## 4 SoftGAN
By smoothing the curve, we can bypass local optimal solutions and approach the vicinity of the global optimal solution. As the value of \(\beta\) decreases, the degree of curve smoothing decreases, eventually approximating the actual objective function curve. At this point, the local optimal solution obtained through gradient descent is equivalent to the global optimal solution. To achieve the desired dynamic control mentioned above, it is important to consider the weight factor, represented as \(\beta\), in relation to the disparity between the distributions of generated data and real data. To measure this correlation, we introduce the Earth Mover's Distance [28]. During the training process, after several iterations, an equal number of samples are randomly selected from both the generated data and the real data. The Earth Mover's Distance between the two distributions is then calculated. The updated value of \(\beta\) is determined by scaling the Earth Mover's Distance between these two distributions. In the experimental section, we scale the Earth Mover's Distance to fit within the range of \([0,1]\). Then, we clamp it by the upper bound \(\frac{1}{2\ln 2}\) to update \(\beta\). The proof of the upper bound of \(\beta\) is in Proposition 4. So far, we have provided a detailed explanation of the proposed method, and we summarize it in Algorithm 1. Regarding the optimization algorithm choice, we uniformly utilize the [PERSON] algorithm [29]. Additionally, we employ a clever strategy to expedite the training process for updating \(G\). When it comes to the minimization problem  
\[\min_{G}\mathop{\mathbb{E}}_{x\sim P_{g}}[\mu_{\
eg D}(x)+\beta H_{\
eg D}(x)] \tag{6}\]  
it shares the same solution as  
\[\min_{G}\mathop{\mathbb{E}}_{x\sim P_{g}}[\mu_{\
eg D}(x)] \tag{7}\]  
However, Equation (6) is more susceptible to being trapped in a local minimum compared to the latter. Therefore, in Algorithm 1 (line 9), we directly calculate Equation (7) to achieve the same objective. ```
Input: the batch size \(m\), the dimension of latent space \(n_{z}\), the number of iterations \(t\), the number of iterations of the discriminator per generator iteration \(n_{disc}\), the number of iterations of updating entropy weight \(n_{\beta}\), Adam hyperparameter \(\alpha\), \(\beta_{1}\), \(\beta_{2}\) Output: Discriminator parameters \(w\) and Generator parameters \(\theta\)
1 Initialize discriminator parameters \(w_{0}\) and generator parameters \(\theta_{0}\) randomly; while\(t<iter\)do
2 foreach\(t_{disc}\) in \([1,...n_{disc}]\)do
3 Sample a batch of \(\{x^{(i)}\}_{i=1}^{m}\sim\mathbb{P}_{d}\) from real data; Sample a batch of \(\{z^{(i)}\}_{i=1}^{m}\sim\mathbb{P}_{z}\) from \(n_{z}\)-dimensional latent space; Compute \(V(D,G)\leftarrow\frac{1}{m}\sum_{i=1}^{m}[\mu_{D}(x^{(i)})+\beta\mathrm{H}_{ D}(x^{(i)})+(\mu_{\
eg D}(G(z^{(i)})))+\beta\mathrm{H}_{\
eg D}(G(z^{(i)}))]\); Update \(w\leftarrow\mathrm{Adam}(-\
abla_{w}L,w,\alpha,\beta_{1},\beta_{2})\);
4 end while
5 Sample a batch of latent variables \(\{z^{(i)}\}_{i=1}^{m}\sim\mathbb{P}_{z}\) from \(n_{z}\)-dimensional latent space; Update \(\theta\leftarrow\mathrm{Adam}(\
abla_{\theta}\frac{1}{m}\sum_{i=1}^{m}\mu_{ \
eg D}(G_{\theta}(z)),\theta,\alpha,\beta_{1},\beta_{2})\); if\(t\mod n_{\beta}==0\)then
6 Sample a batch of \(\{x^{(i)}\}_{i=1}^{m}\sim\mathbb{P}_{d}\) from real data; Sample a batch of \(\{z^{(i)}\}_{i=1}^{m}\sim\mathbb{P}_{z}\) from \(n_{z}\)-dimensional latent space; Compute the discrete Earth Mover's Distance between \(\{x^{(i)}\}_{i=1}^{m}\) and \(\{G(z^{(i)})\}_{i=1}^{m}\); Update \(\beta\) with the scaled Earth Mover's Distance;
7 end while
8 Update t = t + 1;
9 end while
```  
**Algorithm 1**Framework of SoftGANFundamentally, the original GAN also can be interpreted from the perspective of fuzzy concept learning, much like the SoftGAN. The goal of the discriminator in the original GAN is also to learn a fuzzy concept, \(D\), that models the true distribution, as depicted in Equation (1).