 1.3, these solutions may not be directly applicable, as the structure of the sequence space may not be sufficiently well understood. To address this, (Notin et al., 2021) proposed using an estimator of the uncertainty of the decoder, based on the variational approxi- mation to a posterior distribution over the VAE parameters, encouraging LSO to respect the sequence space structure (details are provided Appendix D). Although this approach proved effective, the non-differentiable nature of the un- certainty score required its integration into LSO through heuristic approaches. Additionally, the computation of the uncertainty score is not exact (i.e., it relies on variational approximation and Monte Carlo sampling) and requires sig- nificant amount of time to compute. Therefore, there is a need for robust methods that work across different VAFS and sequence space structures, and can be easily integrated into existing LSO pipelines.
2. Background: Latent Space Optimization
LSO is a method for solving black box optimization prob- lems in discrete and structured spaces, such as the space of valid arithmetic expressions. Formally, let V C R be discrete and structured space, represented as a sequence of L. one-hot vectors of dimension D. We represent sequences of length L of categorical variables with D categories. L is set as the maximum sequence length that we are optimizing for, and one of the D categories is used as an "empty" category (which enables generating sequences shorter than the max- imal length). For instance, in the case of valid arithmetic expressions, V would be the set of sequences that define such expressions. Let M: V + R be the objective function. LSO solves,
arg max M(z). DEV
In this setting, we assume that evaluations of the objective function (M) are expensive conduct. For example, the objective may be the binding affinity of a compound to given protein, measured through a wet lab experiment.
To achieve this goal, we develop LES, score that can be used as a constraint in LSO optimization to increase the number of solutions that respect a given structure. The distinctive characteristics of LES are differentiability and robustness that allow its easy integration into existing LSO pipelines. Specifically, our contributions are as follows:
A popular approach to solve Equation (1) is Bayesian Op- timization (BO), which utilizes first order optimization of a surrogate model for M. However, since the space is dis- crete, first order optimization cannot be directly applied. In an attempt to make BO applicable for solving Equa- tion (1), (Gómez-Bombarce et al., 2018) proposed to trans- fer the optimization problem into that over a domain of the latent space of deep generative model and subsequently perform BO in this space. The main idea is to (1) learn a continuous representation of the discrete objects (c.g., us- ing a VAE) and (2) perform BO in the latent space while
We introduce LES, a score that achieves higher values in regions of the latent space closer to the training data.
'e use the default Inpharmatica rule set comprised of 91 alerts
2

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Mitigating over-exploration in latent space optimization using LES
decoding the solution at each step. Formally, given pre- trained encoder (Eg) and decoder (Go) the initial labelled dataset D { { 3/ Hi= is first encoded into the latent space D= = { = (x4), Yi R- Using the encoded dataset, a BO procedure is conducted, which we describe in Algorithm/ 1. Most commonly, a Gaussian process is used as the surrogate model for M, and the acquisition function the expected improvement (Frazier, 2018), defined as:
The derivation of our score leverages the Continuous Piece- wise Affine (CPA) representation of neural networks, which we briefly review below.
Deep generative networks as CPA Following (Humayun et al., 2022; 2021; Balestriero & Baraniuk, 2018; Balestriero et al., 2024), we consider the representation of Deep Gen- erative Networks (DGNs) as Continues Piecewise Affine (CPA) Splines operators. Let fo be any neural network with affine layers and piecewise affine activations then it holds that
4y(x) = B max(f(z) - max ye. 0)
So(2)= = - (A bw) [(EE])
(5)
where the expectation is with respect to the distribution of the function f. conditioned on D=
WEB
where Q is the input space partition induced by fo, w is a particular region and the parameters A. and b, defines the affine transformation depending on w.
Algorithm
Latent Space Optimization
for { to T do
1. Fit a surrogate model f to the encoded dataset, D
In cases where the neural network So is not composed solely of piecewise affine layers and activations, we leverage the re- sult from (Daubechies al., 2022) to assert that Equation (5) either exactly represents fo provides a sufficiently accu- rate approximation for our practical purposes (Humayun et al., 2022). We therefore argue that all the decoder neu- ral networks included in our study (i.e., GRU, LSTM, and Transformers) can be approximated with high accuracy as continuous piecewise affine (CPA) functions.
2. Generate a new batch of query points by optimizing chosen acquisition function (A4)
(new) arg max A Az(z)
(3)
3. Decode (e(ww) = (((2(()w)) evaluate the correspond- ing true objective values (n)w)) = M(zx(new) and update D= with (z (new), (new)).
3. A Latent Exploration Score to Reduce Over-Exploration in LSO
Over-exploration in LSO Multiple studies (Notin et al., 2021; Kusner et al., 2017) have found that unconstrained latent space optimization (LSO) often yields solutions that disregard the aforementioned structures. For example, when searching for arithmetic expressions, invalid equations like "ssin(zza" frequently occur. Similarly, many solutions in molecule searches fail to pass basic quality filters (Exam- plc 1.3), limiting their practical utility (Maus et al., 2022).
In this section, we introduce Latent Exploration Score (LES), our new score reduce over-exploration in LSO, We begin by motivating LES and proceed to formally derive it in Section 3. In Section 3.2, we provide empirical evi- dence that LES gives higher values in the latent space valid set. The use of LES to regularize LSO is left for Section 4.
Our goal is to develop a meaningful constraint for optimizing the acquisition function within a latent space of given VAE Specifically, we aim to construct con- straint that is a continuous function of z with higher values, indicating that it is more likely that z resides within valid regions of the latent space (Equation (4)).
Motivation
While acquisition functions such as expected improvement (Equation (2)) are designed to balance exploration and ex- ploitation based on the estimated uncertainty from the Gaus- sian process model for M. The frequent generation of invalid solutions during acquisition optimization, which im- plics that the estimated uncertainty can be problematic in this setting, underscores the need for additional regulariza- tion (Tripp et al., 2020), which we aim to address.
Such a score should be higher in regions near training data points, assuming most of VAE training data is valid. To achieve this, we treat the latent space of the VAE a proba- bility space, i.e. ~ Pz. for some prior distribution p (for example standard Gaussian). The prior should reflect our best guess for the distribution of the observed data in the latent space. Solutions are mapped back to sequences by the decoder through a deterministic (we do not consider x to follow a conditional distribution given x) transformation of the latent vectors. Therefore, any distribution on the latent
To mitigate ever-xxploration, we propose adding a penalty to Equation (3). The penalty uses a new score, giving higher values over the latent space valid set, defined as:
{z; Go (z) E V},
(4)
where Go : Z + LL is the decoder network, and V C LLxD is the set of valid sequences.
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Mitigating over-exploration in latent space optimization using LES
space defines a distribution over the space of sequences. Our score the density function of the push-forward measure of æ (z)z), which we call the sequence density, Consequentially, our score depends only on the decoder network, not the encoder, and can potentially he applied to other generative models like GANs or diffusion models.
Lø(2) is a D L logits matrix (D vocabulary size, L sequence length) and Softmax is the softmax operation ap- plied to every column of Lg (z). In order to avoid a violation of the assumption that Go is bijective, we extend the func- tion's output to include the normalizing constant for each column. We find that parametrizing the output to include the inverse the normalizing constant (Equation (8)), helps in avoiding numerical instabilities that are caused by the constant being potentially very large. With this formulation, we can now derive the sequence density function. Theorem 3.1 (DGN sequence density). Let
Why use the sequence density function? We argue that for a well-trained decoder network, the density should be higher in areas of the sequence space close to the training data. To see why, consider decoding model Gg trained a dataset ((z,, Z; 2;)) Ya=r- The average loss (L) at z is
Ga(z) = (p ( ..... (.)((((1)-)) 0x (8)
(()(==)) = L= (Go x).
(6)
As the training process is designed to minimize the popula- tion loss: EE( Go(z)), if successful, we hypothesize that the distribution of Gz(z) puts higher weight in the areas where (=)) is low. Since we expect most of the training data to be valid and to achieve low expected loss, the sequence density should put higher weight on the latent space valid set. In, Section 3.2 we provide an empirical validation for this hypothesis, for Examples 1.1 to 1.3. We highlight that this relationship between the valid set and the sequence density depends on how well the decoder fits the data.
where ( = Softmax(Lo(z)) and c = EEn (Lo()))
Assume that Lug is bijective and can be expressed as a CPA (Equation (5)), and that z ~ Pz, then the density function of Go(z) is given by:
L (1ml
5 det
(9)
for
T
1
1
3.1. Derivation of LES
B. =
diag
(10)
(p
(8) )
(p
(1)
)D
Cz
(8)
Encoder Decoder
'SM GofdMax
A (A
(11)
1
cooc0o1
(.-))((++--)).
Continous Latent space
Gp = SM(LL)
Encoder
Decoder
where (A ..... A! is the Penrose inverse of A (2)
A and Sy is the density function of pz-
CPA representation
LES
The proof is provided in Appendix A. We define LES to be the logarithm of the determinant term,
Logits as CPA
fp
Lx(x)= (A byyl Deil
( det Σ E(AT)(BBBBBA (12) 1=1
LES (pushtowar))
S(z) = log
8(x) = (G,(x))
as the contribution the prior is negligible in magnitude in all the decoders we study. Remark 3.2. LES can be calculated directly without assum- ing the decoder logits follow a CPA function (Ben-Israel, 1999). However, using the expessions derived in Equa- tion (12) has two computational benefits for calculating the derivative of LES. First, using Jacobi's formula and observ- ing that Σ (A))* (B,) A is quadratic formula of the softmax probabilites, we can calculate the derivative of LES in closed form. Second, by the CPA assumption, the matrices ( are a constant function of z and therefore
Figure 2. Derivation of LES. The decoder network (Ga), which maps from the latent space to the output space, assumed to the composition of softmax operation over continuous piecewise affine (CPA) spline operator. LES the density of random variable (z) in the latent space, under the decoder transformation. Calculating LES only requires a pre-trained decoder.
Analytical formula for LES DGNs for discrete se- quences typically output a matrix of logits, transformed into normalized scores by the softmax function:
aA
0. As result, we avoid the need to calculate the hessian of the decoder when taking the derivative of LES.
Go (z) (efimax(Log(])
dz
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Mitigating over-exploration in latent space optimization using LES
Limitations of Theorem 3.1 Our derivation relies on the decoder logits being (i) CPA operator and (ii) bijective be- tween the latent space and the generated manifold in the am- bient space. We argue that (i) is not restrictive assumption, as approximation theory has already demonstrated that any continuous model can be approximated by a CPA network. Therefore, one always recovers Theorem 3.1 even when using non-CPA models. Specificallly, Daubechies et al. 2022 show that ReLU networks (which are CPA spline functions) can approximate refinable functions-a class of non-smooth functions-in addition to known results on other classes of functions (e.g., Weierstrass, polynomials, and analytic functions).
of the decoder (d) with complexity of O(d)).
In Table I we provide the wall clock times for calculating LES for a batch of 20 latent vectors across all architectures and datasets studies in this work. LES is compared with the Bayesian uncertainty score proposed by Notin et al. 2021 (with the default configuration: 10 sampled models and 40 sampled outcomes), which was previously used to regularize LSO. We also compare with a Likelihood score:
((z) -max PG, (X = x Z =)
(13)
where Pc. (2) is the distribution defined by the decoder softmax probabílities, which reflects the likelihood of the most likely æ conditioned on the latent vector 2.
On the other hand, (ii) is a stronger assumption that practi- tioners should be mindful of, as it would invalidate LES as meaningful metric for comparing different samples. For (ii) to be violated, i.e., for Equation (29) to be incorrect, the
For 9 out of the 10 models, particularly when the latent dimension is larger (c.g., SMILES and SELFIES), LES is computed faster than the Uncertainty score, achieving reduc- tions of up to 85% in some cases. As the Likelihood score requires only a single forward pass of the decoder, it offers more computationally efficient alternative, which comes with some performance trade-offs (Sections 3.2 and 4).
Lebesgue measure of the set Ca. {zl 3z" Go(z) = Ga(z") must be larger than 0 (see Lemma A.2 for proof). This would suggest some degeneracy in the decoder func- tion, where large regions of the latent space map to the same output, resulting in a zero gradient of the decoder with re- spect to its imput. However, we believe this is rare in practice. In our experiments, where decoder with respect to the input to calculate LES, we did not encounter instances where the gradient was zero.
Table 1. Wall clack times in seconds (lower is hetter) for calculating LES, the Bayesian uncertainty and the Likelihood scores for sample of 20 latent vectors on a single A100 GPU.
compute the gradient of the
Although we do not formally validate (ii) (the bijectivity assumption), we argue through our empirical analysis in Table 22, conducted across 22 VAEs (including pre-trained models), that (ii) may hold or, at the very least, serve reasonable approximation for real-world VAEs.
Dalasel
Arch. (dim..)
LES
Uncertainty
Likelihood
GRU (25)
0.730
0.823
0.025
Expressions
LSTM (25)
0.164
0.857
0,049
Transformer (25)
0.526
0.481
0.029
GRU (56)
0.663
3.157
0,103
SMILES
I.TTM (56)
0.696
3,990
0.123
Lastly, it is crucial to highlight that the ability of LES to detect out-of-distribution data is closely tied to the decoder's capacity to accurately model the data. Although Theo- rem 3.1 holds for poorly trained decoders under the given conditions, we advise against relying on LES in such cases.
