## 2 Background
### Time-Series Federated Learning  
Federated learning (FL) [1, 2, 3, 6] is a collaborative learning paradigm that allows multiple clients (data owners) to train machine learning models without sharing the raw data. Instead of centralising the training dataset \(D_{train}\), the data remains distributed among the clients, which train local models and exchange intermediate parameters with a central server. The global model \(f_{W}\) is maintained on the central server, which coordinates the training process. At each iteration, the central server transmits the model parameters \(W\) to a selected group of \(k\) clients, which locally compute the gradients for their respective data samples. The server and the clients then perform a federated aggregation protocol [20] to compute the average gradients, which are used to update the global model using gradient descent. A widely used optimisation algorithm in FL is federated averaging (FedAvg) [1, 21, 22], which extends the basic framework of FL by allowing clients to perform multiple local updates before communicating with the central server. The typical case where each client trains its model uniformly is that it trains the model for \(T\) local steps over \(n\) batches and updates its model parameters \(t\) times before sending the update weights \(W_{t}\) back to the central server. Since the reduction in communication overhead is crucial in FL, this approach can reduce communication rounds by a factor of 10 to 100 compared to traditional centralised training. This approach can lead to a reduction in the frequency of data transmission. Some studies report a 12.2-fold reduction in data transmission with comparable model performance [23]. In practical FL scenarios, FedAvg is preferred because clients can process multiple batches within a global round, which reconciles a trade-off between communication efficiency and convergence rate [24]. We define the model update difference as \(\Delta W=W_{t}-W\), which represents the change of model parameters during one round. The FL process consists of multiple global rounds. At the beginning of a global round, the server sends the latest global model to the clients. The clients then train the latest model with their own data and send their training results, i.e., gradients or model updates, back to the server. A global federated round with model updates is shown in Figure 1.  
The local training process usually follows the stochastic gradient descent algorithm, which iterates through the data in batches. A batch may contain the entire local dataset, but in the more general case, a batch is a subset of the local dataset and is also referred to as a mini-batch [25]. During an epoch, clients iterate over their entire training data once. A client can use its entire dataset in a single batch during one round, which then counts as one epoch. We focus on the more general mini-batch case, where the full dataset is split into several mini-batches and the client iterates over these mini-batches in multiple global rounds during an epoch. **Time-series federated learning:** This approach combines the power of federated learning with specialised techniques for time-series analysis. Recent advances in the field of federated learning of time-series have focused on innovative data transformation techniques and modelling architectures to improve prediction accuracy while preserving data privacy. One way is to transform data by using language models for time-series analysis. [PERSON] et al. [26] propose Time-FFM, which transforms time-series into text tokens, enabling the use of pre-trained language models for forecasting tasks. This approach enables the creation of a federated foundation model for time-series forecasting while addressing the problems of data scarcity and privacy. [PERSON] and [PERSON] [27] proposed FedTime, a federated large-scale language model tailored for long-range time series prediction that incorporates channel independence and patching to preserve the semantic information locally. [PERSON] et al.