DECN: Evolution Inspired Deep Convolution Network for Black-box Optimization
Authors: Liu Penghui
Kai Wu
Jing Liu
Venue: Submitted to ICLR 2023 (2023)
Keywords: Learning to Optimize
Black-box Optimization

Abstract: We design a deep evolutionary convolution network (DECN) for continuous black-box optimization to force the random population to move near the optimal solution, which is the goal of population-based optimization, such as evolutionary algorithm and evolutionary strategy. DECN is composed of two modules: convolution-based reasoning module (CRM) and selection module (SM), to move from hand-designed searching strategies to learned searching strategies in population-based optimization. CRM produces a population that is closer to the optimal solution based on the convolution operators, and SM removes poor solutions. We also design a proper loss function to support the training of DECN. The experimental results on unconstrained continuous optimization problems show that DECN can learn searching strategies and surpass population-based baselines. Moreover, DECN obtains good performance when transferred to optimization problems unseen during the training stage. In addition, DECN is friendly to the acceleration with Graphics Processing Units (GPUs) and runs 102 times faster than unaccelerated EA when evolving 32 populations, each containing 6400 individuals.

Under review as a conference paper at ICLR 2023
DECN: EVOLUTION INSPIRED DEEP CONVOLUTION NETWORK FOR BLACK-BOX OPTIMIZATION
Anonymous authors Paper under double-blind review
ABSTRACT
We design deep evolution convolution network (DECN) to overeome the poor generalization of an evolutionary algorithm in handling continuous black-box op- timization. DECN is composed of two modules: convolution-baset reasoning module (CRM) and selection module (SM), to move from hand-designed opti- mization strategies to learned optimization strategies. CRM produces population closer to the optimal solution based on the convolution operators, and SM removes poor solutions. We also design a proper loss function to train DECN as to force the random population to move near the optimal solution. The experimental results synthetic case and two real- world cases show the advantages of learned optimization strategies over human-designed black-box optimization baselines. DECN obtains good performance with deep structure but encounters difficulties in training. In addition, DECN is friendly to the acceleration with Graphics Process- ing Units (GPUs) and runs 102 times faster than unaccelerated EA when evolving 32 populations, each containing 6400 individuals.
INTRODUCTION
Optimization has been an old and essential research topic in history; Many tasks in computer vision, machine learning, and natural language processing can be abstracted optimization problems. Moreover, many of these problems are black-box, such as neural architecture search (Elsken et al.
2019) and hyperparameter optimization (Hutter al. [019)). Various approaches, such as Bayesian optimization (Snoek et al. 2012) and evolutionary algorithms (EAs), including genetic algorithms (Tin et al. 2019: Khadka & Tumer 2018 Zhang & 2007 Such et al. 2017 Stanley et al. 2019) and evolution strategies (ES) (Wierstra et al. 2014; Vicol et al. 2021 Hansen & Ostermeier 2001
Auger & Hansen 2005 Salimans et al. 2017) have been proposed to deal with these problems in the past.
The generalization ability of EAs is poor. Faced with a new black-box optimization task, we need experts to redesign the EA's crossover, mutation, and selection operations to maximize its performance on the target task, resulting in a hand-designe dend-desigeed EA with big application limitation. Most importantly, due to the limitation of expert knowledge, only little target function information is used to assist the design of EA, which makes it challenging to adapt to the target task. How to automatically design optimization strategies according new tasks is crucial. EA generative optimization model that realizes the generation from a random population to an optimal solution by generating potential solutions and retaining good solutions. The task of automatically designing an optimization strategy is learning how to automatically generate and retain potential solutions.
This paper first attempts to develop deep evolution convolution network (DECN) to learn to exploit structure in the problem of interest so that DECN can automatically move random population near the optimal solution for different black-box optimization tasks. DECN uses the process of EAs to guide the design of this new learning-to tornning-tt-opimmieee architecture. Like EAs, we propose two critical components of DECN to generate and select potential solutions: a convolution-base reasoning mod- ule (CRM) and selection module (SM). For CRM, we need to ensure the exchange of information between individuals the population to achieve the function of generating potential solutions. We design lattice-lit like environment organizing the population into the modified convolution operators and then employ mirror padding (Goodfellow et al.] (016) to generate the potential offspring. SM need to update the population to survive the fittest solutions. We design SM based pairwise

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comparison between the offspring and the imput population regarding their fitness, implemented by employing the mask operator. Then, we design the evolution module (EM) based on CRM and SM to simulate generation of EAs. Finally, we build the DECN by stacking several EMs to cope with the first issue.
The untrained DECN does not handle the black-box optimization problem well because it needs information about the target black-box function. In order to better optimize the objective task, we need design a training set containing objective function information and a practical loss function to guide the parameter training of DECN. The characteristics of black-box functions make it difficult for to obtain their gradient information assist the training of DECN. To overcome the second issue, the following questions must be solved: how to design proper loss function and training dataset. We construct differentiable surrogate function set of the target black-box function to obtain the information of the target black-box function. However, the optimal population is usually unknown. The designed loss function to maximize the difference between the initial and output populations to train DECN towards the optimal solution, where the loss function can be optimized by proppappaan
We test the performance of DECN on six standard black-box functions, protein docking problem, and planner mechanic arm problem. Three population-busee optimization baselines, Bayesian method for black-bor optimization (Cao et al. 2019) are employed as references. The results indicate that DECN can automatically learn efficient mapping for unconstrained continuous optimization on high-fidelity and low-fidelity training datasets. Finally, to verify that DECN is friendly to Graphics Processing Units (GPUs)' acceleration, we compare the runtime of DECNs on one 1080Ti GPU with the standard EA.
optimization (Kandasamy et al., 2020), , and Searning-to-optimee
2 RELATED WORK
There many efforts that can handle black-box optimization, such Bayesian optimization (Snoek et al. 2012) and EAs (Mitchell 1998). Since the object of DECN is population, it has a strong relationship with EA. Meanwhile, DECN is new learning-to-optimizz (L.20) framework. Appendix A. shows our detailed motivations.
EAs. EAs are inspired by the evolution of species and have provided acceptable performance for black-box optimization. There are two essential parts to EAs: 1) crossover and mutation: how to generate individuals with the potential approach the optimal solution; 2) selection: how to discard individuals with inferior performance while maintaining the ones with superior performance. In the past decades, many algorithmic components have been designed for different tasks in EAs. The performance of algorithms varies towards various tasks, as different optimization strategies may be required given diverse landscapes. This paper focuses on two critical issues of EAs: 1) Poor generalization ability. Existing methods manually adjust genetic operators' hyperparameters and design the combination between them (Kerschke et al., 2019) Tian et al., 2020); However, its crossover, mutation, and selection modules can only be designed manually based on expert knowledge and cannot effectively interact with the environment (function); that is. they cannot change their elements in real-time to adapt to new problems through the feedback of the objective function. 2) The acceleration of EAS using GPUs is challenging task. The support for multiple subpopulations to evolve simultaneously has paramount significance in practical applications. Besides, many available genetic operators are unfriendly to the GPU acceleration, as GPUs are weak in processing logical operations. DECN overcomes the above issues. It is adapted to different optimization scenarios, based on which DECN automatically forms optimization strategies.
L20. The most related work is about L20 (Chen al., 2022). These methods employ the long short-term memory architecture (LSTM) (Chen et al. | Andrychowicz et al. [016 Chen et al.
