## 3 Methodology
### Efficient Gradient-Based Federated Unlearning (EG-FedUnlearn)  
The Efficient Gradient-Based Federated Unlearning (EG-FedUnlearn) method aims to promptly delete certain points of data from a given federated model, and for this reason, the method exploits gradient information to undo the influence of such data selectively without ever retraining the model. The objectives of this algorithm are to remove the target data influence efficiently without resorting to costly retraining, minimize model degradation when unlearning specific data, and minimize the number of rounds and size of data communicated between federated nodes and the central server. Each federated node would maintain local datasets and train the shared global model. These local updates of the model are computed from the gradients obtained in training on local data. During every training round, each node will store the gradient updates for every data sample or mini-batch of samples in local storage. This stored gradient update is used later to eliminate the effect of a particular data sample when an unlearn request is made. For unlearning given data other than \(d_{i}\), the node calculates the gradient concerning that data and applies this negative update to the local model:  
\[\theta^{n}_{updated}=\theta_{n}-\eta\cdot\Delta\theta^{n}_{d_{i}} \tag{2}\]where \(\Delta\theta_{d_{i}}^{n}\) represents the gradient update associated with data point \(d_{i}\). The negative update essentially reverses the influence of \(d_{i}\) on the model. The local model is updated with this gradient-based unlearning approach. Nodes then communicate the adjusted model parameters to the central server. The central server aggregates the adjusted model updates from all nodes to create a global unlearned model. The aggregation formula is like the standard federated learning process:  
\[\theta_{G}=\sum_{n-1}^{N}\frac{|D^{n}|}{|D|}\cdot\theta_{updated}^{n} \tag{3}\]  
where \(\theta_{updated}^{n}\) are the adjusted model weights received from each node. \(|D^{n}|\) represents the number of data points in the dataset at node \(n\). The server keeps track of summarized gradient information from previous rounds to further improve efficiency. The server then approximates the effects of the target data over the entire training set to minimize the need for repeated recalculations. This summarization acts like a memory of the impact mammoth from big data in the past. The central server applies varying rates of the adaptive learning height to adjust how unlearning affects the gradient. The adaptive learning rate is then made small if the unlearning incurs big losses with respect to accuracy, thus preserving the quality of the model as follows:  
\[\theta_{G}=\theta_{G}+\alpha\cdot\Delta\theta_{G} \tag{4}\]  
where \(\alpha\) is an adaptive scaling factor that adjusts the degree of unlearning. \(\Delta\theta_{G}\) represents the aggregate influence of the data point that is being unlearned across all nodes. As shown in the algorithm methodology in Figure 5, this algorithm uses gradient information to directly reverse the impacts of specific data points on the local models. By applying a negative gradient update, the influence of a data point can be effectively removed. The central server summarizes gradients across multiple rounds, which allows it to understand the impacts of data points throughout the training process. This reduces the need for full retraining or calibration, making the unlearning process more efficient. The algorithm employs an adaptive learning rate during the aggregation process to ensure that unlearning operations do not drastically degrade model quality. If the negative gradient update has a strong negative impact, the adaptive learning rate mitigates this effect to maintain the performance of the model. If we compare the proposed algorithm with FedEraser, we will find that FedEraser relies on storing historical parameter updates and applying calibration training to reconstruct the unlearned model.