Transformer (36)
0
0.726
0.185
GRU (75)
0.498
1.925
0.064
1.5TM (75)
0.525
2.442
0.071
SELFIES
Transfommrr (75)
0.451
0.583
0.085
Transformer (256)
7.422
42,882
0.410
Average
1.226
5.787
0.114
Computing LES LES a function of the matrices B; and Ag.. The matrices B, are a function of the logits and can be calculated using single forward run of Gg. The matrix A is equal to the derivative of Lo at z, and can therefore can be obtained using automatic differentiation ((aszkee et al., 2017)). To avoid the (pseudo) inversion of the matrix A we can exploit the inverse function the- orem which states that JG,1 8 = (JG,) -1 and we can take S(z) -0.5 log det((A,, (Go (A,, "Go(To)) )r), 8L.
3.2. Validating the Relationship Between LES and Valid Generation
To assess LES's ability to identify valid regions (as de- fined in Examples 1.1 to 1.3), we sample data points in the latent space using the twenty-two VAEs studied in Sec- tion 4. Specifically, we sample 500 data points from three distributions: train, prior (A I)), and -ut-of-distribution (N I 5)). We decode each data point and determine if the decoded sequence is valid.
8L.
where (G.(Lo) (derivative of the softmax w.r.t the logits) aL.
admits a simple closed form solution. Ideally, LES can be computed by performing all of the above calculations in parallel using a single forward call to the G2 network. In ad- dition, the computation of the determinant is done via SVD on a square matrix whose dimension the latent dimension
Identifying if a point in the latent space decodes into a valid sequence can be viewed as a classification problem, in which the different scores (i.c., LES or the Bayesian un- certainty score) provide (unnormalized) probabilities for a
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focusing on varying the decoder architectures and the 3 pa- rameter, which controls the trade-off between reconstruction loss and alignment with the prior (KL. divergence term). All models use a convolutional encoder based on the architec- ture proposed by Kusner al. (2017), and were trained for Mitigating over-exploration in latent space optimization using LES
it can easily be used to constrain any optimization problem. Therefore, propose adding an explicit constraint Equa- tion (3), encouraging the solution to achieve a high LES value. We modify Algorithm by penalizing step (2):
Table 2. AUROC values (higher is hetter) for identifying valid data points within the latent space, across datasets and decoder architectures. Data points are sampled from the training data, the VAE prior (standard Gaussian), and out-of-distribution data (Gaussian with std of 5). LES achieves the best performance in most cases (18 out of 22) and on average.
new
arg max A(f(=))+ + AS(z).
(14)
Dataset
Arch.
LES Prior Uncertainty Likelihond
0.94
GRU
0.1 0.94
0.12
0.81
0.94
4.1. Experimental Setup
0.91
0.87
0.92
0.05
0.99
0.07
0.99
0.98
SMILES
LSTM
0.89
0.89
0,90
VAE models To evaluate the effectiveness of LES as reg- ularization method for LSO, we trained twenty-two VAEs,
1.0 0.97
0.12
0.98
0.96
0.05
0.93
0.14
0.84
0.92
Transformer
0.1 0..4
0.14
0.87
0.93
1.0
0.97
0.10
0.89
0.95
0.96
0.38
0.88
GRU
0.1 0.94
0.42
0.94
0.80
0.05
0,96
0.38
0.90
0.96
Expressions
LSTM
0.37
1,0 0,95
0.56
0.91
0.91
300 epochs using the Adam optimizer (Kingma, 2014) with
0.90
learning rate of 1e-3 and batch size of 256.
Transformer
0.1 0,91
0.53
0.87
0.89
0.92
The VAEs for the Expressions dataset, sourced from Kusner et al. (2017), were trained on 80k data points with a latent dimension of 25. The SMILES VAES were trained on the ZINC250K dataset, consisting of approximately 250k drug- like molecules in SMILES format. Following Kusner et al. (2017) and Notin et al. (2021), latent dimension of 56 was used. For the SELFIES VAES, we used subset of approximately 200k molecules from the ZINC250K dataset that passed a set of quality filters (Walters, 2019), using the SELFIES representation (Krenn et al., 2020), with a latent dimension of 75, Additionally, the pre-trained VAE by Maus et al. (2022) had latent dimension of 256.
0,05
1.0
0.02
0.99
0.97
SELFEES
Transformer 0.1
0.99
0.03
0.96
0.98
1,0
0.95
0.06
0.85
0.93
SELFIES
((Maus et al., 2022)) Transformer
0.75
0.69
0.33
0.70
Avemae
0.91
sequence being valid. We measure the performance of these scores using the AUROC metric. Besides LES, the Bayesian uncertainty score and the Likelihood score, we add three additional baseline scores for comparison. The first is the density of a standard Gaussian (Prior), which is the distri- bution the latent vectors are regularized to follow during VAE training. The second is the polarity score (Polarity) (Humayun et al., 2022), based only on the derivative of the decoder logits with respect to the latent vector, which shows the gains due to accounting for the softmax non-linearity in the derivation of LES (Theorem 3. We also consider the average distance to the closest three data points within a random sample of 1000 points from the training data in the latent space (Train distances).
LSO setup We begin by training a single-task Gaussian Process on an initial dataset. Fach sample is mapped to a latent space and paired with its true objective value. For Expressions and SMILES VAEs, we use 500 data points. For SELFIES VAES, we use 1500. Across all tasks, we generate 500 candidate solutions per problem, aligned with prior work (Kusner et al., 2017: Notin et al., 2021) and reflective of real-world wet-lab constraints (Gao et al., 2022). Solutions are proposed in batches of 20.
The results are shown in Tables 2 and 22. LES provides the best performance in 18 out of the 22 VAEs in this analysis, and in all cases provides a clear signal for identifying valid regions, as indicated by AUROC values that are at least 0.75. This is while being much faster to compute than the Uncertainty score (Table 1) and without the need store a potentially large array of latent vectors.
For the SELFIES VAEs (both from Maus et al. (2022) and those trained by us), we employ a deep kernel that reduces the latent space to 12 dimensions before fitting the Gaussian Process. To mitigate vanishing gradients, we use log ex- pected improvemett (Ament et al., 2024) as our acquisition function, which is sequentially maximized.
4. LES-Consttrained LSO
The Expressions dataset consists of arithmetic expressions that are functions of single variable (e.g., (x), + I * z). Our objective is to find an ex- pression that approximates 1/3 + x + sin (x x), as described by Kusner et al. (2017). The optimization target
Optimization tasks
In Section 3.2 we showed that LES is a robust score that obtains higher values in the latent space valid set (Equa- tion (4)). Furthermore, LES is differentiable, which means
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Mitigating over-exploraiiinn in latent space optimization using LES
is defined as M(a)) log(1 + MSE(2)), where MSE(2) is the mean-squared error between the expression æ and
the default approach in the BoTorch package (Balandat et al., 2020). Since recent ofate-of-the-art LSO pipelines (Maus et al., 2022; Lee et al., 2024) have utilized trust regions centered around the best observed value (Eriks- son et al., 2019), we also compare with this approach (TuRBO). Lastly, we implement the Uncertainty Censoring (UC) method proposed by Notin et al. (2021), which sug- gests early stopping of the optimization when the estimated uncertainty exceeds certain threshold. For this threshold, we use the 99th percentile of the observed uncertainty values in the training data, as recommended by Notin et al. (2021).
1/3 + x + sin (x x) evaluated over the range -10 to 10 using grid of 1000 equally spaced points.
For the SMILES dataset, our goal is to maximize the octanol- water partition coefficient, which is calculated using the prediction model developed by Wildman & Crippen (1999). In the case of the SELFIES dataset, following Maus al. (2022), we focus on three objectives: Perindopril MPO, Ra- nolazine MPO, and Zalepion MPO, all of which are part of the Guacamol benchmarks (Brown et al., 2019). While the SELFIES syntax is 100% robust, we consider only so- lutions that pass a set of quality filters for evaluation (see Example 1.3 for more details).
Hyperparameters We calibrate the step size, which af- fects LES, UC, Prior, Likelihood, and LSO (GA), en- sure our gradient ascent procedure (with À 0) improves the acquisition function values across different initializa- tions. The same step size is applied to all models within the same dataset:
LES-constrained LSO For ease of implementation and numerical stability, when applying LES regularization, we adopt simple optimization procedure in which the acqui- sition function is optimized using 10 steps of normalized gradient ascent. This is because we empirically find that the norm of the derivative of the constraint (i.e., S(z)) is typically much larger than the norm of the derivative of the acquisition function. As result, using the gradient ascent update rule z 2(++1) (=()(=())) + XOS(2 leads a numerically unstable optimization process. To address this issue, we propose the following update rule:
Expressions = 0.8, SMILES = 0..003, SELFIES = 0.03, and SELFIES pre-trained = 0.3. The LSO (L-BFGS) method has a single hyperparameter, the facet length, which is set 5. For TuRBO, there are three primary hyperparameters: the initial length, which we set to 0.8, along with the success and failure tolerances, determin- ing when to expand or shrink the trust region, set at 10 and 2, respectively. An ablation study is provided in Appendix B.
4.2. Results
z ((++1_ )A(((Z()))
(8)(2(0)
(15)
(BA(((
(1))2
Optimization results The experimental results, present- ing the average across 10 independent LSO runs for the top 20 and best solutions found, are summarized in Tables 20 and 21 respectively. In both cases, LES achieves the average best performance most frequently (22 and 17 out of 30 times, respectively). Furthermore, LES outperforms other meth- ods in both the average ranking across LSO tasks and the frequency with which it falls within one standard deviation of the best-performing method (Table 3). These findings demonstrate that using LES as a regularization technique generally enhances optimization performance.
We set 0.05 for Expressions. = 0.1 for our SELFIES models, and x 0.5 for SMILES and the pre-trained SELFIES-VAEE VAE. For Ranolazine MPO with pre- trained SELFIES-VAL VAE, we use X = 0.1 to prevent over- regularization, as regularized methods already yield a high percentage of valid solutions (Table 24). We select these values because, without regularization, the SMILES and pre-trained SELFIES models tend to produce lower per- centage of valid solutions (Table 24). Based on our findings, suggest x 0.5 reasonable default value, while leaving the exploration of an optimal choice for future work.
Figures and 4 show the cumulative average top-20 and the best objective values during BO with the pre-trained SELFIES-VAL (Maus et al., 2022), respectiveley. Results align with (Maus et al.. 2022) under our realistic evalua- tion budget. For the average of the top-20 solutions on the Ranolazine MPO and Zaleplon MPO tasks, LES outper- forms TuRBO on valid solutions but underperforms overall, highlighting that LES effectively constrains the optimzation.
Benchmark methods We compare our LES-conntrained method (LES) with five alternative approaches for op- timizing the acquisition function. First, we evaluate a non-regularized version of Equation (15) where = (LSO (GA)), and a two regularization methods that uses the prior density (i.e.,& bg regularization) and likelihood (Equa- tion (13)) instead of LES (Prior, and Likelihood respec- tively), using similar À values described above.
Table 24 shows the percentage of valid solutions found by each method across datasets and VAES. LES improves upon the non-regularized version of gradient ascent by 7% on average and upon TuRBO and LSO (L-BFGS) by 24% and 36% on average respectively. While UC achieves a 2% higher percentage of valid solutions on average, we show
Additionally, we consider optimizing the acquisition func- tion using the limited-memory BFGS method within a sym- metric hypercube centered at O (LSO (L-BFGS)), which is
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Mitigating over-exploration in latent space optimization using LES
Table 3. Summary metrics for 30 Bayesian Optimization experi- ments from Tables 20 and 21. We report the average rank (lower is better) and the count of times each method is within one standard deviation of the best. Results cover both the best solution and the top-20 average. LES outperforms alternatives on metrics.
0.6
Ranolazine MPO
0.5
Top
Top 20
0.4
Method
Avg Rank
# within
Avg Rank
within
0.3
LES
2.15
19
1.9
Likelihood
2.76
0.2
3.15
3.86
0.1
UC
4.17
5
4.3
TuRBO
5.7
2
6.1
0.0
LSO (L-BFGS)
6.17
1
64
100
200
300
400
500
Number of evaluations
0.5
in Table 23 that for the Expressions datasets, where UC excels, the number of valid produced by LES solutions be increased by setting higher À value. However, this did not improve optimization performance.
Perindopri MPO
0.4
0.3
0.2
5. Discussion
We proposed LES to mitigate over-exploration latent space optimization (LSO). LES is differentiable and fully parallelizable. Extensive evaluations demonstrate that in- corporating LES as penalty in LSO consistently enhances solution quality and objective outcomes. Moreover, LES outperforms alternative regularization techniques, proving to be the most robust across diverse datasets and varying definitions of validity. In addition, L.E has only a single hy- perparameter (the regularization strength), and we observe empirically that deploying LES can provide significant per- formance gains. We therefore believe LES offers powerful approach for discovering more realistic solutions, when the criteria for realism are difficult to define or validate.
0.1
100
200
300
400
Number of evaluations
0.4
Zaleplon MPO
0.3
0.2
0.1
Valid
While LES is fully parallelizable, it requires the calculation of the derivative of the decoder as well as the determinant of the change-of varviaal term, which can be computationally expensive. This step of the output and the latent dimension are both large. It is left for future work to develop a fast approximation for this operation in order to enable the use of LES in applications involving large generative models.
100
200
300
400
500
Number of evaluations
LSO (GA)
LES
UC
LSO (L-BFGS)
become bottleneck when the size
Likellhood
TuRBO
Prior
Figure 3. Cumulative objective for the top-20 solutions found during Bayesian optimization with the pre-trained SELFIES VAE (27). Each method is shown in a distinct color. Solid lines represent solutions passing quality filters, while dashed lines include all evaluations. LES outperforms all baselines Ranolazine MPO and Perindopril MPO, achieving competitive re- sults on Zaleplon MPO.
Acknowledgements
Ronen and Yu gratefully acknowledge partial support from NSF grant DMS-2413265 NSF grant DMS 2209975, NSF grant 2023505 on Collaborative Research: Foundations of Data Science Institute (FODSI), the NSF and the Simons Foundation for the Collaboration on the Theoretical Founda- tions of Deep Learning through awards MMMS-203188 and
8

