Not yet reviewed by Pith; the record is open.

Federated multi-task learning (FMTL) aims to simultaneously learn multiple related tasks across clients without sharing sensitive raw data. However, in the decentralized setting, existing FMTL frameworks are limited in their ability to capture complex task relationships and handle feature and sample heterogeneity across clients. To address these challenges, we introduce a novel sheaf-theoretic-based approach for FMTL. By representing client relationships using cellular sheaves, our framework can flexibly model interactions between heterogeneous client models. We formulate the sheaf-based FMTL optimization problem using sheaf Laplacian regularization and propose the Sheaf-FMTL algorithm to solve it. We show that the proposed framework provides a unified view encompassing many existing federated learning (FL) and FMTL approaches. Furthermore, we prove that our proposed algorithm, Sheaf-FMTL, achieves a sublinear convergence rate in line with state-of-the-art decentralized FMTL algorithms. Extensive experiments show that although Sheaf-FMTL introduces computational and storage overhead due to the management of interaction maps, it achieves substantial communication savings in terms of transmitted bits when compared to decentralized FMTL baselines. This trade-off makes Sheaf-FMTL especially suitable for cross-silo FL scenarios, where managing model heterogeneity and ensuring communication efficiency are essential, and where clients have adequate computational resources.