WCC: A Learning-Based Framework for Deciding When to Apply Cooperative Coevolution in Large-Scale Optimization

Wenjie Qiu; Hepei Li; Jianfeng Wang; Jun Zhang; Yue-Jiao Gong

cs.LGOct 27, 2026

WCC: A Learning-Based Framework for Deciding When to Apply Cooperative Coevolution in Large-Scale Optimization

Authors: Wenjie Qiu, Hepei Li, Jianfeng Wang, Jun Zhang, Yue-Jiao Gong

Organizations: School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China · Taiyuan University of Technology, Taiyuan 030024, China · Nankai University, Tianjin 300350, China, and also with Hanyang University ERICA, Ansan 15588, South Korea

Abstract

Cooperative Coevolution (CC) and Non-Decomposition Algorithms (NDAs) are two primary approaches for large-scale global optimization (LSGO). Although both have achieved success, they exhibit markedly different performance across problem types and each faces approach-specific limitations, leaving the question of "when to use CC" unresolved. We address this gap with WCC, a novel learning-based framework that dynamically decides when to deploy CC in LSGO. WCC employs a streamlined design for efficient adaptive learning. We formulate the decision as a Markov decision process, designing a targeted feature set, a robust reward, and a tailored network architecture to provide the agent with rich information about the problem state and optimization progress. Our auto-regressive policy enables WCC to perform fine-grained, dynamic switching between CC and NDAs while conditionally allocating the computational budget without expert specifications. WCC shows strong competitiveness and superior generalization across problems, optimization horizons, and baseline optimizers. It realizes a "train-once and reuse" paradigm under a low budget, significantly curbing LSGO cost. Additionally, in-depth analyses yield insights not captured by existing theory, offering practical guidance on when and how to effectively deploy CC.

Figures & tables

problem

WCC-based Algorithms

CC-based Algorithms

NDAs

Hybrid Algorithms

WCC-MMES

DG2-CMAES

CC-CMAES

LCC-CMAES

ERDG-CMAES

OEDG-CMAES

MMES

sep-CMA-ES

LM-MA-ES

FCMAES

MetaES

HCC-ES

CC-SHADE-SL

2.16E-20

7.36E-20(+)

9.12E+06(+)

1.13E+05(+)

9.51E-19(+)

1.15E-18(+)

7.98E+05(+)

3.90E-17(+)

4.40E+06(+)

5.90E+06(+)

4.49E+11(+)

6.96E-20(+)

1.87E+05(+)

\pm

4.90E-20

\pm

8.97E-21

\pm

2.20E+06

\pm

2.54E+04

\pm

6.35E-19

\pm

3.47E-19

\pm

1.37E+05

\pm

7.95E-19

\pm

3.27E+05

\pm

8.20E+05

\pm

2.35E+10

\pm

8.87E-21

\pm

3.64E+05

2.13E+05

1.94E+05(-)

1.18E+11(+)

3.48E+10(+)

1.49E+10(+)

1.87E+10(+)

4.43E+08(+)

1.38E+11(+)

7.33E+11(+)

3.45E+09(+)

2.37E+14(+)

7.18E+06(+)

1.25E+10(+)

\pm

4.72E+04

\pm

2.45E+04

\pm

5.01E+11

\pm

4.99E+09

\pm

7.32E+09

\pm

5.44E+09

\pm

7.68E+07

\pm

2.55E+10

\pm

1.14E+11

\pm

4.20E+08

\pm

5.40E+11

\pm

2.85E+06

\pm

1.10E+10

1.65E+06

1.70E+06(+)

1.05E+07(+)

1.97E+06(+)

2.33E+06(+)

4.17E+06(+)

1.67E+06(+)

1.75E+06(+)

4.43E+06(+)

1.02E+08(+)

1.81E+06(+)

1.92E+06(+)

TABLE I: Comparing WCC-MMES with comparison algorithms on CEC2013LSGO at 1.5E6 FEs.

Fig. 2: Comparison of WCC-MMES and baselines on AOB at

1.5

6

FEs. Radial values are min–max normalized scores, with larger values indicating better relative performance.

Fig. 3: Generalization experiment of WCC across baseline algorithms.

Fig. 4: Ablation study, including (a) action space ablation, (b) state space ablation, (c) reward function ablation, and (d) network architecture ablation.

