Gradient GA: Gradient Genetic Algorithm For Drug Molecular Design
Authors: Debadyuti Mukherjee
Chris Zhuang
Yingzhou Lu
Tianfan Fu
Ruqi Zhang
Venue: Accepted by TMLR ()

Abstract: Molecular discovery has brought great benefit to the chemical industry. Various molecu-
lar design techniques have been developed to identify molecules with desirable properties.
Traditional optimization methods, such as genetic algorithms, continue to achieve state-of-
the-art results across various molecular design benchmarks. However, these techniques rely
solely on undirected random exploration, which hinders both the quality of the final solution
and the convergence speed. To address this limitation, we propose a novel approach called
Gradient Genetic Algorithm (Gradient GA), which incorporates gradient information from
the objective function into genetic algorithms. Instead of random exploration, each proposed
sample iteratively progresses toward an optimal solution by following the gradient direction.
We achieve this by designing a differentiable objective function parameterized by a neural
network and utilizing the Discrete Langevin Proposal to enable gradient guidance in discrete
molecular spaces. Experimental results demonstrate that our method significantly improves
both convergence speed and solution quality, outperforming cutting-edge techniques. The
proposed method has shown up to a 25% improvement in the Top 10 score over the vanilla ge-
netic algorithm. The code is available at https://github.com/debadyuti23/GradientGA.

Published in Transactions on Machine Learning Research (11/2025)
Gradient GA: Gradient Genetic Algorithm For Drug Molecular Design
Debadyuti Mukherjee"
mukher838purdue..et
Deportment of Computer Science
Purdue Universily
Chris Zhuang"
shuang808@urddue.co edu
Department of Computer Science Purdue University
Yingzhou Lu
ly=660stan edu
Stanford Medicine School Standford University
Tianfan
fut2@rpi..om edu
Department of Computer Science
Rensselaer Polylechmic Instilute
Ruqi Zhang
rugiz&pardue.ccmm edu
Department of Computer Science
Purdue University
Reviewed OpenReview: /ttps:///penreviiee net/ forum? id= kFXckttke
Abstract
Molecular discovery has brought great benefit to the chemical industry. Various molecn- lar design lechniques have been developed to identify molecules with desirable properties. Traditional optimization methods, such as genetic algorithms, continue to achieve state-of- the-art results across various molecular design benchmarks. However, these techniques rely solely on undirected random exploration, which hinders both the quality of the final solution and the convergence speed. To address this limitation, we propose a novel approach called Gradient Genetic Algorithm (Gradient GA), which incorporates gradient information from the objective function into genetic algorithms. Instead of random exploration, each proposed sample iteratively progresses toward optimal solution by following the gradient direction. We achieve this by designing differentiable objective function parameterized by neural network and utilizing the Discrete Langevin Proposal to enable gradient guidance in discrete molecular spaces. Experimental results demonstrate that our method significantly improves both convergence speed and solution quality, outperforming cutting-edge techniques. The proposod method has shown up to 25% improvement in the Top score over the vanilla go- netie algorithm. The code is available at whttts:///// httee://gittuubocom httthab..co/eeeaappps/////eerere
1 Introduction
Designing molecules with desirable biological and chemical properties has become a demanding research topic since its outcome can benefit various domains, such as drug discovery (Huang et al. 2022 material design Yang et al. 2017), etc. However, a limited number of molecules be tested in real-life laboratories
"qqual contribution. First authorship determined by rolling dice.

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Published in Transactions on Machine Learning Research (11/2025)
Altae-Tran et al. 2017) and clinical trials (Chen et al. 2024b a). Therefore, numerous effective techniques for molecule discovery have been proposed to discover favorable molecules throughout the vast sample space.
Some evolutionary algorithms, such as molecular graph-based genetic algorithm (Graph GA) (Jensen 2019a remain strong baselines, often outperforming recently proposed machine learning- based algorithms (Huang et al. 2021; Gao al.] (022b).. Genetic algorithms are cheap, easy to implement, and often regarded as simple baselines for molecular discovery. However, key GA operators, such as selection, crossover, and mutation, are random and do not use knowledge of objective functions. Given the vast molecular scarch space, this random walk approach is like searching for needle in haystack. As result, GA tends to converge slowly, and its final performance can be unstable.
To address this issue, we introdece a novel molecule design method, Gradient Genelic Algorilhm (Gra- dient GA), which leverages gradient information to navigate chemical space efficiently. First, we learn a diffcrentiable objective function using graph neural network (GNN) (Scarselli ct al., (009), which maps the graph-structured information of molecules to vector embeddings. We then apply the Discrete Langevin Proposal (DLP) (Zhang et al. 2022) to incorporate gradient information from this objective, enabling more informed exploration in the discrete molecular space. Our main contributions are summarized as follows.
We introduce Gradient GA, a gradient-based genetic algorithm for more informative and effective exploration in molecular spaces, mitigating randon-walk behavior in genetic algorithms. To the best of our knowledge, this is the first method to leverage gradient information within a genetic algorithm framework.
To enable diffcrentiable objective function for discrete molecular graphs, we graph neural network property predictor to approximale ianl objectives. This allows us to compute gradients by taking derivatives of NN-parameterized objectives with respect to the vector embeddings.
The experimental results demonstrate that the proposed method achieves a significant and consis- tent improvement over a number of cutting-edge approaches (e.g., Graph-GA, SMILES-GA), For example, achieving an improvement of up to 25% over the traditional GA when optimizing the mestranol similarity property.
2 Related Work
Al-aided Drug Molecular Design. Current Al-aided drug molecular design techniques can be primarily classified into two categorics: deep generative models and combinatorial optimization methods.
(T) Deep Generative Models (DGMs) learn the distribution of general molecular structures using deep net- works, enabling the generation of molecules by sampling from the learned distribution. Typical algorithms include Variational Autoencoders (VAEs), Generative Adversarial Networks (GANs), energy-based models,
and flow-based models [Gómez-Bombarell et al. 2018 Jin et al.
2018 De Cao & Kipf 2018 Segler et al.
2018 Fu et al. 2020 Honda et al. 2019 Madhawa et al.) 2019
Liu et al. 2021 Fu & Sun 2022 Chen et al.
2024b Bagal et al. 2021). However, these approaches often require a smooth and discriminative latent space, necessitating careful network architecture design and well-distributed datasets. This requirement can be restrictive in certain scenarios, such as multi-objective optimization. Furthermore, since DGMs learn the distribution of reference data, their ability to explore diverse chemical space is relatively limited, as demonstrated by recent molecular optimization benchmarks (Brown et. al.) 2019: Huang al. 2021 Gao
et al. 2022b).
(II) On the other hand, combinatorial optimization methods directly search the discrete chemical space,
mainly including deep reinforcement learning YYou et al. 2018 Zhou et al. 2019 Jin el. al.) 2020 Gottipati eet al. 2020), evolutionary learning methods (Nigam et al. 2020 Jensen 2019b Fu et al., 2022a). and sampling methods (Xie et al.) 2021 Fu al.) 2021). Specifically, Jensen (0019a) has proposed molecular graph- genetic algorithm. In drug discovery, this algorithm samples two parent molocules and generates child molecules by combining fragments of the parents, with probability of random mutations occurring in
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the offspring. The population is then refined by selecting the highest-scoring molecules. Also, Differentiable Scaffolding Tree (DST) {Fu et al. 2022b) differentiates the structure of discrete molecules and enables gradient computation for efficient optimization; Xie et al. (2021) uses the Markov chain Monte Carlo method (MCMC) to sample potential molccules. Each sampled molecule forms a Markov chain, modeled as a chemical transformation of the previous sample. This transformation occurs through one of two possible actions: (i) the addition of molecular fragments or (ii) the removal of chemical bond. However, both MARS and DST rely on a single selected molecule and fail to incorporate other molecules' properties, unlike GraphGA, hurting their top performance. This motivates us to propose an AL-guided genetic algorithm.
Discrete Sampling. Many applications involve discrete data spaces, such molecular, text, and tabular data. Gibbs sampling has long been the standard method for discrete sampling. However, because Gibbs sampling updates only one variable at a time, it often suffers from slow convergence. To address this, various improvements to Gibbs sampling have been proposed. Tittsiss & Yau (2017) introduces auxiliary variables to enable block updates within the Gibbs sampler. Nishimura Suchard (2023) reformulates the sample space to simplify the sampling process by using prior preconditioning and conjugate gradient techniques. Recently, a growing body of work has explored the use of gradient information to improve sampling in discrete spaces (Grathwohl et al., 2021 Sun et al. 2022 Pynadath et al., 2024). Among these, the Discrete Langevin Proposal (DLP) (Zhang et al. 2022) stands out an analog of the Langevin dynamics adapted to discrete spaces. DLP not only utilizes gradient information but also updates all variables simultaneously at each step, leading to better efficiency.