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Mitigating over-exploration in latent space optimization using LES
814639, NSF grant MC2378 the Institute for Artificial CyberThreat Intelligence and OperatioN (ACTION), and NIH (DMS/NIGMS) grant ROIGM152718.
Humayun and Baraniuk gratefully acknowledge the sup- port from NSF grants CCF191 1094, I1S-1838177, and IIS-1730574; ONR grants N00014-18-1257,, N00014- 20-1-2534, and MURI N00014-20-1-2787; AFOSR grant FA9550-22-1-0066) and Vannevar Bush Faculty Fellow- ship, ONR grant N00014-18-1-2047.
Impact Statement
This work seeks to improve the practicality of latent space optimization, method first used drug discovery. brings no new societal implications beyond those already associ- ated with LSO.

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Mitigating over-exploration in latent space optimization using LES
References
Frazier, P. I. Bayesian optimization. In Recent advances in optimization and modeling of contemporary problems, pp. 255-278. Informs, 2018.
Ament, S., Daulton, S., Eriksson, D., Balandat, M., and Bakshy, E. Unexpected improvements to expected im- provement for bayesian optimization. Advances in Neural Information Processing Systems, 36, 2024.
Gal, and Ghahramani, Z. Dropout as a bayesian approx- imation: Representing model uncertainty in deep learn- ing. In international conference on machine learning, pp. 1050-1059. PMLR, 2016.
Baell, J. B. and Holloway, G. A. New substructure filters for removal of pan assay interference compounds (pains) from screening libraries and for their exclusion bioas- says. Journal of medicinal chemistry, 53(7):2719-21) 2010.
Gao, W. Fu, T., Sun, J., and Coley, C. Sample efficiency matters: benchmark for practical molecular optimiza- tion. Advances in neural information processing systems, 35:21342-21357, 2022.
Balandat, M., Karrer, B., Jiang, D. R., Daulton, S., Letham, B., Wilson, A. G., and Bakshy, E. BoTorch: A Frame- work for Efficient Monte-Carde Bayesian Optimization. In Advances in Neural Information Processing Systems 33, 2020. URL http: arxiv. org/ ats///b///// 19900. 06403.
Gómez-Bombarelli R., Wei, J. N., Duvenaud, D., Hernández-LLobto, M., Sánchez-Lengeling. B., She- berla, D., gquilerra-parraguuin J., Hirzel, T. D., Adams, R. P. and Aspuru-Guzik, A. Automatic chemical de- sign using data-driven continuous representation of molecules. ACS central science, 4(2):268-2) 2018 8.
Balestriero, R. and Baraniuk, R. Mad max: Affine spline insights into deep learning. arXiv preprint 1rXiv:/805.06577 2018.
Griffiths, R.-R. and Hemándezz-Loatoe J. M. Constrained bayesian optimization for automatic chemical design us- ing variational autoencoders. Chemical science, 11(2): 577-586, 2020.
Balestriero, R., Humayun. A. I. and Baraniuk, R. On the geometry of deep learning. arXiv preprint arXiv:2408.000 048009, 2024.
Ben-Israel, A. The change-of-variables variables formula using ma- trix volume. SIAM Journal on Matrix Analysis and Ap- plications, 21(1):300-312, 1999.
Humayun, A. I., Balestriero, R., and Baraniuk, Magnet: Uniform sampling from deep generative network mani- folds without retraining. In International Conference on Learning Representations, 2021.
Bengio, Y., Lahlou, S., Deleu, T., Hu, E. J., Tiwari, M., and Bengio, E. Gflownet foundations. The Journal of Machine Learning Research, 24(1):0000-11060
2023,
Humayun, A. I., Balestriero, R., and Baraniuk, R. Polarity sampling: Quality and diversity control of pre-trained generative networks via singular values. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp. 10641-10650, 2022.
Brown, N., Fiscato, M., Segler, M. H., and Vaucher, A. C. Guacamol: benchmarking models for de novo molecular design. Journal of chemical information and modeling, 1963):009611003 2019.
Corso, G., Stärk, H., Jing, B., Barzilay, R., and Jaakkola, T. Diffdock: Diffusion steps, twists, and turns for molecular docking. arXiv preprint arXiv:2210.0177
Igashov, I., Stärk, H., Vignac, C., Schneuing, A., Satorras, V. G., Frossard, P., Welling, M., Bronstein, M., and Cor- reia, B. Equivariant 3d-conditional diffusion model for molecular linker design. Nature Machine Intelligence, pp. 1-11, 2024.
2022.
Daubechies, I., Vore, R., Dym, N., Faigenbaum-Gollovin, S., Kovalsky, S. Z., Lin, K.-C., Park, J., Petrova, G., and Sober, B. Neural network approximation of refinable functions. IEEE Transactions on Information Theory, (1):482 495,,2-495,90 2022.
Jin, W., Barzilay, R., and Jaakkola, T. Junction tree vari- ational autoencoder for molecular graph generation. In International conference machine learning, pp. 2323- 2332. PMLR, 2018.
Du, Y., Jamasb, A. R., Guo, J., Fu, T., Harris, C., Wang, Y., Duan, C., Liò, P., Schwaller, P., and Blundell, T. L. Ma- chine learning-aided generative molecular design. Nature Machine Intelligence, pp. 1-16, 2024.
Kingma, D. Adam: A method for stochastic optimization. arXiv preprint 4rX.v:1412.6980, 2014.
Eriksson, D., Pearce, M., Gardner, J., Turner, R. D., and Poloczek, M. Scalable global optimization via local bayesian optimization. Advances in neural information processing systems, 32, 2019.
Krenn, M., Häse, F., Nigam, A., Friederich, P., and Aspuru- Guzik, A. Self-referencing embedded strings (selfies): 100% robust molecular string representation. Machine Learning: Science and Technology. 1(4):045024, 2020.
10

---

Mitigating overcpooaton in latent space optimization using LES
Kusner, M. J., Paige, B., and Hernández-Lobato, J. M. Grammar variational autoencoder. In International con- ference machine learning, pp. 1945-1954. PMLR, 2017.
Lee, S., Chu, J., Kim, S., Ko, J., and Kim, H. J. Advancing bayesian optimization via learning correlated latent space. Advances in Neural Information Processing Systems, 36, 2024.
Lipinski, C. A., Lombardo, F., Dominy, B. W., and Feeney, J. Experimental and computational approaches timate solubility and permeability in drug discovery and development settings. Advanced drug delivery reviews, 23(1-3):3-25, 1997.
Maus, N., Jones, H., Moore, Kusner, M. J., Bradshaw, J., and Gardner, J. Local latent space bayesian optimization over structured inputs. Advances in neural information processing systems, 35:34505-34518, 2022.
Notin, P., Hernández-Lobato, J. M., and Gal, Y. Improving black-box optimization in vae latent space using decoder uncertainty. In Ranzato, M., Beygelzimer, A.. Dauphin, Y., Liang, P., and Vaughan, J. W. (eds.), Advances in Neural Information Processing Sys- tems, volume 34, pp. 802-814. Curran Associates, Inc., 2021. URL https::/procceee:
neurips. neurips.cc/papeeee / 0fFAc3381166122rccbaaae2222--2-2 pdf.
Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E.,
DeVito, Z., Lin, Z., Desmaison, A., Antiga, L. and Lerer,
A. Automatic differentiation in pytorch. 2017.
Tripp, A., Daxberger, E., and Hernández-Lobato, Sample-efficient optimization in the latent space of deep generative models via weighted retraining. Advances in Neural Information Processing Systems, 11272, 2020.
M.
Walters, filters, 2019. URL https://github. Com/PatWaaterssrret wtt.teerrrrdd_iitereteeeeeeeeeee Accessed: Jan- uary 14, 2019.
Wildman, S. A. and Crippen, G. M. Prediction of physic- ochemical parameters by atomic contributions. Journal of chemical information and computer sciences, 39(5): 868-877 1999.

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Mitigating over-exploration in latent space optimization using LES
A. Proofs
Lemma A.I. Let fe be a DGN as defined Equation (8) and assume that fo can be expressed as a CPA (Equation (5)) and inevitable, then
B1
0
Jo (x) =
A
(16)
BL
where A the Peerree inverse of the slope matrix, at the knot whose image constrains æ, and
1
B:
diag
(17)
(pb
(pz Jn
Proof. First we write
fo(z) (fo
(18)
Where Softmax is the extension of the column wise Softmax function include the normalizing constants. Specifically, for L by D (z) matrix, we have
(efmmax+(oo (p_), ( ( 1 (c(())-) Ez
(19)
with _ ( ), and c0 - exp(Lo(z)zi)) Next,
1 = (Softmaxa (x:)
(20)
we have that A direct calculation yields,
S (log(p!)) + log(c?")) log(p!") log (o)(c()))
(21)
As we assume tp is bijective and can be written as
(0(2) = Σ
(A + bu) Ize
(22)
WEll
Eg
(Softmax (x)) = (Softmaxi (xx) - b..)).... A!!
(23)
Lastly, as
B
0
@Softmax (x))
(24)
dx
B
for
T
B, = diag
1
(25)
(p2
(6)
(p2
02
we obtain the final result.
12

---

Mitigating over-exploration in latent space optimization using LES
Proof of Theorem 3.1. First, we note that by our invertibility assumption we have that P(a E W) = P(z (w)) We then proceed with a direct calculation
P( E W) P(z E 10- (w))
(26)
Σ
P(z (W)
(27)
Σ
fz(z)dz
(28)
Σ
fz(f0
(1.0--((x)) (x)) det (J16 (x)J5 dx
(29)
WER
/ Σ (=((1-1((0))) det (J5b (x)))((-"((()) ((EEfa(w))
(30)
W WER
Using Lemma A.I, we get that the volume element is
B
0
B
(x)J((-')))
:
A
A
(31)
0
BL
0
B
B
(AL)*
:
A
(32)
B
0
B
Σ
(B.))BBBBBB B.A.
(33)
11
where A! = (A.)................ A f
(i-):(i+1-D)."
Lemma A.2. Let f. Z X be function and define fi X + Z as ft(z {2: f(z) x}. Let be the Lebesgue
measure and assume that μ u({2; 3z' s.t. = f(r))) = 0, then for every B C Z we have
m((z; B)) (B)(
(34)
Proof. We proceed with direct calculation
E B)) s (B)) + u((=;3/ s.t. f(z) = f(z'))
(35)
Now assume that p({2; 32' s.t. f(2) = f(z))) = 0, we have that p({: f(z) E B}) z (B)) The other direction
follows immediately from the definition of ft.
Lemma A.2 implics that Equation (29) can still hold under the assumption that ux{2; s.t. fe(2) fo()')) 0.
B. Ablation studies
B.I. TuRBO hyper-parameters
We provide an ablation study of the initial length and success and failure tolerances. For each setup, the values reported in the main paper are highlighted in bold.
13

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Mitigating over-exploration in latent space optimization using LES
Table Ablation study for the initial length, success tolerance and failure tolerance for the TuRBO method. Average across 10 independent runs. of the top 20 best values across datasets, architectures. Column names indicate the length/success/fall values. Results from the main paper are in bold.
Architecture
0.8/10/10
0.8/10/2
0.8/2/10
0.8/2/2
L6/10/10
1.6/2/10
0.05
-2.03 (0.12)
-2.1
-2.0 (0.09)
(0.1.
-1.93 (0.09)
-1.92 (0.08)
-1.91 (0.06)
-1.93 (0.07)
GRU
(0.15)
(0.18)
-2.11 (0.2)
-2.14 (0.2)
-198 (0.13)
(0.12)
(0.17)
(0.22)
-1.39 (0.12)
-1.34 (0,05)
-1.56 (0.11)
-1.61 (0.11)
-1336 (0.08)
-1.42 (0.11)
-1.45 (0.12)
-1.38 (0.12)
Expressions
0.05
-1.45 (0.09)
-1.39 (0.13)
(0.12)
-1.49 (0.1)
-1.46 (0.08)
-1.52 (0.08)
-1.51 (0.12)
-1.31 (0.07)
LSTM
-1.31 (0.12)
-1.32 (0.13)
-1.37 (0.11)
-1.31 (0.11)
-1.31
-1.36 (0.13)
-1.36 (0.11)
-1.35 (0.09)
-2.16 (0.06)
-2.22 (0.08)
-2.23 (0.08)
-2233 (0.08)
-2.13 (0.07)
(0.06)
-2.15 (0.05)
-2.88 (0.06)
0.05
-2.07 (0.12)
-2.16 (0.09)
-2.03 (0.17)
-2.15 (0.14)
-1,96 (0.13)
-1.74 (0.13)
-2.09 (0.1)
-1.79 (0.11)
Transformer
0.1
-1.45 (0.13)
-1.39 (0.12)
-1.56 (0.13)
-1338 (0.13)
-1.32 (0.11)
-1.43 (0.17)
-1.4 (0.1)
-1.33 (0.12)
-1,96 (0.11)
-1.77 (0.12)
-1.81 (0.11)
-1.85 (0.09)
-1.85 (0.11)
-1.76 (0.08)
-1.78 (0.08)
-1.75 (0.09)
0.05
0.02 (0.01)
0.04 (0.01)
0.04 (0.01)
0.04 (0.01)
0.09 (0.01)
0.09 (0.03)
0.08 (0.01)
0.09 (0.02)
Transformer (pdup)
0.1
0.01 (0.0)
0.01 (0.0)
0.02 (0.01)
0.02 (0.01)
0.03 (0.01)
0.08 (0.03)
0.04 (0.02)
0.05 (0.02)
0.24 (0.01)
0.23 (0.02)
0.21 (0.02)
0.24 (0.01)
0.27 (0.01)
0.26 (0.01)
0.28 (0.01)
0.26
SELFIES
0.05
0.06 (0.01)
0.06 (0.0)
0.06 (0.01)
0.06 (0.01)
0.04 (0.01)
0.02 (0.0)
0.04 (0.0)
0.03 (0.01)
Transformer (rano)
0.1
0.03 (0.01)
0.03 (0.01)
0.03 (0.0)
0.02 (0.0)
0.08 (0.0)
0,05 (0.01)
0.07 (0.01)
0.07 (0.0)
0.05
0.02 (0.01)
0.04 (0.01)
0.02 (0.01)
0.04 (0.01)
0.07 (0.01)
0.07 (0.02)
0.06 (0.01)
0.06 (0.01)
Transformer (rale)
0.1
0.04 (0.01)
0.04 (0.01)
0.04 (0.01)
0.03 (0.01)
0.01 (0.0)
0.02 (0.0)
0.09 (0.0)
0.04 (0.0)
0.16 (0.02)
0.17 (0.02)
0.16 (0.02)
0.17 (0.02)
0.18 (0.02)
0.16 (0.02)
0.18 (0.02)
0.19 (0.02)
SELFIES (27)
Transformer (pdop)
0.44 (0.01)
0.44 (0.01)
0.44 (0.01)
0.44 (0.01)
0.45 (0.0)
0.45 (0.01)
0.44 (0.01)
0.44 (0.01)
former (ranc)
0.20 (0.02)
0.22 (0.02)
0.21 (0.01)
0.20 (0.01)
0.23 (0.01)
(0.01)
0.22 (0.01)
0.23 (0.01)
Transformer (zale)
0.37 (0.01)
0.37 (0.01)
0.36 (0.01)
0.37 (0.01)
0.39 (0.01)
0.38 (0.01)
0.39 (0.01)
0.39 (0.01)
0.05
0.89 (0.19)
0.82 (0.15)
0.64 (0.2)
0.89 (0.14)
1.0 (0.14)
0.98 (0.19)
0.7 (0.24)
1.07 (0.19)
GRU
0.1
0.4 (0.22)
0.41 (0.19)
0.63 (0.26)
0.55 (0.14)
0.83 (0.13)
0.96 (0.33)
0.69 (0.18)
0.75 (0.18)
1.44 (0.0)
-0.16 (0.0)
0.14 (0.65)
1.14 (0.0)
SMILES
0.05
1.18 (0.25)
1.54 (0.41)
1.1 (0.15)
1.1 (0.36)
U.86 (U.27)
1.05 (0.21)
1.07 (0.29)
1.22 (0.27)
LSTM
0.1
0.21 (0.0)
0.61 (0.16)
0.52 (0.26)
0.48 (0.13)
0.7 (0.23)
0.67 (0.23)
1.13 (0.16)
0.97 (0.28)
0.88 (0.23)
0.05
0.88 (0.18)
0.97 (0.19)
0.92 (0.14)
0.67 (0.15)
0.93 (0.17)
1.16 (0.1)
1.09 (0.11)
0.97 (0.24)
Transformer
0.1
0.42 (0.12)
0.62 (0.14)
0.44 (0.16)
0.73 (0.14)
0.91 (0.1)
0.6 (0.14)
0.81 (0.14)
0.7 (0.14)
0.72 (0.18)
0.48 (0.18)
0.61 (0.15)
0.47 (0.16)
0.99 (0.12)
0.74 (0.15)
0.83 (0.16)
0.68 (0.15)
14

