Data-Driven Laplacian-Constrained Personalized Federated Learning

Personalized Federated Learning (PFL) aims to train customized models for clients with highly heterogeneous data distributions while preserving data privacy. Existing approaches often rely on heuristics like clustering or model interpolation, which lack principled mechanisms for balancing heterogeneous client objectives. Serving $M$ clients with distinct data distributions is inherently a multi-objective optimization problem, where achieving optimal personalization ideally requires $M$ distinct models on the Pareto front. However, maintaining $M$ separate models poses significant scalability challenges in federated settings with hundreds or thousands of clients. To address this challenge, we reformulate PFL as a few-for-many optimization problem that maintains only $K$ shared server models ($K \ll M$) to collectively serve all $M$ clients. We prove that this framework achieves near-optimal personalization: the approximation error diminishes as $K$ increases and each client's model converges to each client's optimum as data grows. Building on this reformulation, we propose FedFew, a practical algorithm that jointly optimizes the $K$ server models through efficient gradient-based updates. Unlike clustering-based approaches that require manual client partitioning or interpolation-based methods that demand careful hyperparameter tuning, FedFew automatically discovers the optimal model diversity through its optimization process. Experiments across vision, NLP, and real-world medical imaging datasets demonstrate that FedFew, with just 3 models, consistently outperforms other state-of-the-art approaches. Code is available at https://github.com/pgg3/FedFew.

Federated learning (FL) is a communication-efficient framework for solving distributed optimization problems. Standard FL methods aggregate local updates into a single global model that is shared by all agents, effectively imposing consensus. Under heterogeneous local constraints, however, a common feasible model may be overly restrictive or may not exist. Existing constrained FL methods largely retain this shared-model structure and therefore do not directly address personalization under heterogeneous agent-specific feasible sets. We study a constrained personalized FL problem that assigns a distinct feasible model to each agent while coupling the models through a collaborative objective. We propose Locally Penalized Cross-Estimate Federated Averaging (PCE-FedAvg), where each agent maintains a multi-block vector containing its own model and estimates of the other agents' models while applying the feasibility penalty only locally to its own block. The server aggregates corresponding blocks separately, thereby preserving personalization without requiring agents to explicitly disclose their local constraint sets. For any $ε>0$, we establish finite-time upper and lower suboptimality bounds and agent-wise squared infeasibility bounds, yielding communication complexities of $\mathcal{O}(ε^{-2})$ and $\mathcal{O}(ε^{-1})$, respectively. Experiments on MNIST and CIFAR-10 support our theoretical results.

We propose FedLoRD, a personalized federated learning (PFL) method based on the low-rank structure of the differences between the global and personalized model parameters. PFL is a branch of federated learning that aims to learn models tailored to the local data of each client under biased local data distributions. In its local training, each client learns a personalized model by regularizing the difference between its model parameters and the global ones sent from the server. The regularization function needs to be designed with priors on the model parameter difference; however, the existing PFL methods rely on empirical choices (e.g., $\ell_1$- and $\ell_2$-norms), which do not take into account the actual characteristics of the difference. The resulting mismatch of priors can lead to suboptimal personalized models. To address this issue, we propose a PFL method whose regularizer is built on an observed characteristic of the model parameter differences: they are approximately low-rank at every layer. In our method, we first formulate a local training problem with nuclear norm regularization on the layer-wise differences. We further rewrite this problem via the factorized representation of the nuclear norm so that it can be efficiently solved by a plain stochastic gradient descent method. Image classification experiments on three benchmark datasets demonstrate that FedLoRD outperforms several existing PFL methods.