3 Preliminaries
3.1 Discrete Langevin Proposal
Suppose that the target distribution is T()) & exp(U(u)), where U()) is the energy function, v is n- dimensional variable in the space R". Langevin Dynamics samples from a by iteratively updating v as follows:
o' v'= v
2 VU()) + Vas, N~0
(1)
where a is the step size; Inxn is n-dimensional identity matrix; N(ss)) denotes high-dimensional normal distribution.
From Equation
the probability of selecting v' from V, i.e. p(v'lo),, can be written
p() exp ( 1
|| v 2 11)))
(2)
2a
The distribution p(v'lu) has its mean shifted from toward the optimum due to the gradient term. Therefore, pranahpbbbbbiitty samples from p(o"lo) will be closer to the optimum compared to
To extend Langevin Dynamics to discrete space, Zhang et al. 2022 has suggested the following proposal for the discrete sample space S:
exp (- 1 10 - 0 2 (U)))
p(v'lu) Σ Evrrs (exp
2a
(3)
111! 2 (U))i)
2a
3.2 Genetic Algorithm
The genetic algorithm (GA) is traditional combinatorial optimization method motivated by natural selec- tion and biological evolution processes. Unlike neural network-based methods, GA does not have learnable parameters, is usually easy to implement and tune, and bypasses the overfitting issue. Specifically, a GA process starts by randomly sampling a population of candidates. In drug design, for example, the candidates can be drug molecules. In the t-th generation (iteration), given the population of candidates, GA follows three key steps.
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Randomly Sample Two Parents
Initial Molecule Population
Generate
Sample Parents
Children Population
Update Population
Return
Final
eet Putta
Population
Sample Children based DLP
Feed in Population
Train Graph Embedding Network
L
Embedding
L
Graph Embedding
Update
Predicted Metric Score
Network
Sample Malecule
MPNN MPNN output
Gradient Representation
Intermediate Derivative
Sigmoid Derivative
Figure 1: Gradient GA pipeline.
Crossover, also called recombination, exchanges the structure two parents to generate new children. Specifically, two parents are randomly selected from the population, and their molecular structures are partially swapped create two child molecules. The crossover operators are carriod out multiple times independently, and the generated children are added to the offspring set S(t)
2. Mutation operates single parent molecule (randomly selected from the population) and modi- fies its structure slightly via randomly selecting substructure and flipping it to a new substructure different from its original State. Like the crossover, the mutation is conducted multiple times inde- pendently, and the resulting offspring are retained in the offspring set s(t)
Evolution. Given a population of molecules S(O at the 6-th generation, we generale an offspring pool by applying crossover and mutation operations. Molecules with undesirable properties (e.g.. poor solubility or high toxicity in drug discovery) are filtered out, and the top k candidates are selected to form the next generation population S(++1)
4 Methodology: Gradient Genetic Algorithm
Overview. In this section, we introduce Gradient Genetic Algorithm (Gradient GA). Section 4.1 begins by formulating the molecular design problem. Then, Soction 4.2 describes how to derive gradients by projecting discrete molecular graphs into continuous embedding space. Next, Section 4.3 explains how to use the embedding-derived gradients to efficiently explore the molecular space. For clarity, all mathematical notations are listed and explained in Table The overall pipeline is illustrated in Figure and the corresponding pseudocode is provided in Algorithm
4.1 Formulation: Molecular Design
Drug molecular design aims identifying novel molecules with desirable pharmaceutical properties, which evaluated by oracle. Oracles follows.
objective functions in molecular optimization tasks, formally defined
Definition 4.1 (Oracle). Oracle O(.) : R a blx function that evaluates certain physical, chemical, biological properties of molecule X and yields the ground-truth property O(X).
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Table Mathematical notations and their explanations.
Notations
Explanations
oracle function
Q
molccular space
M
complete MPNN-based model for oracle prediction graph representation
G()
L
multiple-layer perceptron (MLP)
(())
target distribution (normalized score)
U()
energy function population, a set of molecules.
D
In drug discovery, all the oracles can be categorized into two classes based on their accessibility: computa- tional and experimental (wet-lab). Experimental oracles, e.g., in vivo experiment, typically require wet-lab experiments evaluate, which are too expensive and time-consuming. Following most machine learning- aided drug discovery papers, we focus on computational oracles that are easy evaluate in silico, such molecular docking and the quantitative estimate of drug-likeness (QED) (Bickerton et al.. 2012). In real- world drug discovery scenarios, the cost of acquiring oracle evaluations is typically significant and cannot be overlooked. Mathematically, the drug molecular design problem (a.k.a. molecular optimization) can be formulated as
arg max O(X), Xee
(4)
where X is molecule, Q denotes the whole molecular space, i.c., the set of all chemically valid molecules. The size of the whole molecular space is around 1060 (Bohacek et al. 1996) Following MARS (Xie el al. 2021), we regard (-) as an unnormalized probability distribution and introduce a vector embedding v for each molecule X. The target distribution is then defined as =(v) x O(X).
Gradient Definition in Molccular Design. We now define the gradient information used to guide exploration in molecular spaces. To apply (discrete) Langevin dynamics, we need the gradient of the energy function. Since z(v) xx ex)(U(u)), by the chain rule, we have
VU(v)= = Vz(v) T(v)
(5)
Given that T(V) O(X), this leads to
VO(X) Vf(e) VU(0) O(X) O(X)
(6)
where f a differentiable function that approximates the oracle O, i.e. f(v) = 0(x) for all X E Q. We will discuss how to obtain both f and v in the next section.
It is also possible define U(+) = O(X) and (U()) Vf(w). However, this formulation does not incorpo- rate the oracle value into the gradient. Our approach includes O(X) in the denominator, effectively playing the role of adaplive step sizes. We found that this formulation leads to better performance. An empirical comparison is provided in the Appendix
4.2 Gradient Computation
Implementing gradient-based methods molccular discovery is a challenging task due two primary ob- stacles: (1) representing sample molecules in vector format suitable for gradient-based methods, and (2) establishing differrnniialee relationship between the probability distribution and the vector representation.
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gradient representation
parent better oracle
DLP distribution
final sample
parents
V
M
0.70 0.15 0.009 0.008 .....
Oppr
DLP
(U(u)
crossover
M
Otsp'
450
Pir
crossover sample space
Overview of DLP-based sampling procedure in Gradient GA, illustrating how the sampled molecule moves toward the optimum.
Figure 2:
Finding the Embedding. Most gradient-based methods use fixed-length vectors, while molecular data are often represented as graph structures. Therefore, an officient mapping function is needed to convert molecular graphs into fixed-length embeddings. However, directly transforming graph into an embedding using hard-coded functions, such Morgan's fingerprint,
cause unnecessary aggregation and information loss (Chang et al. 2019). To overcome this issuc, we use the Message Passing Neural Network (MPNN) Giimer et al. 2017), which is a Stale-of-the-arl approach for molecular activity prediction. We adopt the MPNN architecture from Xie et al. (2021), which consists of (1) a simple neural network for message passing between neighboring nodes and (2) a Gated Recurrent Unit (GRU) (Cho 2014] for updating node representations. For the readout function, we use Set2Set (Vinyals et al. 2015). which particularly effective for isomorphic graphs due to its set-invariant, property. To reduce the dimensionality of the Set output, introduce two-layer multilayer perceptron (MLP) L1, which compresses the output dimension from 2n to n.
We define the graph representation function, which converts molecular data into graph data (nodes rep- resenting atomic fcatures and edges representing bond fcaturos), as G(.). Consequently, the output of Lı serves as the embedding v for the molecular graph Gx = G(X), as shown below.
h MPNN(G L1(h).
