Decomposable Submodular Maximization in Federated Setting
Akbar Rafiey
TL;DR
This work tackles federated maximization of decomposable submodular functions under privacy and communication constraints. It extends the continuous greedy paradigm to a federated setting (FedCG), proving $(1-1/e)$-approximation guarantees with small additive errors under both full and partial participation, and introduces scalable client-sampling techniques to handle massive client counts. It further develops a discrete federated approach (FedDiscrete Greedy) with provable approximation guarantees for general matroids and uniform matroids, and demonstrates applicability to fundamental problems like Facility Location and Maximum Coverage. The results provide a framework for privacy-preserving, scalable optimization of submodular objectives across distributed data sources, with concrete guidance on communication efficiency and gradient estimation.
Abstract
Submodular functions, as well as the sub-class of decomposable submodular functions, and their optimization appear in a wide range of applications in machine learning, recommendation systems, and welfare maximization. However, optimization of decomposable submodular functions with millions of component functions is computationally prohibitive. Furthermore, the component functions may be private (they might represent user preference function, for example) and cannot be widely shared. To address these issues, we propose a {\em federated optimization} setting for decomposable submodular optimization. In this setting, clients have their own preference functions, and a weighted sum of these preferences needs to be maximized. We implement the popular {\em continuous greedy} algorithm in this setting where clients take parallel small local steps towards the local solution and then the local changes are aggregated at a central server. To address the large number of clients, the aggregation is performed only on a subsampled set. Further, the aggregation is performed only intermittently between stretches of parallel local steps, which reduces communication cost significantly. We show that our federated algorithm is guaranteed to provide a good approximate solution, even in the presence of above cost-cutting measures. Finally, we show how the federated setting can be incorporated in solving fundamental discrete submodular optimization problems such as Maximum Coverage and Facility Location.