2017 & Malik 2016; Wichrowska et al. 2017 Bello et al. (217)) or multilayer perceptron (MLP)
MMetz et al. 2019) as the optimizer to achieve point-based optimization (Sun et al. 2018 Vicol et al.)
2021 2021 Flennerhag et al. 2021 Li & Malik 2016). However, none of the above methods can handle black-box optimization. Swarm-inspired meta-optimizer (Cao et al. 2019) learns in the algorithmic space of both point-based and population- based optimization algorithms. This method does not consider the advantage of EAs and is model-fre ffre method. Existing L20 techniques rarely focus on black-box optimization. Although several efforts like (Cao et al. 2019 Chen et al.) (017)) have coped with these problems, they all deal with small-scale problems in the experimental setting. DECN is
Chen
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new L20 framework that makes up for the performance disadvantage of the current L20 architecture in black-box optimization. This paper makes an essential contribution to the L20 community.
3 DEEP EVOLUTION CONVOLUTION NETWORK
3.1 PROBLEM DEFINITION
An unconstrained black-box optimization problem can be transformed or represented by minimiza- tion problem, and constraints may exist for corresponding solutions:
min f((l()) s.t. E [d,, uz]. Vr; E
where s = (2) F2, ... Tp) represents the solution of optimization problem f while d (d1, dg.... d and u = (n1, "2, : uD) denote the corresponding lower and upper bounds of the solution's domain, respectively. ç is the known parameters of f. We can only use the query-response terminology because the objective function / is black box without closed- form formulation
this setting. Suppose n individuals of one population (S = ,.),""",))))) be $1 (x), xz2- ....................... .xb), ....., s2 (af. 22.... .=p).- Sn x
.²b). This paper aims to make the initial population move near the optimal solution. To be noted, is the parameters (strate-
gies) of G. G is an abstract function remarking the optimization process, So is the initial population, and St is the output population. The procedure of DECN can be formulated as St = f
Based on the optimized 0, DECN optimizes (=((E) by Gg.
3.2 CONVOLUTION- BASED REASONING MODULE
We design CRM to ensure that individuals in the population can exchange information to generate the potential solutions near the optimal solution (similar to the function of recombination operator in EAs). The corresponding correction to the convolution operator can achieve this goal, which is the motivation for our design with convolution. This part mainly describes how to construct CRM to generate new solutions.
Organize Population into Convolution. We arrange all individuals in lattice-like environment with size of L x L. In this case, we can represent the population by using tensor (i,j. d), where (i, j) locates the position of one individual ((i, j) in the L x L lattice and d is the dimension information of this individual. Appendix A.2 gives illustration of the tensor data. The individuals in the lattice are sorted in descending order to construct a population tensor with a consistent pattern (see Figure of Appendix). The number of channels in input tensors is D+1, where D is the dimension of the optimization task, and the fitness of individuals occupies one channel. The fitness channel does not participate in the convolution process but is essential for the information selection in the selection module.
How Design CRM. After organizing the population into a tensor LL, 1), we modified the depthwise separable convolution (DSC) operator (Chollet 2017 to generate new individuals by merging information in different dimensions among individuals. The DSC operator includes a depthwise convolution followed by a pointwise convolution. Pointwise convolution maps the output channel of depthwise convolution to a new channel space. When applied to our task, we remove the pointwise convolution in DSC to avoid the information interaction between channels. Eq. provides the details about how to reproduce offspring, and one example is shown Fig. of the Appendix.
(i,,))
= Σ S(i + k.j 10/++0 +1,
A.J
where 5' (i.j) denotes the individuals in the output population, S(i,j) j) denotes the individuals in the input population, and WK,1 represents the related parameters of convolution kernels. Moreover, to adapt to optimization tasks with different dimensions, different channels share the same parameters. The parameters within convolution kernels record the strategies learned by this module to reason over available populations given different tasks. There are still two critical issues to address here.
1) Since there does not exist a consistent pattern in the population. the gradient upon parameters is unstable as well as divergent. A fitness-sentn sensitive convolution is designed, where the CRM's 's attention to available information should be relative to the quality and diversity of the population. WK. reflects
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the module's attention during reasoning and is usually relative to the fitness of individuals. After that, this problem resolved by simply sorting the population in the lattice based on individuals' fitness.
2) Another vital issue is the scale of the offspring. We conduct padding before the convolution operator to maintain the same scale as the input population. However, filling the tensor of the population with constant values '0' is proper, as usually done in computer vision. Instead, mirror padding copies the individuals to maintain the same scale between the offspring and the input population. As the recombination process conducts the information interaction among individuals, copying the individual is better than extending tensors with a constant value. An implementation of mirror padding to the population is given in Appendix.A.3.
The size of convolution kernels within CRM determines the number of individuals employed to implement reasoning of S' (i. After that, this paper employs convolution kemelss with commonly used sizes. Different convolution kernels produce corresponding output tensors, while the final offspring are obtained by averaging multiple convolutions' output. Then, the fitness of this final offspring will be evaluated.
3.3 SELECTION MODULE
The selection module updates the population so as to survive the fittest individuals. SM updates individuals based pairwise com- parison between the offspring and input pop- ulation regarding their fitness for efficiency and simplicity. S;-1 and 5 are the in- put and output populations of CRM, respec- tively. S and 1 contain D+1 channels. The first channel stores the fitness value of an individual. Thereafter, matrix subtrac- tion of fitness channel corresponding S;-1 and 5/_, compares the quality individuals from S and 5/_ pairwise. A binary mask matrix indicating the selected individual can be obtained based the indicator function (z), where 4 (x) = if x 0 and (x) 0 if < O. To extract selected individuals from S and S/1,, we construct a binary mask tensor by copying and extending the mask matrix to the same shape as S; and 5 The selected information forms new tensor S; by employing Eq. (3) illustrated in Fig.
sile
S1-1
Pitness
C
Channel 6
Jypa
Figure 1: SM. An indication matrix is produced by subtracting the fitness channel (cz ) based which individuals within the input and output populations can be extracted to form offspring.
tile((=>o Mp)) S + lile(1 La>o MF)) SID1
where the tile copy function extends the indication matrix to a tensor with size L, D), My (Mp) denotes the fitness matrix of (i-1(S;_1). and indicates the pairwise multiplication between inputs.
3.4 THE STRUCTURE OF DECN
In vanilla EAs, each generation consists of recombination and selection operations. Like EAs, in Fig. [ learnable module based on CRM and SM is designed to learn opti- mization strategies, termed evolution module (EM). Then, DECN is established by stacking several EMs to simulate generations within EAs. S;11 is the input population of E.,. S(1 is the output of CRM in order to fur- ther improve the quality of individuals the global and local search scopes. Then, SM
DECN
BM4
S
EM,
5
SM
CRM,
SW
Figure A general view of DECN and EM.
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selects the valuable individuals from S and S according to their function fitness. Fig. provides an intuitive description of data flow in EM.
wist comvolution
Parameters shured acress channels
within convolution kernd
x
-1
11
1
1
X2
-0.11
S4
1/9
1/9
1/9
1/9
1/9
population
S. Sg
Sort and
paddlnng
8
5
-0.5
+0.5
0.51
Sg
Sg
8 S1
52
00.
