## 2 Preliminary Knowledge
### Multi-Adversarial Adaptation  
In the realm of cross-condition mechanical fault diagnosis, the domain adaptation techniques that utilizes adversarial networks to perform domain confusion by extracting domain-invariant features has achieved commendable results. Domain adaptation is defined formally as the following: given a source domain \(D_{s}=\left\{\left(x_{i}^{s},y_{i}^{s}\right)\right\}_{i=1}^{n_{s}}\) of \(n_{s}\) labeled samples and a target domain \(D_{t}=\left\{x_{i}^{s}\right\}_{j=1}^{n_{t}}\) of \(n_{t}\) unlabeled samples. The data from both domains are sampled from joint distributions \(P(X^{s},Y^{s})\) and \(Q(X^{t},Y^{t})\), respectively, where \(P\
eq Q\). Utilize the data from the source domain to train a predictive function \(f\) for the target domain, represented as \(f:\,x_{t}\to y_{t}\), so that \(f\) has the lowest prediction error in the target domain.  
The adversarial learning process can be conceptualized as a two-player game. In this game, the two players are the domain discriminator, \(G_{d}\), and the feature extractor, \(G_{f}\). While \(G_{d}\) strives to identify the domain of each sample, \(G_{f}\) aims to extract shared features across both domains, constantly challenging the capabilities of the domain discriminator. For the adversarial learning objective function, the aim is to minimize the classification loss associated with \(G_{f}\) and simultaneously maximize the loss for \(G_{f}\). This process ensures that the parameters of the feature extractor are adjusted specifically to derive domain-invariant features.  
However, a single domain classifier can only align the distributions of the two domains on a global scale. After global feature alignment, it is indeed possible to achieve decent classification results for two domains with similar overall distributions. But merely a global alignment of domains lacks fine-grained information and makes it difficult to distinguish between the structural properties of each category, which potentially leads to a misclassification of samples near the category decision boundary [40].  
Multi-adversarial domain adaption has \(K\) domain discriminators. Each domain discriminator can be called a local domain discriminator \(G_{d}^{k}\) (\(k=1,\,\ldots,\,K\)). \(K\) corresponds to the class of the sample, and the \(K\)th domain discriminator is responsible for distinguishing the domains of the samples associated with the \(K\)th class, as shown in Figure 3. Although the target domain data are unlabeled, the label classifier outputs a predicted label \(\hat{y}_{i}=G_{y}(x_{i})\). \(\hat{y}_{i}\) is actually the probability of sample \(x_{i}\) over all classes. This probability guides the significance of each sample \(x_{i}\) to the specific local domain discriminator \(G_{d}^{k}\). The weight of the feature \(G_{f}(x_{i})\) in relation to the local domain discriminator \(G_{d}^{k}\) is determined by the probability \(\hat{y}_{i}\), reflecting the importance of each sample. The mathematical expression to apply it to all local discriminators is as follows:  
\[L_{d}=\frac{1}{n}\sum\limits_{k=1}^{K}\sum\limits_{\mathbf{x}_{i}\in\mathcal{D}_ {i}\cup\mathcal{D}_{t}}L_{d}^{k}\Big{(}G_{d}^{k}\Big{(}\hat{y}_{i}^{k}G_{f}( \mathbf{x}_{i})\Big{)},d_{i}\Big{)} \tag{2}\]  
where \(G_{d}^{k}\) is the kth local domain discriminator, \(L_{d}^{k}\) is its cross-entropy loss, and \({}_{i}\) is the domain label of \(\mathbf{x}_{i}\), \(n=n_{s}+n_{t}\).