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Mitigating over-exploration in latent space optimization using LES
Table Ablation study for the initial length, success tolerance and failure tolerance for the TuRBO method. Average across 10 independent runs, of the best value across datasets, architectures. Column names indicate the length/sucoess/fail values. Results from the main paper are bold.
Architecture
0.8/10/10
0.8/10/2
0.8/2/10
0.8/2/2
L6/10/10
1.6/2/10
0.05
-0.65 (0.11)
-0.73 (0.12)
-0.71 (0.09)
-0.61 (0.1)
-0.65 (0.12)
-0.59 (0.06)
-0.62 (0.11)
-0.72 (0.13)
GRU
(0.05)
-0.61 (0,07)
-0.59 (0.07)
-0.71 (0.1)
-0.54 (0.04)
(0.05)
-0.62 (0.04)
(0.04)
-0.56 (0.05)
-0.54 (0,05)
-0.54 (0.06)
(0.08)
-0.56 (0.06)
(0.05)
(0.04)
-0.59 (0.05)
Expressions
0.05
-0.46 (0.03)
+0.43 (0.02)
-0.43 (0.02)
-0.38 (0.02)
-0.43 (0.02)
-0.43 (0.02)
-0.47 (0.04)
-0.4 (0.01)
LSTM
-0.38 (0.04)
-0.39 (0.0)
-0.41 (0.01)
-0.42 (0.02)
-0.42 (0.02)
(0.02)
-0.42 (0.02)
-0.42 (0.02)
-0.96 (0.09)
-0.86 (0.0)
(0.0)
(0.11)
-0.88 (0.01)
-0.98 (0.06)
-0.92 (0.04)
(0.01)
0.05
-0.39 (0.04)
-0.44 (0.02)
-0.39 (0.04)
-0.4 (0.05)
-0.44 (0.02)
-0.39 (0.04)
-0.38 (0.04)
-0.42 (0.02)
Transformer
0.1
.0.8 (0.04)
-0.41 (0.02)
.042 (0.02)
0..4 (0.02)
-0.99 (0.04)
-0.41 (0.01)
-0.41 (0.02)
0.37 (0.04)
-0.67 (0.1)
-0.58 (0.1)
-0.52 (0.08)
(0.06)
-0.66 (0.11)
(0.07)
-0.56 (0.05)
-0.54 (0.05)
0.05
0.13 (0.02)
0.15 (0.03)
0.17 (0.03)
0.18 (0.03)
0.23 (0.03)
0.17 (0.04)
0.22 (0.04)
0.2 (0.04)
Transforer (pdup)
0.1
0.08 (0.02)
0.09 (0.03)
0.1
0.09 (0.03)
0.1 (0.03)
0.12 (0.04)
0.1 (0.03)
0.14 (0.04)
0.36 (0.02)
0.31 (0.03)
0.36 (0.02)
0.34 (0.02)
0.38 (0.01)
0.38 (0.0)
0.4 (0.01)
0.36 (0.02)
SELFIES
0.05
0.17 (0.02)
0.2 (0.02)
0.2 (0.02)
0.18 (0.02)
0.11 (0.02)
0.08 (0.01)
0.12
0.13 (0.02)
Transformer (rano)
0.1
0.09 (0.02)
0.1 (0.01)
0.11 (0.02)
0.1 (0.02)
0.06 (0.01)
0.05 (0.01)
0.04 (0.01)
0.06 (0.01)
0.07 (0.02)
0.05 (0.02)
0.11 (0.03)
0.07 (0.02)
0,16 (0.02)
0.16 (0.03)
0.11 (0.02)
0.12 (0.02)
0.05
0.11 (0.02)
0.15 (0.03)
0.12 (0.02)
0.16 (0.03)
0.22 (0.03)
0.23 (0.03)
0.21 (0.04)
0.19 (0.03)
Transformer (rale)
0.1
0.14 (0.01)
0.16 (0.01)
0.15 (0.02)
0.14 (0.01)
0.1 (0.02)
0.12 (0.02)
0.13 (0.02)
0.14 (0.03)
0.36 (0.02)
0.31 (0.03)
0.38 (0.02)
0.36 (0.02)
0.38 (0.01)
0.34 (0.02)
0.31 (0.03)
0.38 (0.02)
SELFIES (27)
Transformer (pdop)
0.48 (0.02)
0.48 (0.02)
0.53 (0.03)
0.49 (0.02)
0.51 (0.02)
0.51 (0.02)
0.49 (0.02)
0.50 (0.01)
former (ranc)
0.38 (0.01)
0.35 (0.01)
0.35 (0.01)
0.32 (0.01)
0.36 (0.01)
0.36 (0.01)
0.37 (0.01)
0.34 (0.01)
Transformer (zale)
0.44 (0.02)
0.44 (0.01)
0.44 (0.03)
0.49 (0.02)
0.47 (0.01)
0.46 (0.01)
0.49 (0.01)
0.47 (0.01)
0.05
2.47 (0.22)
2.47 (0.22)
2.42 (0.21)
2.24 (0.17)
2.35 (0.23)
2.42 (0.28)
1.97 (0.18)
2.25 (0.31)
GRU
0.1
1.76 (0.3)
2.57 (0.31)
2.45 (0.26)
2.07 (0.34)
2.24 (0.28)
2.26 (0.33)
2.24 (0.31)
1.92 (0.24)
2.43 (0.32)
2.48 (0.29)
2.76 (0.24)
2.17 (0.49)
2.43 (0.32)
2.35 (0.37)
1.4 (0.52)
2.04 (0.33)
SMILES
0.05
2.65 (0.32)
2.73 (0.33)
2.89 (0.26)
3.02 (0.34)
2.64 (0.3)
2.91 (0.29)
2.77 (0.23)
2.68 (0.37)
LSTM
0.1
1.97 (0.37)
1.78 (0.44)
1.98 (0.4)
2.16 (0.29)
2.36 (0.41)
1.87 (0.39)
2.34 (0.33)
1.73 (0.41)
3.06 (0.2)
2.71 (0.34)
2.65 (0.16)
2.83 (0.25)
2.75 (0.28)
3.49 (0.26)
2.68 (0.31)
3.16 (0.35)
0.05
2.47 (0.23)
2,88 (0.24)
2.42 (0.17)
2.67 (0.27)
2.91 (0.3)
2.25 (0.2)
2.43 (0.24)
2.48 (0.29)
Transformer
0.1
3.03 (0.3)
2.15 (0.18)
2.51 (0.25)
2.41 (0.24)
2.28 (0.12)
2,28 (0.22)
2.53 (0.26)
2.46 (0.16)
2.37 (0.21)
2.25 (0.18)
2.16 (0.17)
2:21 (0.25)
2.46 (0.2)
2.11 (0.12)
2,31 (0.16)
2.71 (0.32)
B.2. LSO (L-BFGS) hyper-parameters
We provide an ablation study for the facet-length parameter. For each setup, the values reported in the main paper are highlighted in bold.
15

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Mitigating over-exploration in latent space optimization using LES
Table Ablation study for the facet length parameter of LSSO (L-BFGS) method. Average across independent of the average top 20 values across datasets, architectures, and bound methods are displayed for facet lengths of size 1, 5 and 10. Results from the main paper are bold.
Architecture
g
5
10
0.05
-1.79 (0.08)
-1.72 (0.07)
-1.72 (0.07)
GRU
0.1
-1.73 (0.11)
-1.93 (0.09)
-1.89 (0.11)
1
-1.94 (0.07)
-1.97 (0.07)
-2.03 (0.09)
Expressions
0.05
-1.89 (0.09)
-1.78 (0.09)
-1.93 (0.06)
LSTM
0.1
1129 (0.06)
-1.39 (0.08)
-1.37 (0.06)
22.4 (0.05) 1
-2.04 (0.04)
-2.04 (0.05)
0.05
-3.11 (0.14)
-2.93 (0.13)
-3.02 (0.09)
Transformer
0.1
-3.19 (0.28)
-2.69 (0.25)
-2.93 (0.22)
1
-2.44 (0.11)
-2.41 (0.09)
-2.28 (0.11)
0.05
0.22 (0.01)
0.21 (0.01)
0.25 (0.01)
Transformer (pdop)
0.1
0.25 (0.01)
0.19 (0.01)
0.22 (0.03)
1
0.26 (0.01)
0.25 (0.01)
0.27 (0.01)
SELFIES
0.05
0.04 (0.0)
0.03 (0.0)
Transformer (rano)
0.1
0.09 (0.02)
0.07 (0.0)
0.08 (0.01)
0.05
0.12 (0.02)
0.13 (0.01)
0.11 (0.01)
Transformer (zale)
0.1
0.08 (0.01)
0.06 (0.0)
0.07 (0.0)
1
0.18 (0.02)
0.18 (0.02)
0.19
SELFIES (27)
Transformer (pdop)
0.45 (0.00)
0.42 (0.00)
0.43 (0.01)
Transformer (rano)
0.22 (0.01)
0.17 (0.01)
0.19 (0.01)
Transformer (zale)
0.4 (0.01)
0.37 (0.01)
0.38 (0.01)
0.05
0.65 (0.12)
0.52 (0.12)
0.52 (0.13)
GRU
0.1
0.52 (0.14)
0.12 (0.14)
0.36 (0.15)
SMILES
0.05
0.67 (0.15)
0.66 (0.12)
0.56 (0.2)
LSTM
0.1
(0.16)
0.7 (0.2)
0.55 (0.15)
0.05
0.52 (0.18)
0.42 (0.15)
0.55 (0.12)
Transformer
0.1
0.63 (0.18)
0.61 (0.15)
0.56 (0.17)
0.16 (0.15)
0.23 (0.12)
0.39 (0.13)
16

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Mitigating over-exploration in latent space optimization using LES
Table Ablation study for the facet length parameter of I.SO (L.-BFGS) method. Average across 10 independent runs, of the best value across datasets, architectures, and bound methods are displayed for facet lengths of size 1, 5 and 10. Results from the main paper are bold.
Architecture
B
5
0.05
-0.57 (0.07)
-0.56 (0.04)
-0.56 (0.07)
GRU
0.1
-0.53 (0.04)
-0.46 (0.02)
-0.46 (0.02)
1
-0.68 (0.05)
-0.77 (0.02)
-0.73 (0.04)
Expressions
0.05
-0.57 (0.11)
-0.56 (0.04)
-0.67 (0.09)
LSTM
0.1
-0.4 (0.05)
-0.44 (0.04)
-0.47 (0.04)
1
-1.01 (0.1)
-1.02 (0.07)
-0.86 (0.0)
0.05
-1.06 (0.13)
-0.8 (0.1)
-1.11 (0.18)
Transformer
0.1
(0.14)
-0.65 (0.1)
-0.69 (0.09)
-0.85 (0.1)
-0.82 (0.1)
-0.78 (0.12)
0.05
0.36 (0.01)
0.36 (0.02)
0.34 (0.03)
Transformer (pdop)
0.1
0.29 (0.03)
0.34 (0.02)
0.27 (0.03)
0.39 (0.01)
0.36 (0.02)
0.35 (0.03)
SELFIES
0.05
0.06 (0.01)
0.07 (0.01)
0.08 (0.01)
Transformer (rano)
0.1
0.08 (0.02)
0.08 (0.02)
0.07 (0.01)
0.16 (0.02)
0.16 (0.02)
0.16 (0.02)
0.05
0.27 (0.03)
0.27 (0.03)
0.26 (0.03)
Transformer (zale)
0.1
0.15 (0.04)
0.19 (0.03)
0.18 (0.04)
0.38 (0.01)
0.38 (0.02)
0.39 (0.01)
SELFIES 27
Transformer (pdop)
0.51 (0.02)
0.47 (0.01)
0.48 (0.01)
Transformer (rano)
1
0.34 (0.02)
0.37 (0.01)
0.34 (0.02)
Transformer (zale)
1
0.47 (0.0)
0.44 (0.01)
0.45 (0.02)
0.05
2.06 (0.3)
1.71 (0.24)
2.02 (0.22)
GRU
0.1
2.11 (0.28)
1.74 (0.2)
1.94 (0.22)
1
2.4 (0.31)
2.1 (0.32)
2.34 (0.31)
SMILES ES
0.05
2.31 (0.21)
2.32 (0.19)
2.15 (0.18)
LSTM
0.1
1.74 (0.5)
1.85 (0.34)
1.47 (0.49)
1
2.8 (0.25)
3,09 (0.16)
2.42 (0.3)
0.05
1.82 (0.2)
1.79 (0.13)
1.74 (0.17)
Transformer
0.1
2.19 (0.16)
2.24 (0.1)
2.09 (0.17)
1
1.72 (0.18)
2.0 (0.14)
1.96 (0.14)
17

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Mitigating over-exploration in latent space optimization using LES
B.3. Likelihood hyper-parameters
We provide ablation study the effect of changing the regularization strength parameter A. For each setup, the values reported in the main paper are highlighted in bold.
B.3.1. EXPRESSIONS
Table 8. Ablation study for the regularization parameter X for Likelihood. We report the average of the top 20 solutions found for the Expressions VAEs.
Architecture
B
À
0.05
== 0.1
À 0.2
0.05
-1.15
-1.10
-1.11
GRU
0.1
-1.31
-116
-1.14
-0.88
-1.04
-1.15
0.05
-1.00
-105
-1117
LSTM
0.1
-0.75
-0.81
-0.72
-1811
-1.79
-177
0.05
-0.93
-0.9
-0.98
Transformer
0.1
-0.81
-0.74
-0.70
-1.45
-1.42
-1.19
Table Ablation study for the regularization parameter À for Likelihood. Expressions VAEs.
We report the average of the top solution found for the
Architecture
À 0.05
À 0.1
À 0.2
0.05
-0.4
-0.43
-0.48
GRU
0.1
-0.43
-0.42
-0.47
-0.43
-0.58
-0.59
0.05
-0.4
-0.45
-0.44
LSTM
0.1
-0.32
-0.39
-0.36
-0.86
-0.86
-0.86
0.05
-0.38
-0.39
-0.43
Transformer
0.1
-0.41
-0.39
-0.39
-0.65
-0.50
-0.42
18

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Mitigating over-exploration in latent space optimization using LES
B.3.2. SMILES
Table 10. Ablation study for the regularization parameter for Likelihood. SMILES VAEs.
We report the average the top 20 solutions found for the
Architecture
À
0.3
A=
0.5
X
0.8
0.05
2.15
2.31
2.25
GRU
0.1
2.10
2.12
2.22
0.60
1.49
1,40
0.05
2.17
2.28
2.30
LSTM
0.1
0.81
1.43
1,58
0.53
1.67
1.77
0.05
2.43
2.25
2.32
Transformer
0.1
2.26
2.23
2.34
2.01
2.16
1.98
Table 11. Ablation study for the regularization parameter À for Likelihood. SMILES VAEs.
We report the average of the top solution found for the
Architecture
X
0.3
À
0.5
À 0.8
0.05
2.96
3.26
3.27
GRU
0.1
3.02
3.33
3.07
3.12
3.89
3.47
0.05
3.30
3.37
3.19
LSTM
0.1
2.44
3.54
2.92
3.00
3.48
2.94
0.05
3.24
3.19
3.07
Transformer
0.1
3.10
3.16
3.25
3.18
3.2
2.92
19