00.5
Initial
population
0.16
0.16
Filness
0.25
0.25
mask
(1-mask)
0.02
0.02
S 5
S
5
Figure An example to show the data flow in E.M. Suppose f(s (21, {0.0)) (x) (x2 - 0)2, H; E [-1.1]. L is set to 2. We first transfer the initial population with four individuals the tensor and then sort and pad it into the new tensor with 16 channels. The modified DSC operator is employed to generate I1, 22, and the fitness tensor.
and I2 are handled by parameters shared across channels within a x convolution kernel. The fitness tensor is handled by Eq. (3.. The new tensors of Iı, 22, and the fitness tensor averaged to generate the output.
3.5
TRAINING OF DECN
DECN with / EMs generates the offspring S, from the input population So and can be trained based on end-to-end mode. Then, given proper loss function and training dataset, DECN can be trained to learn optimization strategies towards the objective function [((([] by the back pack-propagation. We will generally establish a function set Ftrain to train DECN.
Training Dataset. This paper establishes the training set by constructing a set differentiable functions related to the optimization objective. This training dataset only contains (So, f(({))), the initial population and objective function, respectively. fa represents the ith function in this set. We show the designed training and testing datasets as follows:
ptrein
{fi (s 51,
(etrai)) fm (s (oletrain)) Ftest (slgst)))))
where F1 is not employed in the training stage, and m the number functions in ptrain gtrain Sm.t represents the ith different values of & in mth function fm. which is true for any index pair. The initial population Sp always randomly generated before optimization. ptrain is comprised of different functions and has diverse landscapes from pleat.
How to Train DECN. DECN attempts to search for individuals with high quality based on the available information. The loss function tells how to adaptively adjust the DECN parameters to generate individuals closer to the optimal solution. According to the Adam (Kingma & Ba 2014 method, a minibatch Ω sampled each epoch for the training of DECN upon Feran which comprised by employing K initialized Sp for minibatch Ω for f. ptrain
each f.. We give the corresponding mean loss of
Σ f((s(E
Σ
f((sk(
150
[o)
sES
(S.)
min C,
(5)
SoeR
Σ f((s(E) S
Eq. (5) is to maximize the difference between the initial population and the output population of DECN ensure that the initial population is close the optimal solution. Moreover, Eq.
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Algorithm Training of DECN
Input: Batch size for Adam, Ωi Function set for training, ptrain
Output: Parameters of DECN 0; Randomly initialize 0 of DECN;
Randomly initialize etrain to adjust fy in Ftrain,
repeat
Randomly initialize a minibatch ft comprised of populations So:
for f3 in Ftrain do
Update ! by L; given training data (So:fi)
end for
Update 8 by minimizing- -1/m Ej Lj; Re-initialize parameters Ej of fj in ptrain every 7 epochs;
until training is finished
generally differentiable based the constructed training dataset. Eq. enables DECN to perform the exploitation operation well but does not strongly encourage DECN to explore the fitness landscape. However, have many options balance exploration and exploitation. For example, the constructed Bayesian posterior distribution over the global optimum (Cao & Shen 2020) added
to Eq. (5). Suppose the objective functions hard be formulated, and the derivatives of these functions are not available at training time, then two strategies can be employed: 1) Approximate
these derivatives via REINFORCE (Williams) T992]; 2) Use the neuro-evolution (Such al. 2017]
Stanley et al. 2019) method to train DECN. Algorithm I provides the training process of DECN. After DECN has been trained, DECN can be used solve the black-box optimization problems since the gradient is unnecessary during the test process.
Table 1: The compared results on six functions. The value of the objective function is shown in the table, and the optimal solution bolded. *(*) represents the mean and standard deviation of repeated experiments.
D
F
DECNws3
DE
ES
CMA-ES
L20-swarm
Dragonily
F4
le-6(1e-07)
0.15(0.06)
0.14(0.07)
4c-3(4e-3)
0,30(0.01)
1310(1310)
F5
le-3(6e-06)
4.62(1.07)
0.47(0.13)
0.11(0.05)
0.25(8.70e-4)
48.4(9.58)
F6
8.89(1e2)
244(95.1)
118(235)
1660(2540)
154(243)
4e8(1e08)
10
0(0)
18.5(3.51)
48.1(8.68)
48.3(7.47)
12.8(5.49)
81.1(24.0)
F8
0(0)
0,26(0.12)
0,30(0.17)
0.02(0.01)
0.06(3.39c-4)
35.4(22.0)
F9
1e-3(3e-4)
1.86(0.36)
20.5(0.13)
20.7(0.10)
2.19(0.02)
16.2(3.64)
F4
74-9(4e-11)
8,563(396)
33446.11e3)
7330(1e3)
0.66(0.04)
None
F5
4e-4(0)
28.2(0047)
82)5(2.16)
71.8(9.74)
0.96(0.04)
None
F6
96(0.02)
2.3e8(2.8e7)
2.5e10(3.1e9)
3.4e8(9.5e7)
286(28.5)
None
100
0(0)
9,343(504)
9.3)4(9.133)
8.7e3(1.5e3)
50.6(21.0)
None
0(0)
3.06(0.22)
24.2(2.91)
2.90(0.31)
0.15(1e-3)
None
F9
4e-3(8e-05)
18.9(0.15)
21.4(0.01)
21.4(0.03)
3.06(0.02)
None
EXPERIMENTS
4.1
RESULTS ON SYNTHETIC FUNCTIONS
Results on figh-tidelity Training Dataset For each function in Appendix Table we produce the training dataset as follows: Randomly initialize the input population Sy: 2) Randomly produce shifted objective function fs(g)) ) by adjusting the corresponding location of optima- nomily. adjusting the parameter ç: 3) Evaluate Sn by f((s(E;; Repeat Steps 1)-3) to generate the corresponding dataset. For example, we show the designed training and testing datasets for the F4 function as follows:
ptrain [F F4(s etrain)]. Feest [F4(s]efest)
(6)
Etrain and prast are comprised of the same essential function but vary in the location of optima obtained by setting different combinations of (called by in Table 77). ptrain can be considered

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as the figh-iidlitty surrogate functions of Fresst We train DECN on Ftrain, and then we test the performance of DECN upon Frest, where the values of glest not appearing in the training process.
Here, D {10, 1 and L 10. DECN is compared with standard EA baselines (DE DE/rand/1/bin) (Das & Suganthan 2010), ES ((,,A)ES), and CMA-ES), L20-swarm (Cao et al.
2019
(a representative L20 method for black-box optimization), and Dragonfly (Kandasamy et al.
2020
(the state-of-the-an art Bayesian optimization). DECNws3 contains 3 EMs, and the parameters of
these three convolution kernels are consistent across different EMs (weight sharing). The detailed
parameters of these models can be found in Appendix A.11. The results are provided in Table [ DECN outperforms compared methods by a large margin. This because we use figh-fidelity surrogate function of the target black-box function to train DECN. The trained DECN contains an optimization strategy that is more tailored to the task. Current DE, ES, CMA-ES, and Dragonfly do not use this information to design their element. Even if we constantly adjust the hyperparameters of the comparison algorithm, the results are unlikely to be better than DECN.
Results on Low-fidelity Training Dataset The training of DECN requires a differen- tiable surrogate function for the black-box optimization problem. However, accurate high-fidelity surrogate functions are difficult to obtain. Therefore, this section tests the per- formance of DECNs trained on low-fidelity surrogate functions. Three functions in Table are employed as the low-fidelity surrogate functions for each function in Table 7 Here, the whole functions in Table 6 are employed as Firai in order to train one DECN, and then the results on each function of Table 7 are shown in Table 2 For example, show the designed training and testing datasets for the F4 function as follows:
Table 2: The performance of different DECN.
