## 3 Related Works  
As described in Section 2, the training mechanism of GANs based on game theory is elegant, but it is known to be unstable during training and may not converge at all. Therefore, one must seize the best opportunity of alternate iterations of \(G\) and \(D\), carefully grasping the balance between the two. In addition, another common issue with GANs is that they often produce samples with limited diversity, even when trained on a diverse dataset. Specifically, when the generator \(G\) finds out that one or several generated outputs \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(x\) & the samples in the data set \\ \(\pi\) & the underlying conceptual space the samples belong to \\ \(P\) & prototypes, a set ot prototypical cases of the fuzzy concept \(L\) \\ L & fuzzy concept, which has the form “about \(P\)”, “similar to \(P\)” or “close to \(P\)” \\ \(\
eg L\) & the negation of the fuzzy concept \(L\) \\ \(\mu_{L}(x)\) & the degree of sample \(x\) belonging to the fuzzy concept \(L\) \\ \(H_{L}(x)\) & fuzzy entropy, the degree of \(x\) being a borderline case between \(L\) and \(\
eg L\) \\ \(z\) & random noise fed into the generator of GANs \\ \(G\) & the generator of GANs \\ \(D\) & the discriminator of GANs; in our SoftGAN, it is treated as a fuzzy concept \\ \(P_{d}\) & the target distribution associated with the training data \\ \(P_{z}\) & the prior distribution of \(z\) (usually standard Gaussian) \\ \(P_{g}\) & the generated distribution of the GANs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations quick list. can easily deceive the discriminator \(D\), it might limit itself to only generating those samples without exploring other possibilities. This phenomenon is known as mode collapse. While training instability is a critical factor in mode collapse, there is currently no known method in the literature that addresses both issues simultaneously to improve GAN training. There are a series of WGAN-based works [3; 14; 16; 18] that claim to solve the problem of training instability, which can help reduce mode dropping. However, there has been no detailed discussion or explanation of why mode collapse is reduced in these methods. Additionally, these methods face difficulties in effectively meeting Lipschitz continuity and require complex network architectures (such as the ResNet block [19], Style block [20; 21; 22] and Skip block [23]) to achieve optimal results. Furthermore, as noted in [14], these approaches have a slower convergence speed compared to the original version [1; 2]. In addition to the above-mentioned training instability and mode collapse problems, the theoretical studies on GANs are generally divided into three directions: the improvement of the objective function, the introduction of training techniques and the proposal of an evaluation indicator. The research in this paper belongs to the first direction. In this direction, the most successful research is a series of WGAN-based studies, as far as we know [3; 14; 16; 18]. Among them, the most relevant to ours is WGAN-C [15]. The authors, [PERSON] et al. [15], also use the analogy of teaching and learning to describe the whole training process. The SoftGAN adjusts the tolerability of the discriminator to the generator through the dynamic borderline softening mechanism from the perspective of fuzzy concept learning, while WGAN-C draw lessons from the progressive idea based on WGAN-GP [14] to control the discriminant ability with a set of convex combinations of predefined multiple discriminators. Although both use the analogy to learning, they are different in their implementation and motivation. In addition to the above methods, there are some approaches that reduce the risk of falling into local modes by building multiple discriminators or multiple generators, such as Dropout-GAN [24], D2 GAN [25], GMAN [26] and DuelGAN [27].