(7)
Finding the Gradient. Similar to the vector embedding problem, we can learn the differentiable ap- proximation f(-) in Equation 6 using an MLP-based architecture. After the first two-layer MLP L1, we introduce another two-layer MLP L2, which produces scalar output 7. The gradient Vf(()) then be computed through backpropagation, behavior of the oracle function O so that the entire architecture (from the MPNN to L2), denoted by M, effectively learns both the embedding and the differentiable objective function. To introduce nonlinearity, we apply Lea and sigmoid activation functions before and after Lg, respectively. With v and (U(-) now properly defined for Equation 3 we procoed describe the training procodure. We train the model M to fit. the oracle function O using the initial molecule population D. The loss function is defined as the mean squared error (MSF):
illustrated in Figure 1 Our objective is to make ÿ approximate the
Σ [|M(Gx) O(x)112
D
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The initial molecule population is randomly sampled from drug-like molecule databases, such as ZINC (Ster- ling & Trwin 2015). During the molecular optimization process, continuously expand the training set by adding newly generated molecules that surpass a threshold criterion T. This provides additional information about the distribution of molecules and allows us to retrain the model M. To ensure that the approximation from M accurate, we provide empirical results of the performance using a validation set in Appendix [
4.3 Iterative Sampling
We propose a sampling technique inspired by both Graph GA (Jensen 2019a) and DLP Zhang el al. 2022 The workflow for each iteration illustrated in Figure 2 Similar to Graph GA, we begin by selecting parent molecules based on their scores O(+) and generating child molecules through crossover. We define the sample space S for DLP as the set of all possible crossovers between the selected parents d1 and d2 (E D). Ideally, both parents should be considered as current samples. However, since DL.P is designed to use a single sample, we aggregate the information from both parents into a single embedding v and gradient (u)) using the following equation:
tu, (U(()} = Σ Em - (Us, (U()))
(8)
where {e,, (v:) represents the embedding and gradient information for parent d, Empirically, we found that a simple strategy of using only the best parent as the current sample works well: w; = 1 = arg max()(d))) and wy otherwise; we have provided a comparison of the performance when averaging the parents in Appendix E DLP updates the embedding o' by moving it closer to the optimum, guided by the gradient information. In the final step, DLP generates the next sample set D' of fixed size k from the sample space S. Following the Graph GA approach, each molecule in D' is mutated. Before the next iteration begins, we update both the population and the model. The population D is refreshed by selecting the top [D] molecules based on their oracle scores from the combined set (D, D'}. To further enhance the graph embedding model M's understanding of the larget. molecule distribution TT-), we retrain M using a training set D", which is updated in each iteration according to the following rule:
{d d E D' and T(d)),
(9)
where T(d) is a threshold criterion for adding new samples to the training set. The complete Gradient GA workflow is detailed in Algorithm !
5
Experiments
5.1
Experimental Setup
Baseline Methods. We use the practical molecule optimization (PMO) benchmark (Gao et al. 2022a as our code base to compare results with the state-of-the-art methods. We Welect (1) genetic algorithm, inchuding Graph GA (molecular graph-level genetic algorithm) method (Jensen] 2019b) and SMILES GA (SMILES string-level genetic algorithm), (2) sampling based methods, including MIMOSA Mulli--onsttrinn Molecule Sampling) (Fu et al. 2021), MARS (Markov Molecular Sampling) (Xie et al., 2021 and (3) gradient-based method, DST (Differentiable Scaffolding Tree) (Fu et al.] 2022b We perform a maximum of 2500 oracle calls for each method with early-stopping being enabled when negligible performance improvement occurs a certain number of times.
Datasct. For all mothods, we use the ZINC 250K database TTwwin et al. 2012 to select the initial molccule population, extract chemical fragments, and perform pretraining. ZINC is a free database of commercially available compounds for virtual screening.
Evaluation Metrics. We consider the following evaluation metrics to assess the quality of generated molecules.
1. Average Top-K (K = 10) the top-K average property value, which measures the algorithm's op- timization ability. We limit the number of oracle calls to 2,500 to mimic the real experimental setup, though we expect methods to optimize well within hundreds of calls.
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Algorithm Gradient Genetic Algorithm
Input: oracle function O, step size a, retrainable threshold 7 Initialize D + original population
Initialize new molecule set D' + 0
Train predictive model .M using {(G(d), (dd) Ad D} Initialize retrained molecule set D" O for 1,2,. do
p(d) x O(d) E
Parent molecules d1, d2 p(d)
[d DJ
Get parents' embedding i for each G(d,) using Eq.
Get parents' gradient (U;)) for each dj using Eq.
Evaluate v, (U) using Eq. 8
Get crossover set: S + CROSSOVER(d) d2)
Get sampling probability probs of S using Eq.
3
Evaluate sample set D' +- Sample(S, probs, k)
Mutate each molecule in D' +- MUTATE( D')
Update population D < top_ aop_oracle( ( D, Di DI)
Update training set D" with using Eq. if DD" 2 7 then
Retrain model M with ((d) Vd D")
end if
D" < 0 or + D"
end for
2. AUC top-K (K = 1, 10, 100). Following the setup of practical molecular optimization benchmark Gao et al. 2022b), we assess both optimization ability and sample efficiency using the area under the curve (AUC) of top-K average property value versus the number of oracle calls (AUC top-K) as the primary metric for measuring performance. Unlike using top-K average property, AUC rewards methods that reach high values with fewer oracle calls. We use 10 as is useful identify a small number of distinct molecular candidates progress to later stages of development. The reported values of AUC are scaled from min to max at [0, 1].
3. Diversity of generated molecules defined as the average pairwise Tanimoto distance between the Morgan fingerprints, formulated
diversity(2)=11 =1 12((2-1)
Σ
Szi, z2),
(10)
21 21
where Z is the set of generated molecules to evaluate and sim 22) is the Tanimoto similarity between molccule and 22. Diversity score ranges from 0 to 1, with higher values indicating greater diversity
4. Synthetic accessibility (SA) measures the difficulty of synthesizing the given molecule. The SA score values range from 1 to 10, where lower values indicate molecules that are casier to synthesize.
All these metrics can be calculated via the evaluation function in Therapeutics data commons (TDC) (Huang
et al. 2021 2 !
Implementation Details. We select Mean Squared Error (MSE) and Adam KKngmma & Ba 2014 with learning rale of 0.001 as the loss function and optimizer, respectively, when training M. The model is
trained for 200 epochs for each round of training. The MPNN has 2 layers of convolution and GRU with 1 layer of Sct2Set. There are n = 16 hidden foatures. The upper bound of the generated molecule & is sot to 70. The threshold criterion 7 for a good score of a molecule is set as the maximum metric score 0.0011 We
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have two experimental setups depending on the parameter D The first setup is to clear all the molecules that meet 7 and retrain whenever there is an adequate amount. The second setup is to store the D" and retrain at set oracle steps (default = 500 oracle calls). For simplicity, we take the setup of keeping D" over each oracle call to have more consistent training at each iteration. We provide detailed tables of results for Both selups use the default Graph GA settings from PMO.