DECNws3
DECNws30
DE
F4
53.8(14.3)
1.17(0.58)
0.09(0.02)
F5
4,26(0.61)
0,59(0.16)
0.19(0.03)
F6
4.5(2.3e4)
131(67.1)
17.0(2.88)
F7
24.4(4.01)
0.35(0.22)
5,93(1.40)
F8
1.48(0.14)
0.29(0.08)
0.17(0.04)
F9
4.33(0.41)
0.91(..32)
0.22(0.04)
F4
1.15c((444)
2.19c3(148)
67.4(9.09)
F5
25.0(0.92)
10.4(0.39)
2.22(0.15)
F6
3c8(4e7)
lc7(2c6)
2e4(5e3)
100
F7
776(19.2)
549(18.3)
81.5(11.9)
F8
105(6.12)
20.9(1.17)
1.58(0.08)
11.6(0.21)
6.79(0.14)
3.77(0.16)
ptrain (F(())))))), (2((((22),, (3))),)))), platt [F4(s)))))))
Meanwhile, we also test the impact of different architectures on DECN, including the different number of layers and whether weights are shared between layers. We design three models, including DECNws3, DECNn15, and DECNws30. DECNn15 does not share parameters across 15 EMs. DECNws30 shares parameters across 30EMs. Their parameters are shown in (Appendix)
DECNws30 outperforms DECNws3 in all cases, demonstrating that deep architectures have stronger representation capabilities and can build more accurate mapping relationships between random populations and optimal solutions. DECNn15 outperforms DECNws3 and DECNws30 when = 100. This case is more complex than the case with D 10. Although the number of layers of DECNn15 is lower than that of DECNws30, its representation ability is stronger than that of DECNws30 because does not share weights. However, when the number of layers becomes larger, this architecture is more difficult to train. The transferability of DECN is proportional to the fitness landscape similarity between the training set and the problem. When new problem attributes not available in the training set, DECN can still perform better. However, if extreme attributes are not available, then DECN can be the less satisfactory performance for functions with this attribute. These results show that the optimization strategy learned by DECN has good generality and is transferable to many unseen objective functions.
4.2 RESULTS ON PROTEIN DOCKING
Protein docking predicts the 3D structures of protein- prreein complexes given individual proteins'
3D structures or ID sequences (Smith & Sternberg 2002). We consider the ab initio protein docking problem, which is formulated as optimizing the Gibbs binding free energy for conformation f(s) = GG()). We calculate the energy function in a CHARMM 19 force field as in (Moal& & Bates] 2010). We parameterize the search space
IR 12 in (Cao & Shen 2020 We only consider
100 interface atoms. The training set includes 125 instances (see Appendix A.9), which contains 25
protein-protein complexes from the protein docking benchmark set 4.0 (Hwang al. 2010), each
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of which has five starting points (top-5 models from ZDOCK (Pierce et al. includes three complexes (with one starting model each) of different levels docking difficulty. IATN is the protein class that appeared during training. 1ATN 7 is the No. 7 instance of the IATN class, and it did not appear in the training process. 2JEI. and 7CEI are the No. 1 instances of the two classes of proteins that did not participate in the training process. For example, show the designed training and testing datasets for IATN_7 as follows:
2014 The testing set
ptrain
Flest {f(s|(test)
(8)
where & represents different instances of protein-protede complexes.
In L22--warm, the part of Bayesian posterior distribution over the global optimum (Cao & Shen 2020 is re- moved to keep fair with DECN. The experimental results reported Ta- demonntrattt that DECN outper- forms L20-swarm, Dragonfly, DE, CMA-ES, and ES all three cases. Since L20-swarm has no clite reten- tion mechanism, result is worse than the optimal value of the initial population.
Table 3: The compared results on ab initio protein docking problem. D 12.
Methods
IATN_7
2JEL_1
7CEL
L20-Swarm CMA-ES
2091(25.08)
2766(24.80)
1690(23.64)
-6240(100)
-6260(51.8)
6170(18.4)
ES
-6200(48.1 1)
-6210(5.05)
-180(2.47)
DE
-6260(58.1 1)
-6220(29.2)
-6140(20.5
Dragonfly DECNws3
-6160(4.3)
-6120(2.9)
-6103(2.0)
-6261(96.71)
-6250(84.88)
-6193(84.66)
4.3 RESULTS ON PLANAR MECHANICAL ARM
The planner mechanic arm has been frequently employed an optimization problem assess how well the black-box optimization algorithms perform (Cully et al., 2015 Vassiliades et al., 2018 Vassiliades Mouret 2018 Mouret & Maguire 2020). The planner mechanic arm problem has two (a1, 022, (.,). where n represents the number of segments of the mechanic arm, and I.₃ E (0, 10) and a; E (-IT, TT) rep- resent the length and angle of the ith mechanic arm, respectively. This problem is to find the suitable sets and a such that the distance f(L, a; p) from the top of the mechanic arm to the target posi-
key parameters: the set of L (h,, L2, Len) and the set of angles
V V i cos(a;) p=)² + (2i sin(a;) pw)
tion p is the smallest, where f(L, and (Pa. Py) represents the target point's and y-corddinates. Here, 100. We design two groups of experiments.
Table 4: The results of planar mechanical arm. gen is the number of generations for EAs.
Case
gen
DE
ES
CMA-ES
L20-Swarm
DECNws3
2.96(1.63)
11.2(4.70)
236(46.8)
40.4(3.89)
0.42(0.222
10
300
11.3(14.7)
45.3(43.3)
243(125)
69.5(3.77)
1.04(1.25)
SC
100
1.28(0.60)
10.7(5.91)
2.42(0.65)
40.4(3.89)
0.42(0.22)
50
300
1.54(0.89)
42.0(41.0)
4.06(6.54)
69.5(3.77)
1.04(1.25)
100
1.20(0.64)
10.6(5.58)
1.36(0.35)
40.4(3.89)
0.42(0.22)
100
300
1.38(0.71)
44.9(43.3)
1.38(0.41)
69.5(3.77)
1.04(1.25)
100
0.81(0.47)
8,95(6.42)
0.76(0.20)
31.9(1.78)
0.38(0.25)
CC
100
300
6.15(12.2)
47.8(56.0)
0.87(0.37)
89.1(1.96)
8.27(21.3)
J) Simple Case (SC). We fixed the length of each mechanic arm ten and only searched for the optimal a. We randomly selected 600 target points within the range of S 1000. where r represents the distance from the target point to the origin of the mechanic arm, as shown in Fig. (Appendixx) In the testing process, we extracted 128 target points in the range of S 100 and 300, respectively, for testing. We show the designed training and testing datasets as follows:
ptrain- (J(a))),, ... {(alEeo)))))) plest
[faakttet) ).... "..."...allaaa ((alctest)), E (pa, Pa)
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2) Complex Case (CC). We need to search for L and a at the same time. We show the designed training and testing datasets as follows:
pitrein [f((L, a)l{)), ... .f(LL, a)]6600)) prest = ([((L,a))f (..
setaf) }
(10)
We evaluate the performance of the algorithm by E f 1/128. The experimental results are shown in Table Note that Dragonfly performs poorly due to the high dimensional of this problem (D {100, 200 In simple cases, DECNws3 outperforms all basclines. Nevertheless, for complex cases, DECNws3 outperforms all baselines when r S 100. However, when r S 300, DECNws3 outperforms ES and L.20-Swarm and is weaker than DE and CMA-ES, as shown Table 2 the performance of DECNws3 is worse than DECNws30 and DECNn15, When we use DECNn15 to optimize the complex case, its result is 0.54(0.26), which is better than all baselines.