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Mitigating over-exploration in latent space optimization using LES
B.3.3. SELFIES
Table 12. Ablation study for the regularization parameter À for Likelihood. We report the average of the top 20 solutions found for the
SELFIES VAEs.
Objective
8
0.1
0.3
0.5
0.05
0.37
0.37
0.37
0.1
0.36
0.36
0.34
Pdop
0.34
0.29
0.28
(Maus et al., 2022)
0,42
0.40
0.35
0.05
0.21
0.22
0.21
0.1
0.22
0.21
0.20
Rano
0.21
0.23
0.19
(Maus et al., 2022)
0.22
0.29
0.26
0.05
0.33
0.33
0.32
0.1
0.34
0.33
0.32
Zale
0.30
0.29
0.26
(Maus al., 2022)
0.41
0.39
0.31
Table 13. Ablation study for the regularization parameter À for Likelihood. SELFIES VAES.
We report the average of the top solution found for the
Objective
B
À = 0.1
0.3
0.5
0.05
0.43
0.43
0.42
0.1
0.43
0.41
0.39
Pdop
0.40
0.40
0.37
(Maus et al., 2022)
0.47
0.45
0.42
0.05
0.31
0.31
0.29
0.1
0.33
0.30
0.30
Rano
0.33
0.30
0.33
(Maus et al., 2022)
0.34
0.36
0.37
0.05
0.44
0.45
0.42
0.1
0.44
0.42
0.42
Zale
0.42
0.40
0.37
(Maus al., 2022)
0.49
0.49
0.42
B.4. LES peramterr
We provide an ablation study on the effect of changing the regularization strength parameter X. For each setup, the values reported in the main paper are highlighted in bold.
20

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Mitigating over-exploration in latent space optimization using LES
B.4.1. EXPRESSIONS
Table 14, Ablation study for the regularization parameter À for LES.
We report the average of the top 20 solutions found for the
Expressions VAEs.
Architecture
À = 0.05
À = 0.1
À = 0.2
0.05
-1.43
-1.28
-1.32
GRU
0.1
-1.34
-1.06
-129
-0.44
-0.98
-1.08
0.05
-1.66
-1.09
-1.00
LSTM
0.1
-0.80
-0.72
-0.73
-1.79
-1.79
11.84
0.05
-1.00
-0.66
-0.92
Transformer
0.1
-0.77
-0.75
-0.67
-136
-1.28
-1.05
Table 15. Ablation study for the regularization parameter À for LES. VAEs.
We report the average of the top solution found for the Expressions
Architecture
0.05
À = 0.1
À
0.2
0.05
-0.55
-0.56
-0.44
GRU
0.1
-0.45
-0.44
-0.41
-0.47
-0.47
-0.52
0.05
-0.43
-0.42
-0.41
LSTM
0.1
-0.32
-0.39
-0.39
-0.86
-0.86
-0.86
0.05
-0.43
-0.39
-0.41
Transformer
0.1
-0.36
-0.39
-0.32
-0.52
-0.52
-0.45
B.4.2. SMILES
Table 16. Ablation study for the regularization parameter À for LES. We report the average of the top 20 solutions found for the SMILES VAEs.
Architecture
X = 0.3
À
0.5
À
0.8
0.05
2.20
2.31
2.32
GRU
0.1
2.12
2.30
2.35
0.38
1.64
1.77
0.05
2.34
2.33
2.37
LSTM
0.1
0.94
1.57
1.80
0.59
1.94
2.17
0.05
2.30
2.26
2.35
Transformer
0.1
2.31
2.26
2.29
2.09
2.17
2.30
21

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Mitigating over-exploration in latent space optimization using LES
Table Ablation study the regularization parameter À LES.
report the average top solution found for SMILES
VAEs.
Architecture
B
À = 0.3
À = 0.5
À
0.8
0.05
3.16
3.29
3.18
GRU
0.1
3.19
3.55
3,42
3.13
3.85
3.53
0.05
3.51
3.29
3.35
LSTM
0.1
3.10
3.54
2.93
2.6
3.6
3.29
0.05
3.16
3.21
3.35
Transformer
0.1
3.09
3.23
3.03
2.94
3.2
3.17
B.4.3. SELFIES
Table 18. Ablation study for the regularization parameter À for LES. VAEs.
We report the average of the top 20 solutions found for the SELFIES
Objective
= 0.1
À
0.3
À = 0.5
0.05
0.37
0.37
0.37
0.1
0.36
0.36
0.36
Pdop
0.35
0.32
0.33
(Maus et al., 2022)
0.46
0.48
0.47
0.05
0.21
0.22
0.21
0.1
0.21
0.21
0.19
Rano
0.21
0.21
0.20
(Maus et al., 2022)
0.30
0.29
0.30
0.05
0.33
0.34
0.34
0.1
0.34
0.33
0.33
Zale
0.31
0.31
0.31
(Maus et al., 2022)
0,40
0.40
0.41
22

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Mitigating over-exploration in latent space optimization using LES
Table Ablation study the regularization parameter for LES, report average of the top solution found SELFIES
VAEs.
Objective
A = 0.1
0.3
=
0.5
0.05
0.43
0.43
0.42
0.1
0.43
0.41
0.40
Pdop
0.41
0.42
0.40
(Maus et al., 2022)
0.50
0.50
0.54
0.05
0.33
0.32
0.29
0.1
0.33
0.30
0.31
Rano
0.31
0.28
0.30
(Maus al., 2022)
0.37
0.37
0.39
0.05
0.43
0.45
0.45
0.1
0.44
0.45
0.42
Zale
0.42
0.43
0.39
(Maus et al., 2022)
0.50
0.50
0.47
23

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Mitigating over-exploration in latent space optimization using LES
C. Additional experimental results
Table 20. Average the top solution found during LSO (higher better), across datasets and decoder architectures. bold the best method and underline the second-best. The average ranking for each method (lower is better) is provided, along with the number of times each method within one standard deviation of the best. LES and Prior achieve the highest value most frequently (14 out 30) and outperforms other methods both in terms of the average ranking and the frequency of being within standard deviation of the best result.
Architecture
LES
L.SO (L-BFGS)
UC
1,50 (GA)
Prior
TaRBO
Likelihood
0.05
-0.55 (0.04)
-0.56 (0.04)
-0.59 (0.04)
-0.45 (0.05)
-0.4 (0.07)
-0.73 -0.73 (0.12)
-0.4 (0.07)
GRU
0.1
-0.45 (0.03)
-0.46 (0.02)
-0.47 (0.05)
-0.43 (0.02)
-0.37 (0.03)
-0.61 (0.07)
-0.43 (0.02)
Expressions
-0.47 (0.03)
0.77 (0.02)
.0.51 (0.04)
.0.46 (0.02)
.0.7 (0.03)
0.54 (0.05)
-0.43 (0.01)
0.05
-0.43 (0.02)
-0.56 (0.04)
-0.52 (0.05)
-0.43 (0.01)
-0.41 (0.01)
-0.33 (0.02)
(0.01)
LSTMM
0.1
-0.32 (0.05)
-0.44 (0.04)
-0.39 (0.04)
-0.38 (0.02)
-0.4 (0.01)
-0.39 (0.0)
-0.32 (0.04)
-0.86 (0.0)
-1.02 (0.07)
-0.66 (0.0)
.0.86 (0.0)
-0.86 (0.0)
-0.56 (0.0)
-0.91 (0.04)
-0.43 (0.03)
(0.1)
-0.77 (0.05)
(0.02)
-0.37 (0.05)
(0.02)
(0.04)
Transformer
-0. (0.03)
-0.65 (0.1)
-0.55 (0.04)
-0.39 (0.01)
(0.04)
-0.1 (0.02)
-0.41 (0.02)
-0.52 (0.05)
-0.82 (0.1)
-0.58 (0.05)
-0.58 (0.04)
-0.62 (0.09)
-0.58 (0.1)
-0.65 (0.08)
0.05
0.36 (0.02)
0.42 (0.0)
0.42 (0.0)
0.42
0.15 (0.03)
0.43
Transformer (pdop)
(0.0)
(0.02)
0.42 (0.01)
(0.0)
(0.01)
0.09 (0.03)
(0.01)
(0.01)
(0.02)
0.39 (0.01)
(.4
(0.01)
(0.03)
0.40)
SELFIES
0.05
0.33 (0.01)
0.07 (0.01)
0.36 (0.01)
0.33 (0.01)
0.32
0.2 (0.02)
0.31 (0.01)
Transformer (rane)
0.33 (0.01)
0.08 (0.02)
0.36 (0.02)
0.32 (0.01)
0.32
0.1
0.33
(0.01)
(0.02)
(0.02)
(0.01)
(0.02)
(0.02)
(0.01)
0.05
0.43 (0.01)
0.27 (0.03)
0,42 (0.01)
0.43 (0.01)
0.42 (0.0)
0.15 (0.03)
0.44 (0.01)
Transformer (zale)
0.1
0.44 (0.01)
0.19 (0.03)
0,42 (0.01)
0.42 (0.01)
0.42 (0.01)
0.16 (0.01)
0.44 (0.01)
0.42 (0.01)
0.38 (0.02)
0.39 (0.01)
0.42 (0.01)
0.37
0.31 (0.03)
0.42 (0.01)
SELFIES (27)
Transformer (pdop)
0.54 (0.01)
0.46 (0.01)
0.50 (0.01)
0.49 (0.01)
0.36 (0.02)
0.48 (0.02)
0.42 (0.00)
Transformer (rane)
0.37 (0.00)
0.37 (0.01)
0.36 (0.01)
0.37 (0.01)
0.34 (0.01)
0.35 (0.02)
0.34 (0.01)
Transformer (zale)
0.47 (0.01)
(0.01)
(0.01)
0.47 (0.01)
0.39
0.44 (0.01)
0.42 (0.01)
0.05
3.29 (0.1)
1.71 (0.24)
3.13 3.13 (0.07)
3,26 (0.11)
3.18 (0.06)
2.47 (0.22)
3.26 (0.08)
GRU
0.1
3,55 (0.14)
1.74 (0.2)
(0.1)
3.31 (0.16)
3.15 (0.11)
2.57 (0.31)
3.33 (0.12)
3.85 (0)
2.1 (0.32)
2.24 (0.28)
3.66 (0.16)
3.89 (0.28)
2.48 (0.29)
3,89 (0.2)
SMILES
0.05
3.29 (0.07)
2.32 (0.19)
3.12 (0.08)
3.3 (0.1)
3.28 (0.1)
2.73 (0.33)
3.37 (0.09)
LSTM
D.1
3.66 (0.2)
1.85 (0.34)
2.65 (0.19)
3.52 (0.22)
3.57 (8.18)
1.78 (0.43)
3.54 (0.16)
3.6 (0.14)
3.09 (0.16)
2.6 (0.3)
3.18 (0.17)
3.28 (0.17)
2.71 (0.34)
3.48 (0.11)
0.05
3.21 (0.08)
1.79 (0.13)
3.1 (0.07)
3.14 (0.04)
3.14 (0.04)
2.88 (0.24)
3.19 (0.08)
Transformer
0.1
3.23 (0.04)
2.24 (0.1)
3.28 (0.08)
3.11 (0.05)
3.09 (0.06)
2.15 (0.18)
3.16 (0.06)
3.2 (0.07)
(..1
2.8 (0. 13)
3.13 (0.07)
3.11 (0.1)
2.25 (0.18)
3.2 (0.06)
Average rank
2.15
6.17
4.17
3.15
3.86
5.7
2.76
within I std of best
19
5
8
10
2
16
24

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Mitigating over-exploration in latent space optimization using LES
Table 21. Average of the top 20 solutions found during LSO (higher is better) across datasets and decoder architectures. We bold the best method and underline the second-best. The average ranking for each method (lower is better) is provided, along with the number of times each method is within one standard deviation the best. LES achieves the highest value in most experiments (20 out 30) and outperforms other methods in terms of both the average ranking and the frequency of heing within one standard deviation of the hest result.
Architecture
A
LES
ISO (L-BFGS)
UC
LSO (GA)
Prior
TuRBO
Likelhbod
0.05
-1.43 (0.05)
-1.72 (0.07)
-1.5
-1.25 (0.06)
-1.28 (0.05)
(0.11)
-1.15 (0.09)
GRU
0.1
-1.34 (0.08)
-1.93 (0.09)
-2.11 (0.12)
-1.21 (0.09)
-1,18 (0.11)
-2.14
-1.31 (0.08)
Expressions
1
-0.84 (0.02)
-1.97 (0.07)
-0.91 (0.03)
-0.84 (0.02)
-0.89 (0.05)
-1.34 (0.05)
-0.88 (0.03)
-1.06 (0.06)
-1.78 (0.09)
-1.53 (0.06)
-0.98 (0.05)
-1.02 (0.03)
-1.59 (0.13)
(0.07
LSTM
0.1
-08 (0.03)
-1.39 (0.08)
-1.99 (0.04)
.0.79 (0.03)
.0.99 (0.03)
-1.32 (0.13)
40.75 (0.03)
.1.79 (0.02)
2.04 (0.04)
2.02 (0.03)
-1.81 (0.02)
-1.83 (0.02)
2.22 (0.08)
.1.1 (0.03)
0.05
-1.0 (0.04)
-2.93 (0.13)
-1.64 (0.08)
-1.03 (0.04)
-0.99 (0.05)
-2.16
-0,93 (0.05)
Transformer
.0.77 (0.02)
(0.25)
-1.79 (0.08)
(0.04)
(0.03)
-1.39 (0.12)
(0.03)
-1.36 (0.09)
-2.41 (0.09)
-1.52 (0.09)
-1.5 (0.07)
-1.41 (0.08)
-1.77 (0.12)
-1.45 (0.12)
0.05
0.37
0.21 (0.01)
0.36
0.37 (0.0)
0.37
0.04 (0.01)
0.37
Transformer (pdop)
(0.01)
(0.0)
(0.0)
0.34
(0.0)
(0.0)
(0.0)
(0.01)
0.31 (0.01)
(0.0)
0.28 (0.01)
(0.02)
0.34 (0.0)
SELFIRS
0.05
0.21 (0.0)
0.03 (0.0)
0.22
0.21 (0.0)
0.21
0.06 (0.0)
0.21
Transformer (rano)
0.1
0.21 (0.0)
0.23 (0.01)
0.21 (0.0)
0.22 (0.0)
0.03 (0.01)
0.22
(0.0)
0.07 (0.0)
0.22 (0.01)
0.19 (0.0)
(0.0)
0.21 (0.0)
0.05
0.33 (0.0)
0.13 (0.01)
0.32 (0.0)
0,33 (0.0)
0.32 (0.0)
0.04 (0.01)
0.33 (0.0)
Transformer (zale)
0.1
0.34 (0.0)
0.06 (0.0)
0.31 (0.01)
0.32 (0.01)
0.31 (0.0)
0.04 (0.01)
0.34 (0.0)
0.31 (0.0)
0.18 (0.02)
0.26 (0.01)
0.29 (0.01)
0.23 (0.01)
0.17 (0.02)
0.3 (0.01)
SELFIES (27)
Transformer (pdop)
0.48 (0.01)
0.42 (0.01)
0.44 (0.00)
0.44 (0.00)
0.3 (0.01)
0.44 (0.01)
0.35 (0.01)
Transformer (rane)
0.29 (0.00)
0.17 (0.03)
0.27 (0.01)
0.27 (0.01)
0.25 (0.01)
0.22 (0.03)
0.22 (0.01)
Transformer (cale)
0.4 (0.00)
0.37 (0.01)
0.38 (0.01)
0.39 (0.0)
0.27 (0.01)
0.37 (0.01)
0.31 (0.01)
0.05
2,31 (0.04)
0.52 (0.12)
2.18 (0.05)
2.21 (0.05)
2.2 (0.04)
0.82 (0.15)
2.31 (0.03)
GRU
0.1
2.3 (0.05)
0.12 (0.14)
1.68 (0.04)
2.09 (0.07)
1.93 (0.06)
0.41 (0.19)
2.12 (0.07)
1.64 (0.14)
1.19 (0.2)
0.58 (0.28)
..16 (0.0)
1.49 (0.14)
SMILES
0.05
2.33 (0.03)
0.66 (0.12)
2.02 (0.03)
2.22 (0.05)
2.13 (0.05)
1.54 (0.41)
2.28 (0.03)
L.TTM
0.1
1.57 (0.1)
0.8 (0.12)
0.9 (0.09)
1.43 (0.12)
1.94 (0.1)
0.7 (0:2)
0.95 (0.24)
1.14 (0.21)
0.81 (0.26)
0.52 (0.26)
1.67 (0.13)
0.05
2.26 (0.04)
0.42 (0.15)
2.04 (0.05)
2.22 (0.03)
2.24 (0.03)
0.97 (0.19)
2.25 (0.03)
Transfomer
0.1
2.26 (0.03)
0.61 (0.15)
2.08 (0.04)
2.21 (0.03)
2.17 (0.03)
0.62 (0.14)
2.23 (0.02)
1
2.17 (0.05)
0.23 (0.)
1.32 (0.09)
1.98 (0.06)
1.82 (0.06)
0.48 (0.18)
2.16 (0.05)
Average rank
1.9
6.4
4.3
2,87
3.8
6.1
2.53
# within I std of best
22
3
7
5
0
13
25