both setups in Appendix| A
Method
mostranno simillrity
amllodipine MPO
Avergee Top 10
AUC Top 1
AUC Top 10
AUC Top 100
Averaaee Top 10
AUC Top
AUC Top 10
AUC Top 100
Gradient GA
0.5130=00.092
0.4433=000031
0.4052-0.0315
0.3534:0.0330
0.5687 0336
0.5614=0.0177
0.8858=0.0199 0.5176=00.0187
Grapla GA
1.445220.0241
0.355610000006
0.3208=000019
0.2717:100017
0.5605:00.000
0.5087
0.473420.0215
0. 0142
SMILES GA
0.2582:0000007
0.3777
0.3631=0.0652
0.3344:200.00 02779
0,4480:000118
0.5016:0000.6
0:4956:000144
0.4718=0001188
MIMOSA
0. 1.12622000244
0.41622
0.3019=0.0181
(1286740..0255
0.5215:20.0143
1.533110.0200
0.19531100100
0.4 0075
MARS
0.3411=0.0110
0.3760:200000$
0.3215:0.0099
0.25333000000
0.8843:20.1920
0.4812:0.0146
0.4583:000008
0.3816:0.0015
DST
0.413120.0119
0.41080000022
0.3507=00.088
0.2780:1000029
0.519220.0122
0.5411
0.4906:20.0115
0. 0.1257:10.002
perindoprill MPO
deco hop
Aserage Top
AUC 'Tp
ACC Top
ACC Top 100
Averago Top 10
AUC Top
AUC TTp
AUC Top 100
Gradieut GA
0.4786100002
0.4542:0.0164
0.4361:0.0176
0.388210.0193
0.6026:000003
0.5883=0.0032
0.5763:000050
0,5002:000000 0053
Graph GA
0.4788:00.000 0067
0.4519=0.0055
0.4317=0.0045
0.3770:10.0045
0. 0.0039:10.000
0.5186::
0.5026-10.003
0.4708 0033
SMILES GA
0.3698:000011
0.4
0.4065:10,0102 0.127100.0155
0.534810..00 0.059
0.588240.000
0.5817=000042
0.5733
MIMOSA
0.4620=0.0115 76
0.4500:000011
142299:0.0116
0.3783:10.00 0088
0.6008:00 0053
(.8822200006
0.5773=20.00
0.5600=00 0021
MARS
0.139110.0166
0.453810.0087 0.087
0.4278=0.0085
0.3018:10.000
1.591110.0070
0.5830110.00 0227
0.5711
0.519330.00k 04221
DSTT
0-4615=0000100
0.4530::00000
D-2210:000041
0.3564::0.00 00288
0.6034:000000 0083
0.5860=000007
0.5721
0.5618=00.00 0009
medianl
isomicss
20h10m2rtpp2el
Average Top
AUC Top
AUC Top
AUC Top 100
Average Top 10
AUC Tap
AUC Top 10
AUC Top 100
Gradient GA
0.3033:000074
0.258120.0115
0.1906:10.0183 0.2151
0.7783
0.062820007311
0.564420.0093
0.3033 0614
Graph GA
0.2599:1.00122
0.2815:0.022
0.1959=0.0148
0.1112:0000 0070
0.7222:0.1133
0.684810.0085
0.5196:000770
0.389110.042
SMILKS GA
0.1310=0.01722
0.1832=00022
0.1796:00002
11697:20.0255
0.3180::0 5583
0.824420.0848
0.7825=0.0762
0.7065=0.0668
MIMOSA
0.239110.0080
0.2271::0000
0.1969=0.0043
0.1537.1:00 0030
0.7886:0.0622
0.69052:0
0.50192000060
0.396520.0066
MARS
0.2094:0.0188
0.22390=0000
0:2019=000016
0.1671::00018:
0.6639 1666
0.6751=001008
0.5008 1057
0.4424:0 1499
DST
0.217920.0162
0.2097:10.0080
0.1705_00.0021
0.1331110.002
0.674820.0301
0.6305:00.0135
0.4932:0.0216
0.2288:0.0088
Table 2: Comparison of Average Top 10, AUC Top 1, AUC Top 10, and AUC Top 100 with several Gua-
caMol objectives (mestranol similarity, amlodipine MPO, perindopril MPO, deco hop, medianl, and isomers c9h (9h00222pf)el) under 2500 oracle calls.
5.2 Experimental Results
To evaluate the overall performance of our method, examine various metrics across multiple oracles. Table 2 presents comparative analysis based on Average Top 10, AUC Top 1, AUC Top 10, and AUC Top 100 scores. We observe that for most oracles, Gradient GA outperforms Graph GA, MIMOSA, MARS, and SMILES GA. Our method consistently achieves either the best or second-best, performance. Specifically, in Average Top 10, Gradient GA demonstrates the highest performance, with Graph GA following behind. This highlights the advantage of incorporating gradient information, cnabling more efficient and effective exploration of the local search space. The superiority of Gradient GA is further supported by the AUC Top K scores, which reward methods that reach high values with fewer oracle calls. For AUC Top and AUC Top 10, Gradient GA dominates the rankings, demonstrating faster convergence and better optimization. In AUC Top 100, Gradient GA achieves top score, tying with SMILES GA. However, while SMILES GA performs well in this metric, it significantly underperforms in Average scores. These results confirm that Gradient GA is the most effective method overall for optimizing molecular properties across different oracles.
5.2.1 Oracle Call Efficiency
To demonstrate that using gradient information accelerates convergence, we conduct experiments measuring AUC Top 10 and AUC Top 100 scores as the number of oracle calls increases. All methods are evaluated with 2,500 oracle calls over runs. Our primary focus is on the bio-objective Mestranol Similarity. FiguresB] and [ show that after the initialization phase, where each method achieves a baseline score based on the initial oracle calls, Gradient GA consistently outperforms almost all other methods at each step. This indicates that Gradient GA is not only more effective at finding optimal values but also more efficient, due to its use of gradient guidance rather than random walk exploration.
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Published in Transactions on Machine Learning Research (11/2025)
Mestranol Similarity Oracle Calls
Mestranol Similarity Oracle Calls
0.5
0.5
Gradient GA
Gradient GA
Graph GA
Graph GA
0.4
0.4
AUC Top 10 Scores
MIMOSA
AUC Top 100 Soores
MIMOSA
MARS
MARS
0.3
SMILES GA DST
0.3
SMILES GA DST
0.2
0.2
0.1
0.1
0
0
500
1,000
1,500
2,000
2,500
0
500
1,000
1,500
2,000
2,500
Oracle Calls
Oracle Calls
Figure 3: Comparison of Mestranol similarity AUC Top 10 scores as the number of oracle calls increases.
Figure 4: Comparison of Mestranol similarity AUC Top 100 scores the number of oracle calls increases.
perindopril_mpe
3.5
3.6
3.3
4.8
3.1
5.0
perindopril_
0.57
0.78
0.67
0.75
0.83
0.8
0.85
mestranol_similaittyy
4.3
3.4
3.9
4.2
cestranol_similartyy
D.67
0.77
0.87
0.78
0.86
D.82
45
median1
median1
0.86
0.75
isomers_c9h Siieers_chhhonne 221o222221n
5.2
4.4
3.8
3.6
3.2
isomers_chh 19hoop2p9102222p2ccc
0.79
0.79
0.63
0.82
0.83
0.67
35
deco_ hop-
deco, dero_hop-
amlodi pine mpo
3.8
3.3
2.9
3.7 MARS
2.6
amlodipine mmp
0.62
0.76
0.87
0.74
0.85
0.83
0.60
Gradlant GA
Graph GA
SMILES GA
MIMOSA
DST
Gradient GA
Granh GA
SMILES GA
MIMOSA
MARS
DST
Figure 5: Heatmap of synthotic accessibility (SA) score for all methods and oracles (lower is better).
Figure 6: Heatmap of diversity score for all meth-
ods and oracles (higher is better).
5.2.2 Synthetic Accessibility and Diversity
We further analyze additional metrics, synthetic accessibility (SA) in Figure and diversity in Figure It is important to note that these metrics are not explicitly optimized our objective function. Therefore, their performance byproduct of the discovered molecules rather than a direct outcome our method. From Figure 5 observe that the SA scores of Gradient GA are comparable to those of Graph GA, indicating that there is significant trade-off between improved performance and SA score. Additionally, in terms of overall SA performance, Gradient GA is also close to DST, another gradient--base we observe that the diversity score for Gradient GA is lower than that of other methods. This outcome is expected, as our approach samples molecules near high-performing parent molecules. While this may reduce diversity, can advantageous when the goal is to perform a fine-grained local search over good regions. Compared other variations on genetic algorithms (GA), the diversity for performance tradcoff of Gradient GA is more moderate. For example, there are two methods, Genetic GFN (Kim et al. 2024) which leverages GelowNets to improve molecule selection within GA and MOL GA TTripp & Hernández-Lobato
method. In Figure G
2023
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Published in Transactions on Machine Learning Research (11/2025)
method that introduces optimized version of standard GA aimed at achieving higher AVG and AUC scores on the PMO benchmark. They both perform better than Gradient GA, but at higher amounts of oracle calls, they perform similarly, but their diversity is much worse at all amounts of calls. Detailed results are Appendix
5.2.3 Molecules Generated by Gradient GA
We have included the Top 10 molecules generated by Gradient GA for the mestranol similarity objective in Figure 7 We can observe some structural similarity between the Top-10 molecules as they have been sampled from the same run. Such observation implics that one compatible set parents can produce a high proportion of good molecules. We further extend the study of good molecules for different methods Appendix B
0.57
0.56
0.56
0.56
0.56
0.55
Figure 7: Top 10 molecules generated by Gradient GA for the bio-activity objective mestranol similarity with their associated score underneath each molecule.
5.2.4 Computational Cost Comparison
Gradient computation Gradient GAintroduces additional overhead compared to standard GA. However, the cost remains modest and practical: for 1000-call run, GA required about minute, while our method required 2-3 minutes. In contrast, RT.baaed methods required 4-5 minutes, making our approach about 0.6x their runtime. Thus, while slightly more expensive than GA, our method remains significantly more efficient than RL-based baselines while achieving superior diversity and optimization tradeoffs.
6 Conclusion and Future Work
Genetic Algorithm (GA) is one of the most widely used approaches in drug molecular design, thanks to its flexibility in navigating the molecular space. However, GA usually suffers from slow and unstable convergence duc undirected random exploration. We address this problem by introducing a novel approach called Gradient GA, in which each proposed sample iteratively progresses toward the optimal solution. Our method leverages Discrete Langevin Proposal (DLP) the base sampler, enabling gradient-basse exploration in the discrete molecular space. Extensive experimental results confirm that our proposed approach achieves faster and superior convergence compared to state-of-the-arl methods.
