## 3 Method  
This paper introduces a new improved algorithm, coACO, to solve the BN structure-learning problem. The coACO incorporates several coevolution operators to improve the performance of the basic ACO. Firstly, the entire ant colony is divided into \(S\) independent ant groups, and the number of groups is set as \(n_{s}\), \(s=1\),..., \(S\), as shown in Figure 2. DE is one of the best evolutionary algorithms for solving the real-valued test function suite of the First International Contest on Evolutionary Optimization (1 st ICEO) [22]. Therefore, the DE strategies [26; 27] are used to guide the coevolution process for all ant groups. As is mentioned in the above section, the pheromone plays an important role in the exploration and exploitation of the ant colony for constructing the solution. A reasonable distribution of the pheromone can directly affect ants to explore their solutions. Thereby, the pheromone level is chosen as the cooperative object shared by all ant groups in the coACO. For each ant group, the pheromone level is denoted as the matrix \(\boldsymbol{\tau}^{s}(t)=\{\tau_{ij}^{s}(t)\}\), \(s=1\),..., \(S\). Each ant group communicates with other groups and coordinates its evolution process using the pheromone as the cooperative variable. The coACO introduces coordination operations, which are based on DE, to affect the pheromone, which leads to the achievement of a more reasonable distribution of the pheromone for each ant group. The DE-based coordination operations contain mutation, crossover, and selection operators [21]. The first one is the mutation operator, which is implemented as follows:  
\[\boldsymbol{u}^{s}=\tau^{r_{1}}+F\times(\tau^{r_{2}}-\tau^{r_{3}}) \tag{13}\]  
where \(\boldsymbol{u}^{s}(t)=\{u_{ij}^{s}(t)\}\), \(s=1\),..., \(S\), is the donor pheromone matrix. \(\tau^{r_{1}}\), \(\tau^{r_{2}}\), and \(\tau^{r_{3}}\) are three different pheromone matrixes that are randomly selected from the ant groups, namely \(r_{1}\
eq r_{2}\
eq r_{3}\
eq s\). The real number \(F\) is a positive parameter between [0,2], called the mutation factor, which controls the amplification of the differential variation \((\tau^{r_{2}}-\tau^{r_{3}})\). The second one is the crossover operator, which is performed as follows:  
\[v_{ij}^{s}=\left\{ \begin{array}{ll}u_{ij}^{s}&\text{if }r_{ij}\leq cr\\ \tau_{ij}^{s}&\text{otherwise}\end{array} \right. \tag{14}\]  
where \(r_{ij}\) is a random value generated for each arc \(a_{ij}\) in accordance with a uniform distribution over [0,1], \(cr\) is the given crossover rate within (0,1), and the obtained \(v^{s}(t)=\{v_{ij}^{s}(t)\}\) is the trial pheromone matrix. Figure 3 describes the crossover process of the pheromone matrixes. Figure 2: The grouping operation and cooperative framework of coevolution ACO (coACO). The third operator is the greedy selection, which is performed as follows:  
\[\tau^{s}(t+1)=\left\{ \begin{array}{ll}\mathbf{v}^{s}(t)&\text{if }f(\mathbf{v}^{s}(t)) \geq f(\mathbf{\tau}^{s}(t))\\ \mathbf{\tau}^{s}(t)&\text{otherwise}\end{array} \right. \tag{15}\]  
where \(\tau^{s}(t+1)\) is the pheromone matrix of the \(s\)-th ant group in the \((t+1)\)-th iteration. Ants in each group construct their networks according to the codes from Line 8 to Line 12 in Algorithm 2 using the pheromone matrix \(\mathbf{\tau}^{s}(t)\) or \(\mathbf{v}^{s}(t)\). The fitness of the pheromone matrix \(f(\mathbf{\tau}^{s}(t))\) or \(f(\mathbf{v}^{s}(t))\) is defined as the maximum BIC score of the BNs obtained by the ant group based on the corresponding pheromone matrix \(\mathbf{\tau}^{s}(t)\) or \(\mathbf{v}^{s}(t)\). After the selection operation, all ant groups perform the pheromone global updating procedure using the best BN \(G^{+}\) obtained up to the current iteration. The detailed process of our proposed coACO algorithm in solving the BN structure-learning problem can be described as follows:  
1.