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Mitigating over-exploration in latent space optimization using LES
0.7
0.6
Ranolazine MPO
0.5
0.4
0.3
0.2
0.1
100
200
300
400
500
Number of evaluations
Perindopril MPO
0.4
0.3
0.2
100
200
300
400
500
Number of evaluations
0.5
Zaleplon MPO
0.4
0.3
0.2
Valid
Overall
100
200
300
400
500
Number of evaluations
(GA)
UC
LSO (L-BFGS)
Likelihood
TuRBO
Prior
Figure 4. Cumulative true objective for the best solution found during Bayesian optimization with the pre-trained SELFIES VAE (27).
Each method is shown in distinct color. Solid lines represent solutions passing quality filters, while dashed lines include all evaluations. LES achieves the best performance on Perindopril MPO and is comptetitive on Zaleplon MPO and Ranolazine MPO.
26

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Mitigating over-exploration in latent space optimization using LES
Table 22. AUROC values (higher is better) for identifying valid data points within the latent space, across datasets and decoder architectures. Data points are sampled from the training data, the VAE prior (standard Gaussian) and out of distribution (Gaussian with std of 5). LES achieves the best performance in most cases (I8 out of 22) and on average. In addition, LES achieves AUROC values of at least 0.75 in all cases, indicating can effectively differentiate valid from invalid data points.
Dataset
Arch.
B
LES
Polarity
Prior
Uncertainty
Train distances
Likelihood
0.05
0.93
0.42
0,09
0.85
0.85
0.94
GRU
0.12
0.84
0.86
0.94
1.0
0.91
0.35
0.16
0.87
0.80
0.92
0.05
0.99
0.93
0.07
0.99
0.91
0.98
SMILES
LSTM
0.1
0.89
0.67
0.21
0.89
0.81
0.9
1.0
0.97
0.76
0.12
0.95
0.85
0.96
0.05
0.93
0.89
0.14
0.84
0.77
0.92
Transformer
0.1
0.94
0.91
0.14
0.87
0.86
0.93
1.0
0.97
0.93
0.10
0.89
0.85
0.95
0.05
0.96
0.89
0.38
0.96
0.67
0.88
GRU
0.1
0.94
0.86
0.42
0.94
0.71
0.8
0.94
0.57
0.94
0.75
0.89
0.05
0.96
0.86
0.38
0.90
0.67
0.96
Expressions
LSTM
0.95
0.83
0.37
0.91
0.66
0.95
1.0
0.95
0.79
0.56
0.91
0.72
0.91
0.05
0.91
0.79
0.43
0.86
0.71
0.90
Transformer
0.1
0.91
0.79
0.53
0.87
0.78
0.89
1.0
0.86
0.61
0.70
0.92
0.89
0.92
0.05
1.0
0.99
0,02
0.99
0.62
0.97
SELFIES
Transformer
0.98
0.03
0.96
0.81
0.98
1.0
0.95
0.94
0.06
0.85
0.72
0.93
SELFIES (27)
Transformer
0.75
0.39
0.69
0.33
0.69
0.70
Average
0.93
0.78
0.29
0.88
0.77
0.91
27

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Mitigating over-exploration in latent space optimization using LES
Table 23, Effect of increasing parameter LES, the expressions dataset. Increasing the value of parameter increases the percentage of valid solution in all cases.
Architecture
LES (A = 0.05)
LES (A = 0.5)
0.05
0.92
0.96
GRU
0.1
0.7
0.82
0.72
0.87
0.05
0.93
LSTM
0.1
0.93
0.98
0.76
0.97
0.05
0.87
0.94
Transformer
0.88
0.96
0.83
0.85
D. Background on related work
Bayesian uncertainty (Notin et al., 2021) Under Bayesian viewpoint, the trained neural network parameters (0) follow variational distribution, which we can sample from using MC-Dropout (Gal & Ghahramani, 2016). Based on this distribution, the uncertainty defined as
M(z) = Mpp (p = 2)) - Eg Ep (po ( = =)).
(36)
where H the entropy and pl X = 2) is the posterior predictive distribution. The uncertainty is estimated using MC-Dropout with important sampling (using the posteior predictive as the importance distribution) designed to approximate the expectations over the random variable X, it is typically very large space (i.e., the sample of molecules that can implemented using SMILE string with 120 characters)
28

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Mitigating over-exploration in latent space optimization using LES
Table 24, Proportion of valid solutions found during LSO (higher is better) across datasets and decoder architectures. We bold the best method (higher is better) and underline second best. improves the validity the solutions compared LSO (GA) (which LES with À = 0) across all datasets.
Architecture
LES
LSO (L-BFGS)
UC
LSO (GA)
Prior
TuRBO
Likelihood
0.05
0.92
0.59
1.0
0.91
0.91
0.94
0.89
GRU
0.1
0.7
0.57
1.0
0.64
0.66
0.9
0.67
0.72
0.45
0.99
0.69
0.69
0.83
0.69
Expressions
0.05
0.93
0.62
1.0
0.88
0.89
0.94
0.92
LSTM
0.1
0.93
0.67
1.0
0.9
0.89
0.94
0.92
0.76
0.58
0.99
0.65
0.66
0.73
0.05
0.87
0.37
1.0
0.83
0.85
0.9
0.84
Transformer
0.1
0.88
0.28
1.0
0.8
0.81
0.89
0.85
0.83
0.36
1.0
0.74
0.76
0.79
0.05
0.8
0.05
0.81
0.76
0.73
0.14
0.77
Transformer (pdop)
0.1
0.68
0.04
0.71
0.62
0.56
0.14
0.64
0.59
0.08
0.55
0.49
0.44
0.08
0.53
SELFIES
0.05
0.66
0.02
0.73
0.61
0.57
0.09
0.62
Transformer (rano)
0.1
0.57
0.01
0.67
0.52
0.44
0.04
0.53
1
0.43
0.02
0.49
0.36
0,28
0.01
0.39
0.05
0.71
0.08
0.74
0.66
0.62
0.12
0.69
Transformer (zale)
0.1
0.66
0.01
0.67
0.59
0.51
0.08
0.63
0.54
0.18
0.47
0.43
0.35
0.17
0.46
SELFIES (27)
Transformer (pdop)
0.69
0.45
0.56
0.58
0.75
0.43
0.68
Transformer (rano)
0.26
0.07
0.14
0.19
0.11
0.15
Transformer (zale)
0.65
0.52
0.66
0.66
0.75
0.48
0.76
0.05
0.61
0.14
0.48
0.47
0.44
0.12
0.59
GRU
0.1
0.37
0.07
0.25
0.23
0.22
0.06
0.35
0.09
0.02
0.05
0.05
0.04
0.02
0.08
SMILES
0.05
0.11
0.44
0.42
0.39
0.11
0.57
LSTM
0.1
0.08
0.01
0.06
0.05
0.04
0.01
0.07
0.16
0.09
0.07
0.08
0.07
0.09
0.12
0.05
0.7
0.42
0.7
0.68
0.68
0.31
0.72
Transformer
0.1
0.65
0.67
0.63
0.6
0.55
0.41
0.64
0.48
0.35
0.46
0.34
0.26
0.3
0.45
Average
0.62
0.26
0.64
0.55
0.53
0.38
0.59
29


