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.

翻译：子模函数及其子类——可分解子模函数，以及它们的优化问题，在机器学习、推荐系统和福利最大化等广泛应用中均有体现。然而，优化包含数百万个组件函数的可分解子模函数在计算上几乎不可行。此外，组件函数可能涉及隐私（例如，它们可能代表用户偏好函数），无法广泛共享。为解决这些问题，我们提出了一种针对可分解子模优化的**联邦优化**设置。在该设置中，客户端拥有各自的偏好函数，需要最大化这些偏好的加权和。我们在此设置中实现了流行的**连续贪心**算法，其中客户端并行执行朝向局部解的小步局部更新，然后在中央服务器上聚合局部变化。为应对大量客户端，聚合仅在子采样集合上进行。此外，聚合仅在并行局部步骤之间间歇性执行，这显著降低了通信成本。我们证明，即使采取了上述成本削减措施，我们的联邦算法仍能保证提供良好的近似解。最后，我们展示了如何将联邦设置应用于解决基本的离散子模优化问题，如最大覆盖问题和设施选址问题。