Future work can expand the current method in the following aspects: (1) explore the effect of utilizing gradient information for generated molccule populations; (2) explore better ways to fit both parents into DLP: (3) explore DLP-oriented molecular optimizations that incorporate the Cetropolis-Hastinne criterion; (4) explore multi-objective optimization (preliminary results are given in Appendix G
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A Additional Experimental Results
A.1 Storing molecules for retraining
The tables below are results from the same experimental setup described in Table 2 The higher the score, the better for all metrics except Average SA.
For most oracles, Gradient GA achieves the highest performance in Average Top 1, while for Average Top 100 and AUC Top 1, it consistently outperforms all other methods across all oracles. Graph GA and DST continue to perform well overall, ranking near the top in almost every oracle. MIMOSA and MARS follow closely, occasionally achieving top-tier results. On the other hand, SMILES GA performs the worst in terms of optimization metrics but ranks near the top for diversity and achieves the lowest SA score.
Table 3: mestranol similarity Results
Meshod
Average Tog 1
'Top 10
TTpp 10U
AUC Top 1
Top 10
Top 100
Average SA
Diversity
Grodient CA
0.5367 #0.0421
15633000.0900
0.4678-0. 0.0993
0.4433-0.0310
0.4082-0.0315
0.363440,0356
4,5647 +0,360
0.60822-0 1/294
Grapa GA
0-422600-0241
024452-0-2211
0.355660 0268
0.528++0.1190
0.2717+000147
4,301 1++0.5876
0.7660- 0.980
SMILES
0.201320:0188
0.258210.009
0.77
0.334710.027 0279
1.351010,0524
0.8711 10,0005
MIMOSA
0-4784-0:0466
0-4262-000722 02246
0.55
014162+0.0133
0.3619+00.1181
U. 0.2687 0252
3.93855+-0.115 1149
0.7782-
MARS
0.383720.000
0.311120:0100
0,280310
0205
0.5760110
0.321510.0096
0,252310.
3.9512100
0.858H10
188'
0.450T -0:005
0:113100:017
0.365//000126
04448500.0223
0.3507
0.278000.0029
4.1479:#0.0964
0.8198
Table 4: medianl Results
Meshod
Average Top 1
Top 10
Top 100
AUC Top
Top 10
Top 100
Average SA
Diversity
Gradieut
0.318510.0113
0.305310.0074
0.208810.0000
0.229610.0151 0.258110.0115
0.100610.0188
3.7551=0.2588
0.0974.0
Graph CA
0.277920.0061
0.2092000182
0,2292200012
0.2015
0.190==0.0148
0.142=000000
3.8000=001177 1770
0.7402200.022
SMILES
0.168700.377
01310+00112
00007
0.182 -00211
11170-000022
0.1007 02025
3.3444-0.1011
B.870D+0.007
MIMOSA
011222000126
122012000080
0.195
0.000
0.1900=000044
4,3154=00 1356
0.0088
MARS
02022-00000
0.000
02010-000116
0.1671 -00158
4.2008-0
3080
02168-000196
DST
0:511200:5566
021700000102
0.209F
0.1766-0.0021
0.1331
00002
4.03880023412
O. 18664=000009
Table 5: amlodipine MPO Results
Meshur
Average Top
Top 10
Top 190
AUC Top
Top 10
Top 100
Average SA
Diversily
Gradiont GA
0.688D-0.0307
0.5667 0.0336
0.5400 0.0296
0.5614
0..0867
0-465-000119
3.867+00.3122
0.6176+00.047
Grapia GA
0.5772200010
0.5615 (196)
0.525210 0539
0.506710.0277
0.435110.215
0:112210.1122
3.749610.1500
0.7571
SMILES GA
0-1782-000188
0.440--0.0161
002222-0.057
0.3016+8.0166
0.0066-0.0143
0.4748+0.0158
3.2925+00.061
0.8711 +0.0064
MIMOSA
0.563322..222
0.526520:0143
0:15610001522
0.543110.020 0261
0.0933100100
0.4136.0..00 0075
2.8351 2221
0.7430 0S10
MARS
0.50791000001
0.06610000100
0,441220.00 0295
0.481220.0144
0.453.0.0068
0.381610.0157
3.009010.3345
0.850110.0100
DST
0,501920.0255
0.510220:0122
0.561110.000
0.198100.015
0,125710.001
2.638810.0770
0.839910.0000
Table 6: perindopril MPO Results
Method
Average Top
Top 10
Top 100
AUC Top 1
Top 10
Top
Average SA
Diversity
Gradient GA
0. 1.266
D.178620.0255
D. D.160EE0 0287
0.151220.0161
0:156100:0176
O. 0.3882000119
3,0861200 1956
0.5720:20.060
Graph GA
0.4985
10.0174
0.4788 0006
0.4488 0009
0.4519-0.0055
00.01
0.0770
3.644410 1822
1.7766110.013
SMILES GA
0.3698
17
0.187010.0185
0.434610.0124
10.0115
0.4065 10.0102
1.3328-0.0885
0.8634 10.0080
MIMOSA
0.1199000166
0.336100.166
011
0.128920.0116
00.085
1,10077:00 2006
0.74533
0281
MARS
0.479810.0118
10.061
1.125010.0147
0.453810.005
0.4278-0.0065
O. 10.0042
1.85471100772
0.8272
10.075
DST
1..0101
0.411510.0100
0.4197 10.0177
0.453010.0004
0.4210 10.004
1..064
3.1428 10.1120
0.7997
10.0204
Table 7: deco hop Results
Meshud
Average Top
Top 10
Top 100
AUC Top
Top 10
Top: 100
Average SA
Diversity
Gradiest GA
0.6116200001
0.8026000003
0.5945-00 0.0068
0.3863-0.000
0.366550.0000
0.560210,0064
3.3425-50:1122
0.84111000000
Grapu GA
10:00
0.0039 0.0043
0.59250000221
0.5188100 00S?
1.5022
3.30221011970
0.7870111 (12201
SMILES GA
0.002
0.317850000222
0.386220.00L
0017
0.581720.0042
0.373350.0036
3,284 1.111
0.8699 0.0082
MIMORA
0.607T
0003
0.577350.0055
0.50
2.800G-0 2906
0.742% 0.188
MARS
06004
0000
0.823000.022
1..011000001
0.5453
3,7000 15544
0.8152
TSST
0.6128=0.0118
0..721+0.00025
0.5018H0.0000
2.8446600.1211
0.0060000000
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Published in Transactions on Machine Learning Research (11/2025)
Table 8: isomers c9h10n202112el Results
Mechod
Average Top 1
Top 10
Top 100
AUC Top
Top 10
Top 100
Average SA
Disersity
Gradient GA
0.826K
-0.0940
0.7783-0-0150
0-66280007551
0-403350000014
6.2422 +0.3194
0.700-
Grapa CA
0.7680=00.110 1157
0.7222:0 1119
0.5456E000770
0.3891
4.4457
0.7900=0000616
SMILES GA
0.0587 +0 2014
0.3180-0
19792-00
0,8244 0,0848
0,7056 07,056+000668
3.7066+
0.8390
MIMOSA
081
0.7866-00 0.0624
0.6848
0.0541
O D.696500 0562
0.391910.0440
0.3965 0265
3,6001 10000110..2000
0.8231 10 0300
MARS
0.7208-0.
0.0039-0 1606
0.52080
0.50
0.4484
8.1615+00.N62
0.835--0.0070
DST
0.77082033422
0.67
O. 0.502810 05529
0.691510.0433
0.19221000216
0,2295:00.0095
2.965110.
0.8735.10.00$5
A.2 Retrained only on newly generated molecules between retraining phases
The overall experimental setup includes 10,000 oracle calls, 5 runs, and early stopping enabled. The pa- rameters for Gradient GA are set to 200 epochs, with D" being cleared after each retraining. We have run REINVENT (Oiveeroona et al., 2017), which is a reinforcement learning method, instead of DST. Addition- ally, we evaluate another metric, Diversily, which measures the Tanimolo Similarity belween two molecules. The higher the score, the better for all metrics except Average SA. This setup overall performs worse than when D" is saved; however, it is slightly faster in runtime, due to having fewer molccules for retraining.