Reviews:
[{"review_id": "YNULrs3BIx", "reviewer": "Reviewer_eqXz", "summary": "The authors propose a scoring method, Latent Exploration Score (LES), for Latent Space Optimization (LSO), to mitigate overexploration, which often results in unrealistic solutions in discrete optimization problems. LES uses the VAE decoder’s data distribution without additional training or architectural changes, incorporating molecule density in the latent space. The empirical results show that LES performs better or similar most frequently across 30 tasks compared to baselines.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and effort in reviewing our work, and we are particularly glad to see that the reviewer agrees that our claims regarding the effectiveness of LES are supported by our empirical evidence, which the reviewer described as “extensive”.\n\nFirst, thank you for catching our typos, which have been corrected (`ScaLES → LES`, `Theorem 5 → Theorem 3.1`).\n\nThe reviewer is correct to point out that LES is closely related to decoder performance. As this is an important point, we made sure to include it under the limitations of Theorem 3.1 (lines 247–252):\n\n> “Lastly, it is crucial to highlight that the ability of LES to detect out-of-distribution data is closely tied to the decoder’s capacity to accurately model the data. Although Theorem 3.1 holds for poorly trained decoders under the given conditions, we advise against relying on LES in such cases.”\n\nIndeed, we expect the decoder to struggle with OOD data points in general (this has also been widely observed in many previous works that we cite [1]):\n\n> “...therefore all LSO methods known to us employ some sort of measure to restrict the optimization to near or within the feasible region.”\n\nLastly, we would like to offer some clarification regarding **hyperparameter tuning** (an important clarification which will be added to our submission).\n\nWe acknowledge that some manual selection of the hyperparameter $\\lambda$ was performed based on the level of regularization needed for each task, which could give our method a slight advantage over the baseline. To ensure our improvements are not merely the result of this tuning, we included an ablation study for **TurBO** and **L-BFGS** in our original submission, which demonstrates that LES would out-perform these methods on average even if the best hyperparameter was retroactively selected.\n\nWe also include an additional ablation and analysis of $\\lambda$ selection in our response to Reviewer SqeL.\n\nGiven this, we believe our comparisons are fair and that our conclusions are well supported: LES offers a consistent and meaningful improvement over the baselines, and these gains are not artifacts of hyperparameter tuning.\n\n**Reference**  \n\n[1] Tripp, A., Daxberger, E., & Hernandez-Lobato, J.M.  \n*Sample-efficient optimization in the latent space of deep generative models via weighted retraining.*", "claims_and_evidence": "The authors make several key claims about their method most of which are supported by empirical evidence. Authors, in particular, claim the LES’ robustness in generating valid solutions and achieving high objective values. However, isn’t LES still closely linked to the decoder’s performance? Even if the latent space is optimized, if the decoder struggles with out-of-distribution latent points, will this also impact LES’s effectiveness?", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are mostly well-written and make sense for the LSO problem. The use of top-1 and top-20 averages provides several insights into the performance.", "theoretical_claims": "The paper includes several derivations, including Theorem 3.1, which derives a likelihood-based formula for evaluating latent validity. The derivation appears correct.", "experimental_designs_or_analyses": "The experimental design includes extensive comparisons across various datasets and methods. However, the authors could improve the discussion by addressing potential failure cases of LES, particularly for datasets or tasks involving out-of-distribution latent spaces. Additionally, the number of hyperparameters differs from those used in the benchmark methods. Since a grid search was conducted for these hyperparameters, resulting in an unequal compute budget, I think it is a bit unfair to compare LES with the baselines, as this could potentially lead to misleading performance conclusions.", "relation_to_broader_scientific_literature": "The paper refers to some important previous work and other baselines in latent space optimization (LSO). The authors explain how LES builds on and improves over baselines, especially addressing robustness across benchmarks and tasks.", "essential_references_not_discussed": "The paper provides sufficient relevant background and work to understand the context of the paper.", "other_strengths_and_weaknesses": "* The use of decoder likelihood as a latent validity heuristic is well-motivated.\n* The paper provides detailed derivations and adequate explanations.\n* The method is more efficient than some baseline approaches, though this advantage may diminish for higher-dimensional latent spaces.", "other_comments_or_suggestions": "Minor typos: \n* Line 230: What is ScaLES? I think it’s not defined in the paper.\n* Line 226: Shouldn’t it be Theorem 3.1?", "questions_for_authors": "Please refer to my previous comments.", "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": "DI5FOyIeos", "reviewer": "Reviewer_D91d", "summary": "This paper addresses the problem of latent space optimization (LSO) for combinatorial problems. LSO uses an encoder-decoder architecture to map from a continuous latent space to a combinatorial space. The optimization can be carried out in a continuous space which has the advantage that it allows for continuous optimization.\n\nWhen sampling from the latent space, one often encounters invalid solutions (for instance, invalid arithmetic expressions if the problem at hand is to optimize such).\n\nThe authors propose a score (latent exploration score - LES) that promotes valid solutions and can be incorporated into the optimization objective as a regularization term. The authors show that doing so improves performance. They further validate that their score achieves higher density for valid solutions, which they use as a proxy for LES to avoid over-exploration.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and effort in reviewing our work.\n\nWe address the reviewer's questions and comments chronologically.\n\nThank you for suggesting the addition of the important work by Tripp et al. We would like to point out that we refer to this work in line 155:\n\n> “The frequent generation of invalid solutions during acquisition optimization, which implies that the estimated uncertainty can be problematic in this setting, underscores the need for additional regularization (Tripp et al., 2020), which we aim to address.”\n\nWe thank the reviewer for their suggestion to clarify the meaning of the parameter **beta**, which balances the alignment with the prior and the reconstruction loss. We will add a description of this parameter in the first paragraph of Section 3.2.\n\nDue to margin constraints, **Table 2** is a truncated version of the full **Table 11**, which is deferred to the appendix. In the revised version, we will make sure to clarify this point in the last paragraph of Section 3.2.\n\nThe intuition behind using the log-determinant terms is explained in lines 172–188. In short:\n\n> “Why use the sequence density function? We argue that for a well-trained decoder network, the density should be higher in areas of the sequence space close to the training data.”\n\nAs we mention in line 206:\n\n> “...as the contribution of the prior is negligible in magnitude in all the decoders we study,”\n\nthis is why we only use the log-determinant term.\n\nWe will make sure to be consistent in using the term **score** when referring to LES—thank you for the suggestion.\n\nThe results from Daubechies state that ReLU networks (which are CPA spline functions) can approximate refinable functions—a class of non-smooth functions—in addition to known results on other classes of functions (e.g., Weierstrass, polynomials, and analytic functions), as described in this work. This clarification will be added to the limitations of Theorem 3.1.\n\nThe **EMPTY** category allows generating sequences shorter than the maximal length. For example, if the maximal length is 3, then we can generate the expression `\"x\"` as `\"x<EMPTY><EMPTY>\"`. This example will be added to the first paragraph of Section 2 to enhance its clarity.\n\n**Sequential optimization** means that we generate each solution separately, rather than optimizing all the solutions jointly. This setup was used in the work by Notin et al., which we follow. This clarification will be added to the LSO setup paragraph in Section 4.", "claims_and_evidence": "The main claims of this paper are that 1) a high LES is likely to yield a valid point and 2) that incorporating LES in the black-box optimization improves performance.\n\nClaim 1 is mainly supported by Table 2 (see my questions below regarding this table), which compares LES to other scores (or ‘no score’ = ‘Prior’). LES improves the likelihood of a valid point and is better than the other methods (Uncertainty and Likelihood).\n\nClaim 2 is supported by Figure 3, which shows that LES improves performance or is competitive with the state of the art.\n\nOther claims:\n\n- Recurrent neural networks such as LSTMs or GRU networks can be approximated with high accuracy os PCA functions so that the theoretical results apply. This is supported by references but not shown explicitly in the paper.\n- LES gives high values for valid points in the latent space. This is supported by empirical evidence.", "methods_and_evaluation_criteria": "The evaluation criteria and benchmark datasets are chosen reasonably.", "theoretical_claims": "I did not check the correctness of the proofs.", "experimental_designs_or_analyses": "The experimental designs are comprehensive. The authors conduct a thorough analysis across multiple datasets and VAE architectures, providing a robust evaluation of LES.", "relation_to_broader_scientific_literature": "The LSE is particularly relevant for latent-space black-box optimization problems where the decoder can generate invalid solutions. The domain of problems this method can be applied to is relatively broad and the problem is quite relevant.", "essential_references_not_discussed": "The paper covers the essential references related to its contributions. One paper that should be added is “Tripp, Austin, Erik Daxberger, and José Miguel Hernández-Lobato. ‘Sample-efficient optimization in the latent space of deep generative models via weighted retraining.’ Advances in Neural Information Processing Systems 33 (2020): 11259-11272.\"", "other_strengths_and_weaknesses": "Table 2 is confusing. $\\beta$ has not been introduced before. It is mentioned later in the text, but it does not refer to that table again. The overall section around this table should be revised. I would expect the training data, VAE prior, and out-of-distribution samples to be shown separately in the table, but that doesn’t seem to be the case. It almost seems like something is missing, or I’m misunderstanding something, so the text should be revised.\n\nThe motivation for the definition of LES is very brief. Why is it chosen as the logarithm of the determinant term? What is the intuition behind this choice?", "other_comments_or_suggestions": "There is a broken sentence in the second paragraph of ‘Over-exploration in LSO’ starting with ‘While acquisition functions…’\n\nThe main problem should be introduced with an additional sentence in the abstract.\n\nThe authors sometimes refer to LES as a ‘constraint,’ but it’s not really a constraint. I suggest only using the term ‘score’.", "questions_for_authors": "Below Eq. (5), the authors refer to results from Daubechies et al. What do those say? \n\nRight above Eq. (1), what exactly is the reason for the empty category?\n\nRight above ‘Optimization tasks’ in the right column of page 6: What exactly do you mean when you say the acquisition function is sequentially maximized?", "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": "TU6RwuqkHG", "reviewer": "Reviewer_SqeL", "summary": "The paper proposes LES, a novel method for mitigating the over-exploration phenomenon in latent space optimization for discrete black-box optimization. LES can be easily integrated as a regularizer for acquisition function optimization. Experiment results validate that LES can improve the performance of latent space optimization methods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and effort in reviewing our work.\n\n**Deep Kernel**\n\nThe deep kernel affects only the computation of the kernel in the GP surrogate model—it does not reduce the dimensionality of the latent space on which LES is applied. Based on our results with the SELFIES VAE, we see no indication that achieving improvement becomes more difficult in higher-dimensional settings.\n\n**TuRBO Method**\n\nThe TuRBO method (similar to every other method in this work) uses the LogEI acquisition function, for the same reasons noted by the reviewer. We will add clarification in our revised version, in section 4 under LSO setup paragraph. \n\n**Ablation studies on $\\lambda$**\n\nAs requested by the reviewer, we provide an ablation study of the parameter $\\lambda$ for both LES and the Likelihood score. Due to space constraints, we present the full results for the average of the top 20 solutions—since this metric tends to exhibit lower variance—and summarize the key findings for both the best and top-20 solutions below.\nFor the average of the top 20 solutions, we find that using a lower $\\lambda$ (0.1) for the likelihood score significantly improves performance on the RANO task when using the pre-trained SELFIES VAE, increasing the score from 0.31 to 0.41. We will include this setting in both the appendix and the main text of the revised version.\nFor the best solution, we observe the following notable differences:\nFor the expressions transformer with $\\beta = 1$, increasing the regularization to $\\lambda = 0.2$ improves both LES and the likelihood scores to -0.46 and -0.42, respectively.\n\n\nFor the SMILES transformer with $\\beta = 0.1$, setting $\\lambda = 0.8$ improves the likelihood from 3.16 to 3.25, surpassing the 3.23 achieved by LES at $\\lambda = 0.5$.\n\n*Expressions*\n\n**LES**\n\n| Architecture | Beta |0.05 |0.1 |0.2 |\n|--|--|----------------|--|--|\n| GRU| 0.05 | -1.43| -1.28 | -1.32 |\n| GRU| 0.1| -1.34| -1.06 | -1.29 |\n| GRU| 1| -0.84| -0.98 | -1.08 |\n| LSTM | 0.05 | -1.06| -1.09 | -1.00 |\n| LSTM | 0.1| -0.80| -0.72 | -0.73 |\n| LSTM | 1| -1.79| -1.79 | -1.84 |\n| Transformer| 0.05 | -1.00| -0.86 | -0.92 |\n| Transformer| 0.1| -0.77| -0.75 | -0.67 |\n| Transformer| 1| -1.36| -1.28 | -1.05 |\n\n**Likelihood**\n\n| Arch | Beta |0.05 |0.1 |0.2 |\n|--|--|----------------|--|--|\n| GRU| 0.05 | -1.15| -1.10 | -1.11 |\n| GRU| 0.1| -1.31| -1.16 | -1.14 |\n| GRU| 1| -0.88| -1.04 | -1.15 |\n| LSTM | 0.05 | -1.00| -1.05 | -1.17 |\n| LSTM | 0.1| -0.75| -0.81 | -0.72 |\n| LSTM | 1| -1.81| -1.79 | -1.78 |\n| Transformer| 0.05 | -0.93| -0.95 | -0.98 |\n| Transformer| 0.1| -0.81| -0.74 | -0.70 |\n| Transformer| 1| -1.45| -1.42 | -1.19 |\n\n*SMILES*\n\n\n**LES**\n\n| Architecture | Beta |0.3 |0.5 |0.8 |\n|--|--|--|--|--|\n| GRU| 0.05 | 2.20| 2.31| 2.32|\n| GRU| 0.1| 2.12| 2.30| 2.35|\n| GRU| 1| 0.38| 1.64| 1.77|\n| LSTM | 0.05 | 2.34| 2.33| 2.37|\n| LSTM | 0.1| 0.94| 1.57| 1.80|\n| LSTM | 1| 0.59| 1.94| 2.17|\n| Transformer| 0.05 | 2.30| 2.26| 2.35|\n| Transformer| 0.1| 2.31| 2.26| 2.29|\n| Transformer| 1| 2.09| 2.17| 2.30|\n\n**Likelihood**\n\n| Architecture | Beta |0.3 |0.5 |0.8 |\n|--|--|--|--|--|\n| GRU| 0.05 | 2.15| 2.31| 2.25|\n| GRU| 0.1| 2.10| 2.12| 2.22|\n| GRU| 1| 0.60| 1.49| 1.40|\n| LSTM | 0.05 | 2.17| 2.28| 2.30|\n| LSTM | 0.1| 0.81| 1.43| 1.58|\n| LSTM | 1| 0.53| 1.67| 1.77|\n| Transformer| 0.05 | 2.43| 2.25| 2.32|\n| Transformer| 0.1| 2.26| 2.23| 2.34|\n| Transformer| 1| 2.01| 2.16| 1.98|\n\n*SELFIES*\n\n\n**LES**\n\n| Objective | Beta |0.1 |0.3 |0.5 |\n|--|--|--|--|--|\n| Pdop| 0.05 | 0.37| 0.37| 0.37|\n| Pdop| 0.1| 0.36| 0.36| 0.36|\n| Pdop| 1| 0.35| 0.32| 0.33|\n| Rano| 0.05 | 0.21| 0.22| 0.21|\n| Rano| 0.1| 0.21| 0.21| 0.19|\n| Rano| 1| 0.21| 0.21| 0.20|\n| Zale| 0.05 | 0.33| 0.34| 0.34|\n| Zale| 0.1| 0.34| 0.33| 0.33|\n| Zale| 1| 0.31| 0.31| 0.31|\n\n**Likelihood**\n\n| Objective | Beta |0.1 |0.3 |0.5 |\n|--|--|--|--|--|\n| Pdop| 0.05 | 0.37| 0.37| 0.37|\n| Pdop| 0.1| 0.36| 0.36| 0.34|\n| Pdop| 1| 0.34| 0.29| 0.28|\n| Rano| 0.05 | 0.21| 0.22| 0.21|\n| Rano| 0.1| 0.22| 0.21| 0.20|\n| Rano| 1| 0.21| 0.23| 0.19|\n| Zale| 0.05 | 0.33| 0.33| 0.32|\n| Zale| 0.1| 0.34| 0.33| 0.32|\n| Zale| 1| 0.30| 0.29| 0.26|\n\n**SELFIES (Maus et al)**\n\n**LES**\n\n| Objective | Beta | 0.1 | 0.3 | 0.5 |\n|-----------|------|---------------|---------------|---------------|\n| Pdop      | –    | 0.46          | 0.48          | 0.47          |\n| Rano      | –    | 0.30          | 0.29          | 0.30          |\n| Zale      | –    | 0.40          | 0.40          | 0.41          |\n\n**Likelihood**\n\n| Objective | Beta | Lambda = 0.1 | Lambda = 0.3 | Lambda = 0.5 |\n|-----------|------|---------------|---------------|---------------|\n| Pdop      | –    | 0.42          | 0.40          | 0.35          |\n| Rano      | –    | 0.22          | 0.29          | 0.26          |\n| Zale      | –    | 0.41          | 0.39          | 0.31          |", "claims_and_evidence": "The authors present their claim with clear and convincing evidence.", "methods_and_evaluation_criteria": "The authors follow (as far as I know) the conventional procedure for latent-space Bayesian optimization.", "theoretical_claims": "I checked that the Theorem 3.1 is valid.", "experimental_designs_or_analyses": "The authors follow (as far as I know) the conventional procedure for latent-space Bayesian optimization.", "relation_to_broader_scientific_literature": "Discrete black-box optimization can be used for scientific discovery problems.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "**Strengths**\n\n- As a result, LES can be implemented as a regularizer when we optimize the acquisition function during the Bayesian optimization procedure, which is straightforward. \n- The authors try to validate that LES can mitigate the over-exploration problem by toy experiment setting and try to validate that LES does not suffer from high time-complexity.\n\n**Weakness**\n\n- According to the experiment setup, the authors employ a deep kernel to reduce the dimension of latent space into 12 dimensions, which is quite low compared to the original setup of [1]. Is there any reason why such kind of dimension reduction is required? Otherwise, is it hard to achieve improvement using LES when the latent dimension is high?\n\n- I find that there are several ablation studies on hyperparameters of baselines. However, it is hard to find the ablation of hyperparameters such as $\\lambda$ in terms of performance. While I find that using LES can lead to a high proportion of valid solutions, I wonder if the suggested method is robust to different hyperparameter configurations in terms of performance.\n\n[1] Maus, Natalie, et al. \"Local latent space bayesian optimization over structured inputs.\" Advances in neural information processing systems 35 (2022): 34505-34518.", "other_comments_or_suggestions": "Here are some questions:\n\n- Authors use TuRBO as a baseline. When I read the experiment setup part, the authors used logEI as a base acquisition function. Do authors also apply LogEI for TuRBO as stated in [1]? For a fair comparison, it seems that all baselines use the same acquisition function.\n\n- For Table 10, it seems that regularizing the acquisition function with likelihood score is the second-best baseline. Could authors also provide ablation studies on $\\lambda$ for the likelihood baseline? I think it is crucial, as the likelihood score also has a single hyperparameter and is faster to compute. If we can achieve a similar performance by tuning the hyperparameter for the likelihood score, the significance of the proposed method greatly decreases. I hope the authors verify this part carefully.\n\n[1] Ament, Sebastian, et al. \"Unexpected improvements to expected improvement for bayesian optimization.\" Advances in Neural Information Processing Systems 36 (2023): 20577-20612.", "questions_for_authors": "Please see above.", "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": {"ethical_review_concerns": "N/A"}}, {"review_id": "6aeTroUlsY", "reviewer": "Reviewer_BNj6", "summary": "The paper proposes a latent exploration score method to mitigate over-exploration in latent space optimization, utilizing VAEs' properties for continuous optimization. It also demonstrates cases where solutions may become impractical.