4.4 ACCELERATING DECN WITH GPU
We show the surprising performance of DECN with GPU-accelerated CRM and SM. To display the adapt- ability of DECN to GPUs, we offer the average runtime (second) of DECN and unaccelerated EA for three gen- erations in Table 5 See Appendix A.6 for more results. DECN and EA optimize K = 32 populations, each containing L L individuals (number of individuals: K x L x L). Similarly, we employ the runtime of EA with SBX crossover and Breeder mutation operator without acceleration as a reference in this experiment. In the unified test environment, the function estima- tion time consumed by DECN and EA is basically the same. As can be seen, with the increase of L, the ad- vantage of acceleration based on GPUs is clear. DECN is around 102 times faster than EA when acceleration GPUs and can be
Table
DECN's calculation efficiency upon one 1080Ti GPU.
L
D
Algorithm
10
80
DECN
0.0054
0.0500
EA
0.6942
72.3664
DECN
0.008026
0.237182
50
EA
0.6820
71.4425
DECN
0.0412
2.7274
500
EA
0.7208
74.9417
D E (10, 50, 500). This case indicates that DECN is adapted accelerated sufficiently. However, GPU cannot accelerate EA's crossover, mutation, and selection modules. In the case of large population of individuals, these operators take up a high running time.
4.5 VISUALIZATION
We take two-dimensional F4 function an ample to verify that DECN can indeed advance the optimization. In Fig.
the iteration pro- ceeds, DECN gradually converges. When passing through the first EM module, the CRM is first passed, and the offspring 8/- are widely dis- tributed in the search space, and the offspring are closer to the optimal solution. Therefore, the CRM generates more potential offspring and is rich in diversity. After the SM update, the generated S; is around the optimal solution, showing that the SM update can keep good solutions and remove poor ones. From the population distribution results of the 2nd, 3rd, and 15th EMs, DECN continuously moves the population the vicinity of the optimal solution.
Figure 4: Visualization of the optimization pro- cess.
CONCLUSIONS
We successfully designed DECN to learn optimization strategies for black-box optimization automat- ically. The better performance than other human-de -demande methods demonstrates the effectiveness of DECN. DECN be well adapted to new black-box optimization tasks. Moreover, DECN has outstanding adaptability to GPU acceleration due to the tensor operator. The limitations are discussed in Appendix A.8.

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REPRODUCIBILITY STATEMENT
The source code of Pytorch version of DECN can be downloaded in supplementary materials. The parameters of DECN are shown Table in Appendix. Nine synthetic functions are shown in Appendix A.5 (Tables and 77. The 25 protein-protein complexes used for training DECN are shown in Appendix A.9.
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A APPENDIX
A.1 BACKGROUND
Recombination. The subtraction is applicable during the production new individual, such as DE (Das & Suganthan| 2010) and recombination operator in EAs, usually conducted individuals. DE can reproduce a unique individual 8" based on I1 with 81, 82,
8n.
8k 7[88-8441
(11)
where F; is a scaling factor and Sy the best solution is selected from $1, $2... Sn- expression expansion, this process can be summarized by weighted recombination process as given Eq. 12 These operators are manually designed with different parameters (a;).
After an
01
S1 +
02 x 82 + ar ST
0g 8/.
(12)
Selection. Many selection operators exist, such as the binary tournament mating selection operator in Eq. [13). The selection operator is retain individuals of higher quality for the next generation, which can be regarded as an information selection process.
p: { (8)) < f(Kk) f(s.) > f(sk)* (s;, sx) S,
(13)
where P: reflects the probability that 8; is selected for the next generation, and (8t, 8k) in Eq. randomly selected from the population S. The selection process will be repeated until the number of individuals chosen.
13
A.2 HOW TO ORGANIZE A POPULATION INTO A TENSOR
As shown in Fig. 5 individuals in the lattice are sorted in descending order to construct a population tensor with consistent pattern. Suppose population S 81, "*. ,L L and f(s)) f(22) <
< (SLLL), where f(s) is a minimization task. $1, $2, ... SLXL are arranged descending order within the L x L lattice.
Ms) descend
Stxt
:
:
:
+
:
As) dosccas
S4+2
S2+1
S2
$1
Figure 5: Organizing population into a tensor.
Figure 6 gives an illustration of the tensor data. As can be seen, the number of channels of input tensors D+1, where D is the dimension of the optimization task, and the fitness of individuals occupies one channel. The fitness channel does not participate the convolution process but is essential for the selection module in DECN for the information selection.
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Si
rox
CODY
Figure 6: The realization of similar functions as recombination operators based on convolution operator. Convolution kernels slip over the whole L lattice and conduct the information interaction within the neighborhood of (ž,j). For the picture on the left, the small red square with many channals represents St.
A.3 POPULATION ARRANGEMENT
Figure 7 gives an example of population arrangement and padding for the problem min f(s) 21 Xx1, where 21; 21 E [0, 10). The blue part marks the population arranged in a 10 lattice, while the gray region marks the mirror padding part.
10.
13.
9.9
0.8
9.7
9.5
9.5
9.4
9.3
9.2
9.1
9.1
9.9
9.8
9.5
9.3
9.2
80
8.0
2.4
7.7
7.5
2.5
2.4
7.2
2.1
2.1
5.5
5.0
5.0
4.9
4
4.7
4.5
4.5
4
2
4.1
4.1
3.5
2.0
2.0
1.9
88
1.7
1.5
1.5
1.4
13
1.2
1.1
0.9
88
6.5
0.5
0
0.>
0.1
Figure 7: Population arrangement and padding.
A.4 SEVERAL ESSENTIAL ISSUES ABOUT CRM
There several essential issues necessary to be considered.
1) How many individuals should participate the CRM reasoning progress. remains a challenge implement information reasoning over multi-individuals in EAs. In most recombination operators, the participant number is usually set to 2. However, based on the gradient information provided by the back- propagao,,,,,,,, is easy to control individual's element by adjusting Wx.1-
2) How to integrate the offspring produced by different convolution kernels. Since the convolu- tion operation can be transformed multiplication between matrices, simply averaging over the
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Offssring
Average
DSC
Padding
Population Tensor
Figure 8: Reproduction of the offspring based on CRM.
results output by different convolution kernels does not influence the training process. For example, a1 of + azCo²zh + a3Coo ++ Co x + Co + Co Sxy. where z' zÿ. and zÿ are input elements of s; a1. d2, and 0g are the constant, and Co denotes the convolution matrix.
3) How many convolution kernels should be used within CRM. We suppose that these are three convolution kernels for æ. can find that the outcome on Co + 02Co + agCooxx3 is equivalent to a Coaxa.. The output of multiple convolution kernels can be replaced by one convolution kernel. Thus, the number convolution kernels the same size has no apparent influence DECN.
4) The impact of neighborhood recombination operation. The neighborhood recombination operation has been commonly accepted in EAs to alleviate the selection pressure and prevent the premature convergence of populations. Moreover, the receptive field of convolution kernels expands as the number of layers increases. Thus, DECN can Iearn efficient optimization strategies across generations.
A.5 NINE SYNTHETIC FUNCTIONS AND PARAMETERS
Table Training functions.