From the results, can see that Gradient GA and Graph GA are near the top in terms of performance with REINVENT, These results follow the results gathered from PMO (Gao et al. 20226). We notice that Gradient GA usually performs better than Graph GA whenever it is dcaling with a molecular objective that is on the lower end, so the exploration near optimal molecules matters more than a random walk behavior may have a harder time finding the optimal molecules. Overall, the best-performing models are REINVENT, Gradient GA, and Graph GA.
Table 9: perindopril mpo Results
Method
Average Top 1
Top 10
Top 100
AUC Top
Top 10
Top 100
Average SA
Diversity
Graciout GA
200100
0:001310010
03513100113
4.522820.0115
0.310210.0109
0.487410.0111
4,7355110 1388
Graph GA
0.536020.0012
0.324520.0585
0.30010000528
0.301010.0083
0.481810.0871
0.451510.0039
3.989644.1300 1366
0,0085 1100
SMILES GA
0.44660000044
0,4784600145
0.4433-00012
0:113--000121
4.7030 4997
0.4083=0 11G3
MARS
0.478020.0137
0006412000128
0.455700.0135
0,77L10000122
0.450010.0068
0.4.73.0.0000
5.02022002177
0,825110.0055
MIMOSA
0.700-00017
0.31
0,2856000000 DORS
3,9169+0,28 2835
0.8004==0 0,230
REINVENT
0.6196-0.0460
0.6164
0.0485
0.6132
0.0511
044635000764
0:4428H0.0765
0.4188+0.0746
4.1524-90.3064
0.3480-0.0008
Table 10: mestranol similarity Results
Method
Average Tog: 1
TTpp 10
Top 100
AUC Top
Tog 10
Tog: 100
Average SA
Diverslly
Gradient GA
0.731810:0821
0.718010.0252
0.861810.0288
0.6170-0000001
0.583310.0034
0.527310:0018
4.801910.3707
0.532910.
Graph GA
0.7835:00 1325
00R92GE 1161
20:022
0511250000166
5.372750:0429
0.3173110:0077
4. 27800100.5000
0.6509
SMILES
0.008810.0:50
0.687710.0654
0.119910.0119
5.132L10.00
0.5278.1
MARS
0.11220006622
0.378220:0P18
0.3202200758
0.409910.003
0.36/1110.06 0665
0.304710.044
4.0455
0.85-46:00011
MIMOSA
0.523910.0115
1077
04415010.0180
0.507910:0010
1.0080
0.009910.0011
32..2227
0,805210
REINVENT
0.7835=000823
0.7809=0.068
0.7627 0.084
0.39201000899
0.370910.0833
03334661.0224
3.692910.49119
(.206
Table 11: medianl Results
Method
Average Top 1
Top 10
Top 190
AUC Top
Top: 10
Top 100
Average SA
Diverslly
Graciinn GA
73
0:3714-0.0185
0.3378+0.00 0005
0.3376=0.0215
0.3108=0.0155
0.2789+0.011
4.005++0.1277
0.6181 U228
Graph GA
0.291620.0225
0:2585000088
09111
0,268010.000 0225
0.231810.0158
4.006310.1091
0,709110 0366
SMILES GA
0.000
0.299100.0288
0. 0.180100.299 0293
1.1977.0.00 0296
0224
0.188910.0208
6.0024111.009
0.6300 0982
MARS
022222-0.0201
01201110.0111
0.177710.023
0,223910.00 0140
0.201910.0116
0.167119.0158
4,2508 10.,00
0.815810.00
MIMOSA
0.0275
0.801120.0036
0.268610.0000
0013
0.227810.0013
0.165110.0013
3.970110.0735
0.0013
REINVENT
0.4079
0.4984
0.4151=0.0334
01257
0,22220000447
0.18/240.0333
4.711060.6151
0.136
16

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Published in Transactions on Machine Learning Research (11/2025)
Table 12: isomers c9h10n2o2pf2el Results
Method
Average Top
Top 10
Top 100
AUC Top
Top 10
Top 100
Averagee SA
Diverity
Gradient GA
0.939410.0000
6.9297
0.020
0.7778
0.701820.8808
6.4589 2025
17.81100.088
Graph GA
19.15719.02291
1.889910.000
01171610:0177
0.00110000000
0.8021
0.717120.0141
1.527310.3671
0.7985 0265
SMILES GA
0.931010.0000
0.8896=0.0457
0.8055
0.8640000.0352
0.806310.0450
5.831020.2002
0.718140.0019
MARS
0.7200001120
D.OGSE 1O06
0.526840 2389
0.675T 1032
0.8089 1067
0.44244-0.
3.1615+0 8G26
0,83355 0.0990
MIMOSA
8.808T +U00637
0.7564 0666
0.002-000248
0.024
0.5077+0.120
4.1175+0.8822
D. 7000+000722
REINVENT
0,9106+000012
0.356++0077
-00067
3.16774077M4
0.067340.0776
Table 13: deco hop Results
Method
Average
Top
Top
Top
Top
SA
Diversity
Gradien GA
06475-50.0022
6.6413+00001
0865350 00R2
0:620=000060
0.6124-0.0099
20024:0000000
3.9402:0 4004
05652 0921
Graph
SMILES GA
06147
0.66
061161000000
0,892
102771000000 0060
0.508-4.0000
0.584 00087
1.016:20.634
0.026810.000
0.5711
MIMOSA
010011200122
0.0002200013
0,389810 0100
0.8428 0003
0.513
0.4726.
0.099
4.131520.2665
0.671220.0811
1.801440.1476
0.7915=0.1430
0.7853-0.1413
0.6377 7 20.002
0.6415=00.0643
#0,0478
3.0143=0.223
Table 14: amlodipine mpo Results
Method
Average Top
Top 10
Top 100
AUC Top
Top
Top 100
Average SA
Divensity
Gradieut GA
11992110199
0.054810001000
0.65
U.GRIBLO 0311
0.1200000833
TOU1S1B011
0.401210.055
Graph GA
6777
026
0.656910.0220
0.6350=0.0174
0.600410.0133
3.8835200 1383
0222
SMILES CA
0378
0.5079-0.0922 0.0224
14061+0.222
3.697510
MIMOSA
0.045-HLLTTS 18
0155
0.
0.5420-0.0114
40.000
REINVENT
0.798210.0468
0.7451
0.5670
0674
0.5409-
3.2323:0.222
0670
17

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Published in Transactions on Machine Learning Research (11/2025)
B
Molecules Generated by Various Methods
Figures show the top 10 molecules generated for the mestranol similarity by Gradient GA, SMILES GA, Graph GA, MARS, and MIMOSA. The molecules proposed by Gradient GA, SMILES GA, and Graph GA show strong structural resemblance. Although MARS and MIMOSA-proppset molccules bear more uniqueness, they yield lower score for the product metric, denoting poor choice of molecules. From the figures, it is evident that Gradient GA is exploring similar molecules to Graph GA, while further exploring an area around its top-performing molecules.
0.25
0.22
0.41
0.41
0.24
0.41
0.21
0.32
0.41
0.41
Figure Top molecules generated by SMILES GA for the bio-activity objective restr similarity with their associated score underneath each molecule.
64
0.49
0.49
0.50
0.48
0.48
0.48
0.47
0.47
0.46
0.46
Figure 9: Top 10 molccules generated by Graph GA for the bio-activity objective mestranol similarity with their associated underneath each molecule.
18

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Published in Transactions on Machine Learning Research (11/2025)
0.38
0.38
0.38
0.37
0.37
HN
OH
0.35
0.36
Figure 10: Top 10 molecules generated by MARS for the bio-activity objective mestranol similarity with their associated score underneath each molecule.
0.54
0.49
0.49
0.47
0.47
0.46
Figure 11: Top 10 molecules generated by MIMOSA for the bio-activity objective mestranol similarity with their associated score underneath each molecule.
19

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Published in Transactions on Machine Learning Research (11/2025)
C Comparison of Including (x) in Gradient
In the table below, we show the results of using (U(v) = Vf(1) vs VU(o) results, using (xx) leads to better score throughout all metrics.
Vf(e) O(X)
We notice from the
Table 15: Comparison of Average Top 10, AUC Top 1, AUC Top 10, and AUC Top 100 with GuacaMol objective, mestranol similarity, under 2500 oracle calls. The best Gradient GA setup bolded. We conduct five independent runs using different random sceds for both versions of Gradient GA, and report the average scores and their standard deviation.
