A Locally Penalized Cross-Estimate Federated Method with Guarantees for Constrained Personalized 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.

Personalization addresses one of the defining challenges in federated learning: statistical heterogeneity among clients. We show that the dissimilarity between the ideal personalized models of two clients can be upper bounded by the dissimilarity of their distributions. Motivated by this, we propose Karula, a data-driven regularization method that constrains pairwise model dissimilarities between clients according to their distributional differences. Distributional similarity is measured using a surrogate for the 1-Wasserstein distance adapted for the federated setting, allowing the method to adapt to highly complex interrelations between clients that approaches based on hard clustering may fail to recover. We propose an approximate projected stochastic gradient algorithm to solve the resulting constrained optimization problem, and prove that it converges at a rate of $O(1/K)$ to a neighborhood of a stationary point.

Personalized federated learning (pFL) aims to adapt models to client specific data distributions, yet it often fails to reliably preserve personalized information. Local training is hindered by high variance gradients induced by limited and heterogeneous client data, while aggregation further distorts client specific optimization directions. To address these challenges, we propose pFLAlign, a gradient alignment framework to maintain client specific information during both local training and aggregation. pFLAlign consists of two complementary mechanisms: one adapts local gradient directions to reduce variance during client side optimization, and the other mitigates aggregation induced distortion by realigning the global model with each client's personalized direction. Theoretically, we derive pFLAlign from a PAC Bayesian analysis, which reveals how personalized gradient alignment preserves client specific information. Our experiments and ablation studies show that pFLAlign consistently improves personalization performance and training stability, achieving state of the art results.