Fig. 5: Proportion of NDA selections across problem types and optimization stages. Each cell reports the mean proportion of $a_{\mathrm{mode}}=\mathrm{NDA}$ , where darker colors indicate more frequent NDA usage. Rows denote four equal intervals divided by MaxFEs, and columns denote test functions grouped by separability.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Group	Index	Formula	Explanation
$\mathcal{S}_{pro}$	1	$\frac{1000}{D}$	The relative size of the problem dimensions $D$ .
	2	$1-\frac{\sum_{i=1}^{D}\sum_{j=1}^{D}\mathbb{I}\left(\Theta[i,j]=0\right)}{D^{2}}$	The sparsity of DSM $\Theta$ characterizes the interactive complexity of the problem.
	3	$\frac{\left\|\{i\|\sum_{j=1}^{D}\Theta[i,j]=1\}\right\|}{D}$	The proportion of completely separable variables.
	4	$\frac{\left\|\bigcup_{i=1}^{\|\mathbb{S}\|}\bigcup_{j\neq i}^{\|\mathbb{S}\|}\left(\Omega_{i}\cap\Omega_{j}\right)\right\|}{D}$	The proportion of overlapping variables, $\Omega_{i}$ is the variable set of subspace $i$ .
	5	$\frac{\|\mathbb{S}\|}{D}$	The relative size of the number of subspaces $\|\mathbb{S}\|$ .
$\mathcal{S}_{conv}$	6	$\frac{cost^{}_{0}-cost^{}_{t-1}}{cost^{*}_{0}}$	The decrease ratio of optimal cost $cost_{t-1}^{*}$ at step $t-1$ .

Appendix

TABLE I: Complete definitions of the 16 WCC state features.

Setting	Value
Training epochs	90
Training MaxFEs	$10^{6}$ per episode
Optimizer	Adam
Initial learning rate	$10^{-4}$ ; exponential decay
Discount factor $\gamma$	0.999
Rollout length $n_{step}$	5

Appendix

TABLE II: Reported WCC-MMES training and evaluation settings.

Problem	WCC-based Algorithms	CC-based Algorithms			NDAs					Hybrid Algorithms
Problem	WCC-MMES	DG2-CMAES	CC-CMAES	LCC-CMAES	MMES	sep-CMA-ES	LM-MA-ES	FCMAES	MetaES	HCC-ES	CC-SHADE-SL
1	2.03E-20	6.68E-20(+)	7.38E+06(+)	1.09E+05(+)	1.18E+04(+)	1.69E-17(+)	1.63E+05(+)	1.95E+05(+)	9.09E+09(+)	6.31E-20(+)	2.58E-05(+)
	$\pm$ 1.78E-20	$\pm$ 3.78E-21	$\pm$ 3.92E+06	$\pm$ 2.71E+04	$\pm$ 1.80E+03	$\pm$ 7.95E-19	$\pm$ 4.33E+04	$\pm$ 4.04E+04	$\pm$ 7.69E+08	$\pm$ 3.45E-21	$\pm$ 1.22E-04
2	1.35E+03	3.47E+03(+)	2.13E+03(+)	2.09E+03(+)	1.26E+04(+)	5.34E+03(+)	5.98E+03(+)	1.38E+04(+)	4.05E+04(+)	3.26E+03(+)	1.98E+03(+)
	$\pm$ 3.55E+02	$\pm$ 3.49E+02	$\pm$ 1.31E+02	$\pm$ 1.82E+02	$\pm$ 7.61E+02	$\pm$ 1.59E+02	$\pm$ 2.74E+02	$\pm$ 8.52E+02	$\pm$ 5.53E+03	$\pm$ 3.25E+02	$\pm$ 5.73E+02
3	2.00E+01	2.10E+01(+)	2.00E+01( $\approx$ )	2.03E+01(+)	2.00E+01( $\approx$ )	2.14E+01(+)	2.15E+01(+)	2.04E+01(+)	2.16E+01(+)	2.09E+01(+)	2.08E+01(+)

Appendix

TABLE III: WCC-MMES and comparison algorithms on CEC2013LSGO at

3\times 10^{6}

FEs.

Fig. 1: Ablation study, including (a) action space ablation, (b) state space ablation, (c) reward function ablation, and (d) network architecture ablation.

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