Gradient GA
mestranol similarity
Average Top
AUC Top
AUC Top 10
AUC Top
With ((r)
0.5130=0.00 0393
0.4433=0.0310
0.4082=0.0311
0.353440.0033
Without (xz)
0.5064=0.0311
0.443310.0319
0.4072=0.0367
0.350110.044P
D Surrogate M Error Analysis
In this section, discuss the effects of surrogate loss introduced by M. We consider two different sets for validation: (i) Validation Training Set: usual validation set, kept from training samples of M, (ii) Validation Known Samples: validation set constructed from the current population. The size of the validation set is 10% of the respective sets. We observe the training loss, both validation losses with respect to oracle calls in Figure 12 While Validation Training Set Loss shows a similar trend to Training Set Loss, Validation Known Samples Loss decreases steadily over time. This shows that the approximation from M is relatively accurate in more unknown situations for the neural network, demonstrating good estimation capability.
Figure 12: Comparison of the total training loss, total validation loss from the training set, and also the validation loss from all known samples with 10000 oracle call runs.
10
1.2
Training Validation Training Set
Validation Known Samples
0.8
Loss
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.J
Oracle Calls
-10
20

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Published in Transactions on Machine Learning Research (11/2025)
E Comparison between Parent Selection Methods
In the table below, we show the results of using the best parent and averaging the parents for the gradient computation. We notice from the results, using the best parent leads to a better score throughout all metrics except diversity.
Table 16: Comparison of Average Top 10, AUC Top 1, AUC Top 10, AUC Top 100, and Diversity with GuacaMol objective, Amlodipine MPO, under 1000 oracle calls. The best Gradient GA setup is bolded. We conduct five independent runs using different random seeds for both versions of Gradient GA, and report the average scores and their standard deviation.
Gradient GA
Amlodipine MPO AUC Top 10
Average Top 1
AUC Top 1
AUC Top 100
Diversity
Best Parent
0.5294+0.0065 0.064
0.4681 0.0165
0.3929+0.01- 49
0.6855+00.00
Averaging Parents
0.5337=0.0174
0.4979+0.0198
0.4616=0.0057
0.3908-0.008
0.7425=0.0257
F Variations of Graph GA Algorithms
In the tables below, we show Gradient GA compared to other recent GA methods under the same experi- mental setup in Table 2 From the results, it is clear that our method achieves higher diversity compared to Genetic GFN and MOL GA, though it lags behind in terms of AVG and AUC scores. A notable trend emerges when comparing performance at 2,500 versus 10,000 oracle calls: the gap in AVG and AUC scores between Gradient GA and other GA-based mothods narrows significantly at 10,000 calls. While all methods experience a drop in diversity as the number of oracle calls increases, Gradient GA consistently maintains a higher diversity at both checkpoints.
We believe that Gradient GA offers balanced trade-off, achieving competitive AVG and AUC scores while preserving diversily, especially at higher oracle call budgets. This is particularly important given that, as shown in the previous section, GA-based methods generally exhibit lower diversity than state-of-the- non-GA approaches. Maintaining molecular diversity is crucial for practical applications, as it ensures the recommended molecules are not overly similar in structure.
Metric
Method
Top 1
Top 10
Top 100
AUC Top
AUC Top 10
Diversity Top 100
Mestranol Similarity
Genetic GFN
0.688 0.024
0.668
0.024
0.650 + 0.044
0.614 0.053
0.545
0.021
0.458 + 0.027
MOL GA
0.728 + 0.040
0.687 + 0.051
0.622 + 0.083
0.572 +: 0.058
0.538 # 0.053
0.571 4 0.113
Gradient GA
0.537 1 0.042
0.513 : 0.039
0.467 1 0.029
0.413
0.031
0.405 :: 0.032
0.669 1 0.029
Amlodipine MPO
Genetic GFN
0.671 0.041
0.660
0.050
0.635 1 0.035
0.603
0.530 1: 0.135
MOL GA
0.656 + 0.041
0.663
0.036
0.639 + 0.033
0.605
0.038
0.592 = 0.086
0.492 + 0.015
Gradient GA
0.588 0.040
0.567 +: 0.034
0.540 4: 0.030
0.561
0.018
0.518 + 0.019
0.618 + 0.048
Perindopril MPO
Genetic GFN
0.588
0.039
0.573
0.028
0.553 1. 0.025
0.517
0.017
0.502
0.017
0.486 0.122
MOL GA
0,582
0.053
0.575 = 0.0119
0.551 + 0.045
(1326
0.011
0.513 = 0.039
0.188 +: 0.021
Grudient GA
0.486 0.027
0.479 1 0.026
0.462 1 0.029
0.454
0.016
0.436 1 0.018
0.572 1 0.069
Do0o Hop
Genetic GFN
0.641
0.009
0.636 0.011
0.626 + 0.011
0.609
0.010
0.601 + 0.010
0.556 + 0.084
MOL GA
0.649 + 0.028
0.647
0.029
0.610 + 0.027
0.619
0.026
0.616 = 0.024
0.485 = 0.032
Gradient GA
0.612 0.001
0.005 0.603
0.595 + 0.007
0.588
0.003
0.576 + 0.005
0.644 + 0.042
Median
Genetic GFN
0.387
0.030
0.365
0.027
0.311 4: 0.020
0.314
0.027
0.291
0.027
0.563 0.108
MOL GA
0,319 1 0,031
0.325
0.010
0.298 1 0.006
01300
0.015
0.279 :: 0.007
0.642 1 0.049
Gradient GA
0.318 + 0.011
0.303 = 0.007
0,269 + 0.009
0.258 + 0.012
0.230 = 0.015
0.697 + 0.037
Isomer 20h10n222p22el
Genetic GFN
0.919
0.022
0.900
0.024
0.858 + 0.020
0.800
0.067
0.705 = 0.060
0.671 + 0.070
MOL GA
0.905
0.031
0.597
0.021
0.855 4: 0.012
0.845
0.021
0.808
0.019
0.735 4: 0.048
Gradieut GA
0.826 1 0.094
0.778 = 0.096
0.670 1 0.099
0.663 +: 0.073
0.544 = 0.066
0.745 +: 0.039
Comparison of Average Top 1, Average Top 10, Average Top 100, AUC Top 1, AUC Top 10, and AUC Top 100 with several GuacaMol objectives (mestranol similarity, amlodipine MPO, perindopril MPO, deco hop, medianl, and isomers c9h (01102222/2cl) under 2500 oracle calls.
Table 17:
21

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Published in Transactions on Machine Learning Research (11/2025)
Metele
Method
Top
Top
Top
AUC
AUC 10
AUC
Divcesity
Top
Mestrnnll Similarity
CPPN
(17706 1: 0,0724
1,7796
0.0744
0.7770
0.0095
0.7127
0.0443
0.7057
0.0337
0.2314
0.0001
MOL GA
0.8179 1. 0.0974
0,8033
0.1019
0,7084
4. 0.3018
0.6382
8.0899
0.6774
1. 0,0864
0.336
0,0771
0,4189
0.0442
Gescliol GA
0.7365 + 0.0321
0.7140
# 0.0282
0.6615
+ 0.0218
0.6179
# 0.0001
0.0833
+ 0.0349
0.5273
= 0.0318
0.6829
0.0594
Amlodiytne MPO
Gezstti GPN
0.7416 1: 0.0838
0.7366
0.0236
0.7225
1. 0.0633
0,6850
0,0457
0.6737 1. 0,0079
0.6477
1. 0,0318
0.42% 1: 0,0715
MOL
0.8534 1: 0.0238
0.8644
0.0216
0.8150
0.0165
0.7480
0.0158 0.6309
0.7329
0.0130
0.8089
0.0000
0.406
0,0038
Gradien GA
00042 0.0499
118848
0.0530
0,6205
0.0456
0,0348
0.5845
0.0850 0.4402
Perindognl
Groutie
03871
0.0785
MOT. GA
0.6109 + 0.031a
0.005
0.0328
0.5917
0-08
0.5009
0/0002
0.5033
0.0248
0.5444
0.0214
0.1523 0.0.05
Gradient GA
0.5741 1: 0.0104
0,5613
0.0100
0,5445
1 0:0113
0,5228
0.0113
0.5102
1. 0.0110
0.4874 0,0111
0.5477
0.0094
Decn Hop
Genatie GPN
0,8076 + 0.0080
0.6058
0.0044
0.6000
+ 0.0090
0.0026
0.0082
0.0569
+ 0.0087
0.6458
+ 0.0098
0.1323 + 0.0041
MOL GA
0.6731
1: 0,0219
0.6713
0.00
0,8089
1 0.0038
0,6544
0:0001
0.6250
1. 0.0018
0.6218
1: 0.0052
0.4156 0.0087
Gradhent GA
0/6452 0.0101
0.63M
0.0088
0,6311
0.005
0,0077 0.0130
0.6085
0.0006
0.6031
0.0001
0,0515 0.4622
Mediiu
Geartic GPN
0,4000 0.0000
0,4000
0.0000
0.3336 0.0040
0,3639
0.8768
0.0074
0.3116
0.0075
0.4038
0,0729
MOL.