ID
Functions
Range
Fl
Σ,
[wisin (a;: 6.)
x 10, 10), E [-00, 10
Σ, |mg by]
J. 10, 10 E
-10, 10
B Σ, (xi - h)) (1++1 bi+1)l + (z,
b.l
x 10, 10), b E
-10, 10
A.6 ACCELERATE DECN WITH GPU
The acceleration of EAs using GPUs is challenging, and lots of research has contributed to this prob- lem. The support for multiple subpopulations to evolve simultaneously has paramount significance in practical applications. The efforts (Jin & Qin 2017, Qin et al.) 2012) accelerated the K-Means process within the brain storm optimization algorithm through GPUs and proposed an improved CUDA-based implementation of differential evolution GPUs. Many other EAs have benefited from the computing performance of GPUs (Huang et al. 2021 Cheng & Gen 2019). However, all of them just parallelized the current EAs. Besides, many available genetic operators the GPU acceleration, as GPUs are weak in processing logical operations. As both CRM and SM are comprised of operations upon tensors, they can be sufficiently accelerated by GPUs.
unfriendly to
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Table 7: Testing Functions.
ID
Functions
Range
F4(Sphere)
E b, max 1sl1ss D), 2; -b:
2: | 100, 100), E 50, 50] 100, 100), E | 50, 50
F5
F6(Rosenbrock)
(100(2) 2++1²² + (z: - 1)²), 21
# 100, 100). E 50, 50]
F7(astiig)
[2]
c0coss2Tzs) + 10), 24 = I2 - b,
r
[ 5, 5j.
E [- 2.5, 2.5]
F8(Griewank)
Σ
x?
-1 cos( 3t +1, Hz bb
E [-600, 600 b | 300, 300]
4000
(9(ckleyy)
-20 -20expp-0022 1 Σ +
E (-32 32 b | 1-16, 16]
exp
1 Ee cos((2222)) + 20 + exp(1),
Table 8: Investigation of DECN's calculation efficiency when accelerated upon one 1080Ti GPU. The results in this table are the average time (second) of algorithms conduct the evolution of 32 input populations for three generations.
D
Algorithm
20
40
DECN(s)
0.004627
0.005978
0.007449
0.015492
2
EA(s) 0.700342
2.863495
12.67563
71.74005
Rate(DECN/EA)
0.006607
0.002088
0.000588
0.000216
DECN
0,005487
0.007838
0.01664
0.049973
10
EA
0.694213
2.879791
12.87636
72.3664
Rate(EM/EA)
0.007904
0.002722
0.001292
0.000691
DECN
0.007052
0.013323
0.039877
0.138544
30
EA
0.693307
2.87876
13.00381
71.67544
Rate(EM/ RA)
0.010171
0.004628
0.003067
0.001933
DECN
0.008026
0.01875
0.062079
0,237182
50
EA
0.681967
2,868
12.80662
71.44253
Rate(EM/ EA)
0.011769
0.006538
0.004847
0.00332
DECN
0,011725
0.033109
0.117593
0.478518
100
EA
0.699074
2.865546
13.12218
71.33443
Rate(EM// EA)
0.016772
0.011554
0.008961
0.006662
DECN
0.041167
0.147843
0.610056
2.727426
500
EA
0.720847
2,966926
13.59977
74.9417
Rate(EM// EA)
0.057109
0.04983
0.044858
0.036394
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DECN mainly containing operations upon tensors and is easily accelerated by GPUs. Current distributed EA methods usually separate a population into multiple subpopulations that evolve simultaneously. Such separation is also commonly accepted operation in many EAs However, none of them can accelerate the genetic operators. Here, we show the surprising performance of DECN with GPU accelerated CRM and SM. Moreover, Tensorflow has provided mature solutions for the acceleration upon GPUs, and DECN implemented by Tensorflow is supportable to load multiple populations the input.
To show the adaptability of DECN to GPUs, we offer the runtime of DECN and unaccelerated EA in Table[8] within which both DECN and EA optimize K 32 populations with each containing L individuals (number of individuals: K x L L). Similarly, we employ the runtime of EASBX without acceleration as reference this experiment. acceleration based on GPUs is clear. When the dimension D) E {2, 10), DECN runs 103~104 times faster than EA. DECN is still around 102 times faster than EA when D E (30, 50, 100, 500). This case indicates that DECN is adapted to the acceleration of GPUs and can be accelerated sufficiently. However, with increasing D, DECN increases the proportion of evaluations in the runtime and ultimately weakens the advantage of acceleration. These cases indicate the acceleration advantage of DECN when optimizing a larger population.
can seen, with the increase of L., the advantage of
A.7 THE CONVERGENCE OF Loss FUNCTION IN TRAINING PROCESS
This part is the change curve of the loss function of the training process of DECNnWs15 on F4-F9. The results are shown in Fig. Here, D=10.
A.8 LIMITATIONS
However, DECN has many drawbacks. We hope to address these deficiencies in future work. I) The designed loss function enables DECN to perform the exploitation operation well but does not strongly encourage DECN to explore the fitness landscape. However, we have many options to balance exploration and exploitation. For example, the constructed Bayesian posterior distribution (Cao & Shen 2020) over the global optimum is added to Eq. 7. In addition to adding items that focus on the exploration ability of the loss function, new modules can also be designed to be added to the EM to help DECN jump out of the local optimum.
2) For the constructed training dataset, DECN does not have an advantage if it is utterly irrelevant to the optimization objective. Thus, establishing a suitable training dataset is essential.
3) DECN only focuses on continuous optimization problems without constraints. For problems such as expensive optimization, combinatorial optimization, constrained optimization, and multi-objectiv optimization, DECN needs to be adjusted according to the characteristics of the problem. We can think of DECN as standard optimizers like vanilla DE, ES, GA, and PSO. In order to deal with different types of problems, we need make different corrections to DECN. For example, to deal with expensive problems, we need to build surrogate models to assist DECN. We need to redesign the CRM module to generate new feasible solutions for combinatorial optimization problems. For example, for TSP tasks, GNN may be feasible option to generate new solutions instead of CRM. We can redesign the CRM module for constrained optimization problems to generate feasible solutions. Of course, the casiest way is to use constraint violations and fitness functions criteria for selecting the next generation in the SM module.
A.9 TRAINING DATASET FOR PROTEIN DOCKING
The training dataset contains 25 protein- protein complexes from the protein docking benchmark set 4.0 (Hwang et al.) 2010). The detailed information is shown as follows: IATN, 1AVX, 1AY7, 1BJ1, IBVN, ICGI, IDFI, TEAW, 1EWY, 1EZU, IGRN, 1IBR, 1IJK, 1IQD, 1JPS, 1KXQ, 1M10, IMAH, IN80, IPPE, 1ROR, IXQS, 2B42, 2COL,, and 2HRK.
A0 MOTIVATIONS
The generalization ability of current evolutionary algorithms (EAs) is poor. Faced with a new black- box optimization task, need experts to redesign/select the EA's crossover, mutation, and selection
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(b)
(c)
(d)
(c)
(f)
Figure 9: The convergence of loss function in training process. (a) F4, (b) F5, (c) F6, (d) F7, (e) F8.
and (f) F9.
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target
r
1
a2
a1
03
12
Figure 10: Planar Mechanical Arm.
operations (including their hyperparameters) to maximize its performance on the target task, resulting in hand-designed EA with big application limitation. Most importantly, due to the limitation of expert knowledge, only little target function information is used to assist the design of EA, which makes challenging to adapt to the target task. How to automatically design optimization strategies according to new tasks is crucial. To the best of our knowledge, there is currently no work to address this issue. We think EA is generative optimization model that realizes the generation from a random population to optimal solution by manually designing crossover, mutation, and selection operations. The purpose of these operations is to generate potential solutions and retain good solutions. The task of automatically designing an optimization strategy is learning how to automatically generate and retain potential solutions. This paper is to show how DECN finish this task.
