Statistical and system heterogeneity are inherent characteristics of many federated learning scenarios. As discussed earlier in this chapter, a single global model, even one trained using advanced aggregation methods, might struggle to perform optimally for every client when local data distributions or device capabilities vary significantly. Viewing the federated learning problem through the lens of Multi-Task Learning (MTL) offers a powerful alternative approach specifically designed for such situations.

In the MTL paradigm, the goal is not to learn a single function but rather multiple related functions simultaneously. Applied to federated learning, we can consider learning a personalized model for each client (or group of clients) as a distinct, yet related, "task." Each client $k$ aims to minimize its own local loss function $F_k(w_k)$ based on its local data $D_k$ . Instead of forcing all clients to agree on a single global model $w$ , the MTL approach seeks to find a set of personalized models $\{w_1, w_2, ..., w_K\}$ for the $K$ clients.

The key insight is that while personalized, these tasks (client models) are not entirely independent. They often share underlying structures or features learned from the collective data. MTL formulations leverage this relatedness, typically through regularization, to allow models to borrow strength from each other. This prevents individual models from overfitting solely to their local data, especially if a client has limited data, while still allowing specialization.

Formulating FL as an MTL Problem

A common way to formulate federated learning as an MTL problem is to augment the sum of local client losses with a regularization term $\Omega$ that enforces relationships between the client models:

\min_{w_1, ..., w_K} \sum_{k=1}^K p_k F_k(w_k) + \lambda \Omega(w_1, ..., w_K)

Here, $p_k$ represents the contribution weight of client $k$ (often proportional to its dataset size), $F_k(w_k)$ is the local loss for client $k$ using its personalized model $w_k$ , and $\lambda$ is a hyperparameter controlling the strength of the regularization term $\Omega$ .

The regularization term $\Omega$ is where the task relatedness is encoded. Various forms exist, for example:

Mean Model Regularization: Penalize the distance of each client model $w_k$ from a central model (e.g., the mean model $\bar{w} = \sum_k p_k w_k$ ). This encourages models to stay close to an average representation while allowing deviations. $\Omega(w_1, ..., w_K) = \sum_{k=1}^K \|w_k - \bar{w}\|^2$
Pairwise Regularization: Penalize differences between pairs of models, potentially weighted by some measure of client similarity. $\Omega(w_1, ..., w_K) = \sum_{j,k} S_{jk} \|w_j - w_k\|^2$ where $S_{jk}$ could encode prior knowledge about client similarity (e.g., based on location, device type, or inferred data distribution).
Graph-Based Regularization: If client relationships can be represented as a graph (e.g., social network, geographical proximity), the regularization can enforce smoothness over the graph structure.
Shared Representation: Assume each model $w_k$ consists of shared parameters and task-specific parameters. Regularization might act only on the shared part or encourage similarity in the task-specific components.

Comparison between standard FL with a single global model and federated multi-task learning where client models share knowledge but retain local components.

Example Algorithm: MOCHA

One well-known algorithm that explicitly frames federated learning as multi-task learning is MOCHA (Multi-Task Federated Learning). MOCHA aims to solve the regularized objective function shown above. It tackles this potentially complex, high-dimensional optimization problem by leveraging its structure and employing techniques like stochastic dual coordinate ascent within the federated setting.

In essence, MOCHA involves iterative updates at both the client and server level. Clients perform local updates related to their data and the current regularization constraints, while the server coordinates these updates and manages information related to the task relationships (often encoded in dual variables). While the details of the optimization are intricate, the core idea is to find a balance between minimizing local losses and satisfying the global regularization constraints that link the models together.

Advantages of the MTL Approach in FL

Direct Personalization: By design, MTL approaches learn personalized models $w_k$ for each client, directly addressing statistical heterogeneity.
Principled Knowledge Sharing: The regularization term provides a structured way for clients to share knowledge and benefit from the collective data, preventing overfitting to purely local patterns.
Flexibility: Different regularization terms allow encoding various assumptions about how client tasks are related.
Improved Local Performance: For heterogeneous datasets, MTL formulations often result in better average performance across client-specific models compared to a single global model.

Relationship to Other Personalization Techniques

The MTL approach is related to other personalization methods discussed in this chapter:

Clustered FL: You can view clustered FL as a specific instance of MTL where clients within a cluster are assumed to share the same task (model), and the regularization enforces this grouping. MTL offers potentially finer-grained personalization down to the individual client level.
Meta-Learning (e.g., Per-FedAvg): Meta-learning aims to find a good model initialization that can be rapidly adapted to each client's data with a few local gradient steps. MTL directly optimizes the final personalized models $\{w_k\}$ , although the objectives can be mathematically related under certain conditions. The focus differs: meta-learning emphasizes fast adaptation, while MTL emphasizes the structure of the related tasks.

Implementation Considerations

While powerful, the MTL framework introduces its own considerations:

Optimization Complexity: Solving the joint optimization problem over all $w_k$ can be more computationally demanding than standard FedAvg. Algorithms like MOCHA are designed to handle this in a distributed way, but they may involve more sophisticated communication protocols.
Communication Overhead: Depending on the specific algorithm and regularization used, communication might involve more than just model weights or gradients (e.g., dual variables).
Hyperparameter Tuning: Selecting the appropriate form of regularization $\Omega$ and tuning the balance parameter $\lambda$ is important for good performance and requires careful validation.

In summary, framing federated learning as a multi-task learning problem provides a theoretically grounded and effective way to handle client heterogeneity and achieve personalization. By explicitly modeling the relationship between client tasks, MTL algorithms can learn specialized models that leverage shared knowledge, often leading to improved performance in diverse federated networks.