# Graph-Augmentation-Free Self-Supervised Learning for Social Recommendation
## 3 Proposed Model
### Graph-Augmentation-Free Self-Supervised Learning  
Data augmentation is the premise of self-supervised contrastive learning models, and they can obtain a more uniform representation by perturbing the structure and optimizing the contrastive loss. With regard to the learning of graph representations, a study on SimGCL [10] showed that it is the uniformity of distribution rather than dropout-based graph augmentation that has the greatest impact on the performance of self-supervised graph learning. Therefore, we considered adding random noise to the embedding to create a self-supervised signal that could enhance the performance of the contrastive learning.  
\[\widetilde{\mathbf{Z}}^{(l)}=\mathbf{D}^{-1}\mathbf{A}\widetilde{\mathbf{Z}}^{(l-1)}+\varepsilon ^{(l)}\]  
where \(\varepsilon\) is the added random uniform noise vector, and the noise direction is in the same direction as the embedding vector, \(\varepsilon=x\odot sign(z)\), \(x\in\mathbb{R}^{d}\sim U(0,1)\). The embedding representation added with perturbation retains most of the information of the original representation, as well as some invariance.  
By applying different scales of embedding vectors to the current node embedding, the perturbed embedding vectors can then be fed into the encoder. The embedding vector of the final perturbed node is expressed as:  
\[\widetilde{\mathbf{Z}}=\frac{1}{L+1}\sum\limits_{l=0}^{L}\widetilde{\mathbf{Z}}^{(l)}\]  
We regarded augmented views from the same node as positive examples and augmented views from different nodes as negative examples. Positive auxiliary supervision promotes the consistency of predictions among different views of the same node, while negative supervision strengthens the divergence between different nodes. Formally, we adopted the contrastive loss InfoNCE [24] to maximize the consistency of positive examples and minimize the consistency of negative examples:  
\[\mathcal{L}_{cl}^{U}=-\sum\limits_{u\in\mathcal{U}}\log\frac{\exp(sim(\tilde{ z}_{u}^{(k)}\cdot\tilde{z}_{u})/\tau)}{\sum_{c\in c}\exp(sim(\tilde{z}_{u}^{(k)} \cdot\tilde{z}_{u})/\tau)}\] \[\mathcal{L}_{cl}^{I}=-\sum\limits_{i\in\mathcal{I}}\log\frac{\exp (sim(\tilde{z}_{u}^{(k)}\cdot\tilde{z}_{i})/\tau)}{\sum_{j\in\mathcal{I}}\exp (sim(\tilde{z}_{i}^{(k)}\cdot\tilde{z}_{j})/\tau)}\]  
where \(\tilde{z}_{u}^{(k)}=\tilde{z}_{u}^{(k)}/\|\tilde{z}_{u}^{(k)}\|_{2}\) represents the \(k\)-th layer L2-regularized embedding vector compared with the final layer embedding, \(sim(a,b)\) represents the dot-product cosine similarity between normalized embeddings, and \(b\). \(\tau\) is the temperature parameter. The total loss function for self-supervised learning includes the contrastive loss for each view user and the item contrastive loss, as shown in the following equation:  
\[\mathcal{L}_{\text{ssl}}=\sum_{q\in\{r,\ell,i,j\}}\mathcal{L}_{cl}^{q,II}+ \mathcal{L}_{cl}^{r,I}\]  
To improve recommendation through contrastive learning, we utilized a multi-task training strategy to jointly optimize the recommendation task and the self-supervised learning task, as shown in the following equation:  
\[\mathcal{L}=\mathcal{L}_{BPR}+\lambda\mathcal{L}_{ssl}\]  
where \(\lambda\) is the hyperparameter used to control the auxiliary task.