By constructing a of differentiable surrogate functions of the objective black-box function, DECN can allow the designed CRM and SM to learn the strategy of optimizing the objective function. At this point, DECN effectively utilizes the information of the target black-box function assist the construction of the optimization strategy. The degree of fit of DECN with the target task is much higher than that of the numan-designe EA. The following statement may be one-sided Bayesian optimization also suffers from poor generalization. For example, how to choose/design appropriate acquisition functions for different problems.
We use the process of an evolutionary algorithm to guide the design of DECN and realize the mapping from a random population to the optimal solution.
First, we need to design module to ensure the exchange of information between individuals in the population to achieve the function of generating potential solutions (similar to the recombination operators in EA). We can achieve this function by modifying the convolution operation accordingly, which is our motivation for using convolution to design. The designed CRM module achieves this purpose (see Section 3.1).
Second, to survive good individuals for the next layer of DECN, we design the selection module (SM) based pairwise comparison between the offspring and input population regarding their fitness (see Section 3.3, Equation 3). We can clearly observe that Equation can indeed keep good individuals.
Third, the untrained DECN docs not handle the black-box optimization problem well because needs information about the target black-box function. In order to better optimize the objective
19

---

Under review as a conference paper at ICLR 2023
task, need to design training set containing objective function information and practical loss function to guide the parameter training of DECN (see Section 3.5). The characteristics of black-box functions make it difficult for to obtain their gradient information to assist in the training of DECN. We construct a differentiable surrogate function set of the target black-box function to obtain the information of the target black-box function. The designed loss function is to maximize the difference between the imitial population and the output population of DECN to ensure that the initial population close to the optimal solution.
There are few box optimization problems, and their performance is weak. From the experimental results, DECN makes up for the performance disadvantage of the learning to-optimize architecture in the black-box optimization problem. We also strongly believe that this paper makes an essential contribution the Garning-to-optime community.
architectures (Chen el al. 2022 currently dealing with black-
A.11 PARAMETERS
DECN is compared with standard EA baselines (DE (D/rran////bin) (Das & Suganthan, 2010). ES ((,A)-ES), and CMA-ES), 1.20-swarm (Cao et al. (0199) (a representative 1,20 method for blox
optimization), and Dragonfly (Kandasamy et al. 2020p (the state-of-the-art Bayesian optimization).
DE and ES are implemented based on Geatpy (et.al. 2022)) and CMA-ES is implemented by Pymoo
(Blank & Deb 2020). The parameters of DE, ES, CMA-ES, and Dragonfly are adjusted to be optimal each problem. L20-swarm and DECN use the same training set and loss function. All algorithms are run ten times for each function. DECNws3 contains EMs, and the parameters of these three convolution kernels are consistent across different EMs (weight sharing). The population sizes of DE, ES, CMA-ES, and DECN are 100. DE, ES, and CMA-ES run for 100 generations. For DECNws3, its architecture determines that DECN has only been iterated for three generations. DE, ES, and CMA-ES have 100/3 times as many function evaluations as DECN, which is highly unfair DECN. Both Dragonfly and L20-Sward run to convergence.
Table 9: Experimental setup for DECNws30, DECNws3 and DECNnWs15. In DECNws3, parameters of these three convolution kernels consistent across different EMs (weight sharing). Moreover, during the training process, the 2-norm gradients is clipped to be not larger than 10, and the learning rate (Ir 0.01) shrinks every 100 epochs. The shrinking rate is set to 0.9. The generation of these reference algorithms is set to 100, while DECNwS3 only evolves the population with EMs. 5000 epochs conducted during the training process. All experimental studies are performed Linux PC with Intel Core 17-10KOKK CPU at 3.80GHz and 32GB RAM.
Model
L
K
EMs
Convolution kemelss
by
Epochs
T
Weight Gradient share
norm
3 u 0, 0.5
DECNws30
10
32
5: u 0, 0.5
0.01
10000
10
True
10
7 7: u = 0, o = 0.5
3 x 3: u = 0, 0.5
DECNWs3
10
32
3
5 x 5: u
0, 0.5
0.0005
5000
10
True
10
7 7: u 0,, 0.5
x 3: u = 0, 0.5
DECNa15
10
30
16
15
x 5: u = 0, 0.5
0.0005
2000
10
False
10
7 x 7: u
0, 0.5
20


Reviews:
[{"review_id": "q_5vYksjofy", "reviewer": "Reviewer_Rucq", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper introduces a deep evolutionary convolution network (DECN) for continuous black-box optimization. DECN is composed of two modules: convolution-based reasoning module (CRM) and selection module (SM). The paper describes both modules and shows how to integrate them. It also contains a description of the process of training including the design of the loss function. The experiments carried out on unconstrained continuous optimization problems show that DECN surpasses some black-box optimization baselines and obtains good performance when transferred to optimization problems unseen during the training stage. The paper also presents experiments on accelerating DECN using GPU suggesting its good adaptability to GPU acceleration due to the tensor operator. The main article is also followed by the Appendix providing more details, including limitations of the model.", "summary_of_the_review": "In general, I like the paper and the idea - it seems to be quite original and the quality of the results seems to be very good. The weaknesses of the article that I see are related mostly to clarity and lack of reproducibility. The weaknesses of the method are clearly explained in the appendix. The quality of the results seems to be very good.", "strength_and_weaknesses": "Strengths:\n- Originality: the idea seems to be quite novel.\n- Comprehensive description: many details are provided, and the description of the method and experiments seem to be comprehensive.\n- Quality of results: the experiments carried out on unconstrained continuous optimization problems show that DECN surpasses some black-box optimization baselines and obtains good performance when transferred to optimization problems unseen during the training stage. The paper also presents experiments on accelerating DECN using GPU suggesting its good adaptability to GPU acceleration due to the tensor operator.\n- The authors are honest about the limitations of DECN.\n\nWeaknesses:\n- In some cases, writing could be improved.\n- It doesn't look that the code is available, so the reproducibility is limited.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "nAql0ZCCnkw", "reviewer": "Reviewer_j4Dd", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper introduces a new evolution based search technique inspired by convolutional networks used for black box optimisation.  The paper introduces a novel mechanism for generating and then selecting individuals in a population based on learnable convolutions.  The paper empirically test the new approach on a number of problems and demonstrates superior performance to some baseline optimisers both in terms of accuracy and speed. They further show that the method benefits from transfer between problems.", "summary_of_the_review": "The authors have clearly worked hard to develop a novel method to perform black-box optimisation.  However, the field of EAs are full of such novel methods each claiming to be SOTA based on a rather limited empirical test.  For me, I need to see a convincing argument why the approach should work and this seemed missing.  I also need some strong empirical comparison on competitive problems and I felt that despite some effort in this direction it was still not sufficiently convincing.", "strength_and_weaknesses": "The strength of the approach is that it proposes a novel technique that are shown to have better performance than other well known methods.\n\nFor me there are a number of weaknesses.  The approach is motivated by a rather unconvincing (at least to me) analogue between recombination operators and convolutions.  Convolutions in deep neural networks have advantages such as locality and translational invariance that do not seem to carry over to optimisation.  The model of recombination operator is rather strange as it looks like a mixing operator that is likely to result in rapid convergence.  This seems to addressed by a diversity mechanism that is described in a single line.  For me the paper needs to make a more convincing case why the convolution being proposed is in anyway useful.  The resulting model seems to me very complicated and ad hoc with no real motivation.  Maybe I am missing something, but for me I need much more convincing that the proposed method is well motivated.\n\nOf course, the results seem impressive and may well justify the method.  However, the test functions used are a very old set of problems that are often solved efficiently by algorithms that are not generally very competitive.  They are not particularly representative of the type of continuous variable optimisation problem that most people are interested in solving.  