0.0394
0.0350
Gesclient GA
0.3023 + 0.0173
0.5714
0.0883
0.5378
0.0096
0.3370
0/0215
0.3108
0.0153
0.2759
0.0125
0.0191
0.0278
bomme dthil (bckstp12e)
Genatii GPN
0,98994
0.0000
0
0.0153
0.9081
0.3170
0.9138
0.06-49
0,8956
0.0818
0.8460
0.0525
07101
0.0.62
MOL GA
0.9066
+ 0.0084
0.9215
0.0266
0.9028
+ 0.0237
0.9006
+ 0.0210
0.8854
+ 0.0207
0.8818
+ 0.0116
0.7850 # 0.07332
Gesdlinn GA
0.9394 =: 0.0000
1152337 0.0196
118773
+: 0.0264
0,8256 0.0704
0.7878
+: 0,0778
0.7118
0,0808
0.7481
0.0389
Table 18: Comparison of Average Top 1, Average Top 10, Average Top 100, AUC Top 1, AUC Top 10, and AUC Top 100 with several GuacaMol objectives (mestranol similarity, amlodipine MPO, perindopril MPO, deco hop, medianl, and isomers c9h10n2o2pf2cl under 10000 oracle calls.
G Preliminary Multi-Objectien Results
Despile the proposed work being heavily focused on single-obeective optimization, we present our future idea about multi-objective optimization in this section. We envision the multi-objective approach for Gradient GAby providing the aggregate of objectives as a single objective to the current framework. We have chosen a simple product as the aggregate function, due to its simplicity and applicability over previous works, such as MARS (Xie et al. (2021)). We have conducted sample run with 2500 Oracle calls for multi- objective mestranol similarity and amoldipine MPO and have seen comparable results similar to those of single-objective. Tables 19 20 provide the comparison of the sample run with the original results. This experiment shows that the proposed framework yields similar outcomes for individual objectives even when the target is a joint distribution of the said objectives.
Table 19: mestranol similarity Results: singlo-objective multi-objective
Method
Average Top
Top
Top 100
AUC Top
10
Top 100
Average SA
Diversity
Gradinst GA
0.0011001111
031301000000
0. 0.403310.0233
0.4533_0.0510
1964720 3033
0.869210.0294
Geadied Gimniii-b ubjjcttt)
0.6328
0.3497
0,4285
0.8398
0.5366
0,4158
4.4750
0.1552
Table 20: amoldipine MPO Results: single-objective vs multi-objective
Method
Average Top
Top
Top 100
AUC Top
Top
Top 100
Average SA
Diversity
Grddent CA
0.58800 0.0507
0.3 000206
0.561 470.017
U. 0.008 0.3176-0.0188
S.0077 0.3122
0.81961000075
Gradicnt CAlariii oboective)
0.3641
0.50083
0.4360
0.4937
0,4738
4.9944
0.5476
22


Reviews:
[{"review_id": "B5Q91fPTwy", "reviewer": "Reviewer_DK6J", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors introduce Gradient GA, a genetic algorithm for molecular graph generation that includes gradient information to guide the generation process. To handle the difficulty linked to the discrete nature of graphs, the authors leverage an embedding method with a GNN. The graphs are embedded in some latent space, these representations are used to approximate an oracle objective function on molecules, and this latter part is used to compute gradients from the latent space to the objective. This gradient is then used to guide the sampling process, akin to Discrete Langevin. The authors report good experimental results on several datasets, with several classical metrics in molecular graph generation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper is clear, and the claims are convincing.", "audience_yn": "Yes", "audience_explanation": "Molecular graph generation is a very active area of research, and primary field of application, for graph generative models.", "requested_changes": "I found the paper quite clear and well-written, and I generally like the proposed method, which I find clever and well-put together.\n\nI may have minor remarks concerning notations; it seems that some notations may be redundant and/or slightly useless, which results in a multiplication of them that may obfuscate a bit the main ideas, underselling one strength of the proposed method, which is to be quite simple and straightforward (the authors even give a table of notations, which I find a tad too much here). For instance, the ``objective'' function/mapping appears under many form: $\\mathcal{O}$, $f$, $L_2$, $\\mathcal{M}$... and it can be a bit tedious to identify that some of those are the same object. It may be possible to gain clarity by being a bit more straighforward here.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "tpHuGINfOF", "reviewer": "Reviewer_Ttib", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Summary: \nThis paper presents Gradient GA, a new genetic algorithm for drug molecular design that uses gradient guidance from neural network-based objectives to efficiently explore chemical space. Traditional genetic algorithms operate by random operations, which can be slow and inefficient. Gradient GA replaces random changes with informed moves guided by gradients in a differentiable space, made possible by a message passing neural network (MPNN) that predicts molecular properties. The approach demonstrates faster convergence and higher-quality molecule generation across several benchmark tasks when compared against state-of-the-art optimization methods like Graph GA, SMILES-GA, MIMOSA, MARS, and DST.\n\nStrength: \nThe integration of gradient information into a GA framework for molecular discovery is innovative and clearly explained. The manuscript presents extensive comparative data across multiple baselines and metrics, notably using the number of oracle calls. Results span multiple molecule optimization tasks and metrics (Average Top-K, AUC Top-K, diversity, and synthetic accessibility) against other algorithms. Proves the benchmarks by showing fewer oracle calls needed for molecular property evaluation, compared to other algorithms.\n\nWeakness: \nGradient GA relies heavily on neural embeddings to represent molecules for optimization. However, the paper does not fully address how the embedding space is defined for multiple features or objectives. If embeddings for different molecular features are not aligned or mapped consistently, the optimization process may become biased and less effective, especially for multi-objective molecular design. This technical gap could lead to gradients that do not truly guide the search towards molecules that are optimal across all desired properties.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper's claims are strongly supported by quantitative data. Improvements in convergence speed and solution quality are shown through direct comparisons of AUC Top-K scores versus the number of oracle calls. Tables and figures illustrate that Gradient GA requires fewer expensive oracle calls to reach superior results on common molecular benchmarks compared to other algorithms.", "audience_yn": "Yes", "audience_explanation": "Gradient GA will be valuable for researchers in computational chemistry, drug discovery, and machine learning for scientific applications. Specifically, those working on molecular optimization, virtual screening, graph neural networks, and efficient algorithmic search will benefit from these results", "requested_changes": "Clarifying whether embeddings for distinct objectives are jointly normalized or aligned, and how gradients are mapped and interpreted when objectives have different semantics or scales, would increase the transparency of the method and guide future extensions to multi-objective tasks. Even a brief note on this point, such as discussing embedding coordination or normalization across features, would not require major changes to the algorithm design, yet would augment the paper’s clarity and its value to researchers working on general-purpose molecular optimization.", "broader_impact_concerns": ".", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wLxsBHy6JZ", "reviewer": "Reviewer_vDAC", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a gradient genetic algorithm for drug molecular design. The algorithm learns a differentiable objective function with a GNN to map the graph-structured information of molecules to vector embeddings. Then it applies discrete langevin proposal to incorporate gradient information from the objective, which accelerates convergence concerning tranditional genetic search.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims are well described and supported by experimental results.", "audience_yn": "Yes", "audience_explanation": "Researchers from the molecular design and evolutionary algorithm communities may be interested in the paper.", "requested_changes": "1. The gradients are derived from an NN surrogate trained to approximate the oracle. If the surrogate is inaccurate, especially in early iterations or in noisy oracle settings, the gradient direction can be misleading. The paper would benefit from a robustness analysis on surrogate error and its impact on optimization performance.\n2. One significant advantage of genetic algorithm is its diversity discovery ability. While gradient-based search helps fine-grained local exploitation, it risks overfitting to a narrow region of the search space and lossing solution diversity. How can the algoirthm make a balance between onvergence and diversity?\n3. Similiar to point 2, only the best-scoring parent contributes to the gradient update, discarding information from the other parent. This limits the potential benefits of crossover and may reduce structural innovation. Exploring gradient aggregation from both parents could improve exploration.\n4. The algorithm requires frequent retraining of the surrogate and gradient computations. Although the number of oracle calls is capped, the paper does not discuss the additional computational overhead from model training and whether it remains practical in more expensive real-world scenarios.", "broader_impact_concerns": "n/a", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)