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and effort in reviewing our work.\n\n### Typos\n\nThank you for pointing this out—we will update the text to replace \"quadratic formula\" with \"quadratic function.\"\n\n### Additional Visualizations\n\nWe thank the reviewer for their great suggestion to add more visualizations to our manuscript.  \nAs Figure 1 provides a visualization of the dynamics within the latent space, we would be happy to include additional figures using different starting points in the revised version.\n\n### Empirical Rigor\n\nOur main claim is that LES can be used as a robust method (across VAEs) to improve the number of valid solutions discovered through latent space optimization. We consider a rigorous empirical evaluation of this claim to be one that explores multiple VAE models, multiple notions of validity, and multiple latent space optimization tasks. This definition aligns with the goal of LSO, where we aim to use a VAE to find realistic solutions for a given objective.\n\nBy this definition, we believe that our empirical evaluation is rigorous for the following reasons:\n\n- We study **22 VAEs**, varying both the decoder architecture and the beta parameters that control the trade-off between the prior and the reconstruction. This goes beyond previous related work, which studied only 4 VAEs [1].\n- We study **three notions of validity**, ensuring our empirical results are not specific to a single definition.\n- We evaluate **five different latent space optimization tasks**, all considered standard benchmarks in this literature [1,2].\n\n**References**  \n[1] Notin, P., Hernández-Lobato, J.M., & Gal, Y. (2021). Improving black-box optimization in VAE latent space using decoder uncertainty. *Advances in Neural Information Processing Systems*, 34, 802–814.  \n[2] Maus, N., Jones, H., Moore, J., Kusner, M.J., Bradshaw, J., & Gardner, J. (2022). Local latent space Bayesian optimization over structured inputs. *Advances in Neural Information Processing Systems*, 35, 34505–34518.\n\n### Methodological Validity\n\nWe believe that the validity of **Theorem 3.1** in our setup is well justified through the approximation results we cite. In addition, our extensive empirical evaluation further supports the soundness of our methodology.\n\n### Failure Cases of Other Methods\n\nThe failure mode we aim to address is that existing baselines tend to generate unrealistic solutions [1]:\n\n> “It has been widely observed that optimizing outside of the feasible region tends to give poor results, yielding samples that are low-quality, or even invalid (e.g., invalid molecular strings, non-grammatical sentences).”\n\n**Table 13** systematically demonstrates this widely known failure mode by comparing the percentage of valid solutions generated by each optimization method. In short, LES produces on average **62% valid solutions** more than any unregularized method—L-BFGS (26%), TurBO (38%), and GA (55%).\n\n**Reference**  \n[1] Tripp, A., Daxberger, E., & Hernandez-Lobato, J.M.  \n*Sample-efficient optimization in the latent space of deep generative models via weighted retraining.*\n\n### Discussion Section\n\nWe thank the reviewer for the valid suggestion to revisit Examples 1.1–1.3 in the discussion and to explicitly articulate the challenges being addressed, which we will happily do. A revised version of the first paragraph is included below:\n\n> “We propose LES to address over-exploration in latent space optimization (LSO). Desirable properties of LSO include stability—across tasks and VAE architectures—and the ability to consistently discover high-quality solutions that optimize a given objective. However, existing approaches often suffer from instability and excessive exploration, which can lead to unrealistic or invalid outputs without careful regularization. LES mitigates these issues by encouraging optimization to remain near the data manifold, thereby stabilizing LSO and improving the realism of generated solutions. LES is differentiable and fully parallelizable. Extensive evaluations demonstrate that incorporating LES as a penalty in LSO consistently enhances solution quality and objective outcomes. Moreover, LES outperforms alternative regularization techniques, proving to be the most robust across diverse datasets and varying definitions of validity—including syntactically valid expressions, chemically valid SMILES, and molecules that pass domain-specific property filters. In addition, LES has only a single hyperparameter (the regularization strength), and we observe empirically that deploying LES can provide significant performance gains. We therefore believe LES offers a powerful approach for discovering more realistic solutions, especially when the criteria for realism are difficult to define or validate.”", "claims_and_evidence": "The paper begins by examining cases of over-exploration in latent space optimization and proposes a method to efficiently identify regions that preserve the structure of the sequence space. The authors assert that all decoder neural networks in their study can be accurately approximated as continuous piecewise affine functions.", "methods_and_evaluation_criteria": "The paper introduces a latent exploration score to constrain the acquisition function optimization in a VAE's latent space. This constraint is designed as a continuous function of z, where higher values indicate greater likelihood that z lies within valid latent space regions.", "theoretical_claims": "The authors establish a theoretical framework representing Deep Generative Networks (DGNs) as Continuous Piecewise Affine (CPA) spline operators. Based on this formulation, they derive theorems characterizing the density function.", "experimental_designs_or_analyses": "The authors experimentally validated the relationship between the Latent Exploration Score (LES) and valid generation through multiple trials, with results averaged for statistical reliability.", "relation_to_broader_scientific_literature": "The method could potentially be valuable for drug discovery applications.", "essential_references_not_discussed": "Given that this analysis focuses on latent space dynamics, would it be valuable to include additional visualizations correlating latent space movement with variations in generated outputs?", "other_strengths_and_weaknesses": "While the proposed method presents an interesting approach with potential for scientific discovery, the experimental results currently lack sufficient empirical rigor to fully support its claims. Additionally, Theorem 3.1 appears to function more as a formulation or assumption rather than a demonstrably proven theorem. Due to these limitations, the methodological validity remains unconvincing in its current form.", "other_comments_or_suggestions": "On Page 4, Line 213, the term should be corrected to \"quadratic function\".", "questions_for_authors": "The paper would benefit from (1) a more explicit articulation of the specific challenges being addressed, and (2) a systematic demonstration of comparative limitations (e.g., failure cases of existing methods) to better contextualize its contributions. The discussion could be enhanced by explicitly revisiting Examples 1.1–1.3 in the conclusion, systematically comparing the proposed method’s generated expressions with those of baseline approaches. This would reinforce how the technique addresses the specific limitations identified earlier, providing a cohesive demonstration of its advancements.", "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": "LOLkItQ5kM", "reviewer": "Reviewer_p6GH", "summary": "This work targets the problem of over exploration in latent space optimization. The overarching problem is that when we optimize over discrete data in the latent space of a generative model e.g. a VAE, we often select points that decode poorly or are otherwise invalid. This work proposes a score (LES) to guide the latent space optimization process as a constraint. LES is informed by the decoder loss in order to score points higher when they're closer to the training data. Formally, LES is defined to be the output log-density of the decoder \n\nThis method is evaluated on more than 20 VAE models over 5 problem domains such as SMILES, SELFIES, expressions, etc. \nThe LES score is evaluated against baselines such as a \"max probability\" score, an uncertainty-based approach, the density of the prior, the polarity score, and also a score based on distance to training data.\n\n----------------------------\n### update after rebuttal\n\nI appreciate the additional clarifications and experiments. I will keep my (positive) score then same.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and effort in reviewing our work. We are glad to see that the reviewer finds our work to address an appropriate hole in the research landscape.\n\nWe would like to clarify that we did not claim that LES is faster than all alternative approaches, but rather that it is faster than Bayesian uncertainty methods (lines 231–236):\n\n> “LES is computed faster than the Uncertainty score, achieving reductions of up to 85% in some cases. As the Likelihood score requires only a single forward pass of the decoder, it offers a more computationally efficient alternative, which comes with some performance trade-offs.”\n\nWe believe that this claim is well-supported by Table 1.\n\n### Is Gaussian with std = 5 OOD?\n\nWe believe that a standard deviation of 5 is sufficient to qualify as out-of-distribution (OOD). To substantiate this claim, we provide the proportion of valid solutions for each VAE on a random sample of 500 data points generated with std = 5:\n\n| Architecture | Beta  | % Valid | Dataset     |\n|--------------|-------|---------|-------------|\n| Transformer  | 0.05  | 0.00    | SELFIES     |\n| Transformer  | 0.1   | 0.01    | SELFIES     |\n| Transformer  | 1     | 0.14    | SELFIES     |\n| Pre-trained  | —     | 0.09    | SELFIES     |\n| GRU          | 0.05  | 0.07    | SMILES      |\n| GRU          | 0.1   | 0.01    | SMILES      |\n| GRU          | 1     | 0.00    | SMILES      |\n| LSTM         | 0.05  | 0.00    | SMILES      |\n| LSTM         | 0.1   | 0.01    | SMILES      |\n| LSTM         | 1     | 0.00    | SMILES      |\n| Transformer  | 0.05  | 0.02    | SMILES      |\n| Transformer  | 0.1   | 0.03    | SMILES      |\n| Transformer  | 1     | 0.00    | SMILES      |\n| GRU          | 0.05  | 0.79    | Expressions |\n| GRU          | 0.1   | 0.67    | Expressions |\n| GRU          | 1     | 0.61    | Expressions |\n| LSTM         | 0.05  | 0.81    | Expressions |\n| LSTM         | 0.1   | 0.80    | Expressions |\n| LSTM         | 1     | 0.64    | Expressions |\n| Transformer  | 0.05  | 0.67    | Expressions |\n| Transformer  | 0.1   | 0.46    | Expressions |\n| Transformer  | 1     | 0.29    | Expressions |", "claims_and_evidence": "Claims: \n1. The LES score increases the number of valid solutions when used as a constraint in LSO\n- This is quantified in Figure 3 and Table 3, where LES produces the highest percentage of valid solutions\n2. LES is 80% faster than alternative approaches\n- This is not well supported by Table 1. It looks like the likelihood score is 5-10X faster; the likelihood score also performs reasonably well on OOD data. It is worth nothing that the likelihood score underperforms in real tasks. \n3. LES-constrained DSO performs better than six baseline approaches. \n- Given the BO and OOD experiments, (tables 2 and 3 respectively), this is well supported by the experimental evidence.", "methods_and_evaluation_criteria": "I checked the benchmark datasets e.g. SMILES, expressions, SELFIES. \nThese datasets all feature discrete optimization settings that are reasonable for evaluating and using a score like LES.", "theoretical_claims": "I checked the proof of lemma / theorem 3.1, I did not find any issues.", "experimental_designs_or_analyses": "I checked the following experiments:\n* AUROC of OOD data shown in Table 2\n- The setup for OOD data seems reasonable, if not a little contrived. Part of the motivation for this work was that other approaches find solutions that are invalid under the decoder. Is a Gaussian with std=5 enough to map to this scenario? \n* Score computation time shown in Table 1\n- No issues with timing, though likelihood is quite fast for the amount of performance that you lose. \n* Bayesian Optimization experiments shown in Figure / Table 3. \n- I don't have issues with the BO setup.", "relation_to_broader_scientific_literature": "This work found an appropriate hole in the research landscape. Namely, that many samples returned by black-box optimization in the latent space of deep generative models do not decode into valid samples. Other methods that tackle this problem are either simplistic (sample from the prior) or else use the decoder in a round-about way such as using the uncertainty of the decoder to score samples [1]. This method sits in between the two, by using the log-determinant of the decoder density directly we can see if a sample is valid. \n\n[1] Notin, Pascal, José Miguel Hernández-Lobato, and Yarin Gal. \"Improving black-box optimization in VAE latent space using decoder uncertainty.\" Advances in Neural Information Processing Systems 34 (2021): 802-814.", "essential_references_not_discussed": "I did not find any additional references that need to be included.", "other_strengths_and_weaknesses": "* I appreciate that this method is \"decoder aware\" given that any sample has to plausibly map back to a training-like sample under the decoder, the proposed method ensures that we're much more likely to get a valid sample -- which is what we really care about in applications. \n\n* I found this paper to be well-written.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": {"ethical_review_concerns": "N/A"}}, {"review_id": "9XjpDmbGqS", "reviewer": "Reviewer_A6Zc", "summary": "The paper develops Latent Exploration Score (LES) to mitigate the over-exporation in Latent Space Optimization (LSO). LES can be applied to any VAE decoder, and the paper develops a numerical procedure to incorporate LES as a constraint in LSO. The paper evaluates LES on multiple benchmark tests and demonstrates that LES improves the existing results.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and effort in reviewing our work.\n\n### Comments and Suggestions\n\nWe thank the reviewer for their comments and suggestions, which we will incorporate into the camera-ready version of our work.\n\n### Question\n\n**All code and trained VAEs will be released if the paper is accepted.**\n\nWe believe the UC early stopping procedure is too conservative, and essentially gets stuck around the initial data point—especially if the landscape is not smooth in terms of valid and invalid regions. Figure 1 provides some evidence that this is the case for the expression dataset. We will add this explanation in our revised version (lines 400–405) for further clarity.\n\nIn practice, we believe that for the more interesting molecular tasks, $\\lambda$ = 0.5 should generally perform better than what is currently used (e.g., Turbo). We provide this guideline in lines 370–372:\n\n> “Based on our findings, we suggest $\\lambda$ = 0.5 as a reasonable default value, while leaving the exploration of an optimal choice for future work.”", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation on multiple benchmarks based on AUROC values make sense to me.", "theoretical_claims": "Yes, the proof looks correct to me.", "experimental_designs_or_analyses": "Yes. The analyses look sound.", "relation_to_broader_scientific_literature": "The paper is related to the broader literature on VAE, latent space optimization, and the over-exploitation issue. The paper has discussed it in details in the background section.", "essential_references_not_discussed": "not I'm aware of.", "other_strengths_and_weaknesses": "The paper is clearly-written, theoretically sound, and has novel idea. The experiment results also look good. I just have few questions/comments below.", "other_comments_or_suggestions": "1. it might be better to add a link to the appendix D Bayesian uncertainty score on line 220 (LES is compared with the Bayesian uncertainty score...)? \n\n2. Table 3 on line 367-368 is linked to wrong place.", "questions_for_authors": "1. would the code used for experiments in this paper be released if the paper is accepted? \n2. could you provide some insights on why UC finds more valid solutions (especially for expressions task) in table 13, but this doesn't seem to transfer to better AUROC values?\n3. could you provide some guidelines on how one should tune the λ parameter in practice?", "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": "iGzvJMDXdu", "reviewer": "Reviewer_oSTV", "summary": "This paper proposes a new score for optimizing the encoded latent space from pre-trained VAEs. The proposed new score employs the pre-trained decoder as a constraint to mitigate the over-exploration problem.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and effort in reviewing our work.\n\n### Experimental Design and Analysis\n\nWe calculate Equation 13 for a given **Z** because we are interested in a score that can be attributed to a single latent vector, rather than to a distribution over the output space \\( p(x) \\). Indeed, \\( p(x) \\) is not a function of a single latent vector and cannot be used to guide the optimization. We will make sure to emphasize this point in the revised version of our manuscript, under Equation 13.\n\n** likelihood vs LES **\n\nThe reviewer is making a great point, and we are glad to see that the reviewer agrees with us (lines 234–237) that:\n\n> “As the Likelihood score requires only a single forward pass of the decoder, it offers a more computationally efficient alternative, which comes with some performance trade-offs.”\n\nIt is important for us to provide the full picture so that practitioners can use LES in cases where they feel the juice is worth the squeeze, but can also opt for the likelihood score if they have a limited computational budget.", "claims_and_evidence": "The author is motivated to use a pre-trained decoder (Sec.3, why use the sequence density function)  and provides empirical validation for the motivation. With my limited experience in this specific area, I believe no theoretical evidence is made. As such, I think it would be better to further validate the motivation.", "methods_and_evaluation_criteria": "The author conducts various evaluations based on standard benchmarks and performs ablation studies for different VAE backbones.", "theoretical_claims": "N/A.", "experimental_designs_or_analyses": "1. Despite most parts beyond my expertise, why does Eqn.13 compute likelihood for only p(x|z) instead of the log-likelihood over p(x)?\n\n2. By comparing the performance of likelihood and LSO, it seems Likelihood can render competitive performance (sometimes better) while requiring much less computational cost. How does the author address such a balance?", "relation_to_broader_scientific_literature": "The proposed idea is to address the problem of over-exploration issues in prior LSO methods.", "essential_references_not_discussed": "Due to my limited knowledge, I think the author conducted extensive experiments to demonstrate and discuss the model for baselines.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "The paper is, in general, well-written, while it can be organized better (e.g., the three examples in the introduction can be in the background).", "questions_for_authors": "n/a", "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:

Meta-review: To address the issue of over-exploration in existing Latent Space Optimization (LSO) methods, the authors introduce the Latent Exploration Score (LES). LES serves as a constraint within the LSO framework to guide the optimization process more effectively. Formally, it is defined as the output log-density of the decoder, helping to regulate the search in latent space and prevent overly aggressive exploration.

Almost all reviewers agree that the paper merits acceptance, and after carefully reading the rebuttal and discussion, I tend to concur. Please incorporate the reviewers’ feedback into the final version of the paper, with particular attention to the following concerns:

1- Incorporate the hyper-parameter tuning experiments 

2- Conduct additional ablation studies for the extra factors and parameters 

3- Provide further clarification regarding the comparisons with certain baselines
Decision: Accept (poster)