To make a convincing empirical case for this approach I would like to see this tested on a modern set of test problems.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "iLo5NSVrW7", "reviewer": "Reviewer_Tk68", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "The paper proposes the use of convolutional neural networks to act as a recombination operator in a population-based black-box optimization algorithm (aka evolutionary algorithm). That operator is combining the solutions making the population, ranked according to their fitness, with weights provided by the learned convolutions making the neural network. The recombination is made both between individuals, through some kind of mix-up between all individuals of the population through a depthwise separable convolution. The weights are modulated according to the fitness of the individuals used, and multiple convolutions are averaged to produce the output individuals. The neural network model is trained on some problems to generate a combination operator to use for similar problems. Results are provided by comparing the proposed approach with some evolutionary algorithms, with competitive performances.", "summary_of_the_review": "The overall proposal is not well motivated, the explanations are not sufficiently precise and clear, the experiments reported appear good, but still do not bring high confidence in them given the lack of clarity and precision of the experimental configuration. ", "strength_and_weaknesses": "Strength\n- The proposal is relatively straightforward and carries interesting ideas for the  learning-to-optimize context.\n- The results show the approach is able to outperform some black-box optimization baselines for the problems tested.\n- The approach appears to be GPU-friendly.\n\nWeaknesses\n- The approach proposed relies on some intuitive idea that convolutions in neural networks make some computations that can be useful to replace combination operators in an evolutionary algorithm. This intuition appears really rough to me, and the overall conceptual proposal is not well principled.\n- The writing clarity of the paper is low, there is a lack of precision and details, several elements remain unclear and reproducibility appears not obvious to me.\n- Problems tackled are a subset of what can be tested for real-valued black-box optimization (although they are the common baselines).\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "_YbPJcLh36L", "reviewer": "Reviewer_g29J", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors propose a deep convolution network (DECN) that mimics the operation of evolutionary search techniques, specifically recombination and selection, for black-box optimization problems. The paper first describes the general challenges in creating a convolution operator to mimic evolutionary, as well as related work, and then provides definitions of the operations the authors aim to achieve in DECN design (recombination and selection). Subsequently, the authors describe their method, including how they construct a tensor from a population of solutions for a given black-box optimization problem and arrange it to make it amendable for learnable convolutions. The convolution reasoning module (CRM) serves to produce the offspring for the population, whose components are described in detail by the authors, while the selection module (SM), based on pairwise comparisons of solution fitnesses, serves to select the solutions of the evolving population. Taken together CRM and SM make up an evolution module (EM), which can then be used for end-to-end gradient based optimization of the DECN modules. \n\nThe authors then test DECN on six black-box optimization functions and compare DECN's performance to a set of black-box optimization algorithms with the Table 1 showing outperformance of DECN. Subsequently, authors show results on a protein docking task, provide a study of generalization of DECN module across the six aforementioned optimization functions, as well as GPU runtime study. ", "summary_of_the_review": "Overall, I would say that the weaknesses of the paper in its current form outweigh the strengths leading me to vote for rejection. My most pressing concerns include the lack of clarity in the way the paper is currently written (which affects many other parts of my assessment), as well as the quality and relevance of the current experiments (would like to see more challenging optimization problems and the inclusion of more modern baseline algorithms). I would say that the core novelty proposed in the method holds promise and would encourage the authors to continue to refine their research to make the paper stronger. \n\n---\nUpdating my score during discussion phase given the changes and responses provided by the authors.", "strength_and_weaknesses": "**Strengths**\n\n* The paper proposes a novel method to introduce learnable convolution operations into an evolution inspired setting. The results shown in the paper indicate that the proposed method outperforms some common black-box optimizers on a given set of tasks in addition to providing some generalization benefits.\n* The papers provides a cohesive data structure for applying the method and outlines its different component in much detail.\n\n**Weaknesses**\n\n* The paper studies only limited setting (six standard black box optimization functions) with limited experimental settings. Given that the authors claim scalability and speed of their method as an advantage, it would have been nice to see more challenging optimization settings with greater compute costs. \n* The protein docking experiments is not very well described. How does this setting turn into continuous optimization and what is the search space? It would also be helpful to have a consistent set of baselines across all experiments.\n* The setting of the transferability study is unclear. Is this zero-shot learning or few-shot learning? It's hard to asses how meaningful the results are without those details.\n* The authors dedicate a large part of the paper to describing many details of their methods, which then ends up taking up the majority of space. I think the paper could be strengthened by focusing the method on the important parts and re-using the rest of the paper to perform a more thorough analysis.\n\n**Additional Questions**\n\n* Are you initializing multiple populations at the same time? In Algorithm 1, you say \"initialize a minibatch comprised of K populations\", which appears to imply that there are multiple populations solving the same problem?\n* What is $\\zeta$ in equation 7? I did not find a definition.\n* It would be good to clarify what DE is in your baselines.\n* Did the GPU implementation rely purely on deep learning frameworks? Your general description seems to indicate that. \n\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "1MwL1jsHua", "reviewer": "Reviewer_9TVi", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "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": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes a black-box optimization algorithm by incorporating convolutional neural networks into an evolutionary algorithm. Convolutional layers are used for both generating the offspring as well as selecting the offsprings that survive. Experiments are performed on standard black-box functions and protein docking.", "summary_of_the_review": "The paper proposes to incorporate convolutional networks into evolutionary algorithms for solving black-box optimization problems. Both recombination and selection operations are modeled through learned convolutional networks. The motivation for modeling the operators through convolutional layers is not well justified, given that the population members do not necessarily have spatial relations. Experimental evaluation is limited in some respects.\n\nPost Rebuttal Update: \n\nI read the response of the authors, the discussion between reviewers and authors, and the updated paper.\n\nThe original submission certainly had some writing and presentation issues. The author's rebuttal pointed to at least two reviewers who misunderstood the paper. The paper has since been revised significantly. I think the revised form is certainly better, but there are still some claims that are too strong, and I do not agree with them. For example, the claim that evolutionary algorithms have poor generalization.\n\nI have slightly increased my rating, but I still lean toward rejection. I feel the paper has potential but needs further revision.", "strength_and_weaknesses": "Strengths:\n1. Adopting neural networks within evolutionary algorithms for black-box optimization is interesting.\n2. The method seems to be effective, especially for the protein docking problem, task-specific optimization methods have not been compared against.\n\nWeaknesses:\n1. While the idea of modeling the recombination and selection operators through neural networks .\n2. The approach may suffer from the common problems with current learning based methods, e.g., overfitting, lack of domain generalization etc. If the optimization landscapes used for training and deployment are not similar, the approach may lead to sub-optimal solutions. Furthermore, there is no mechanism to identify such scenarios. The paper acknowledge this limitation.\n\nOther Comments:\n1. The topic of the paper, evolutionary black-box optimization, may be more suitable and better appreciated by evolutionary optimization community.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject