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——的优化问题广泛存在于机器学习、推荐系统和福利最大化等应用中。然而，当可分解子模函数包含数百万个分量函数时，其优化在计算上是不可行的。此外，分量函数可能涉及隐私（例如可能代表用户偏好函数）而无法广泛共享。为解决这些问题，我们为可分解子模优化提出了一种联邦优化框架。在此框架中，各客户端拥有自身的偏好函数，需要最大化这些偏好的加权和。我们在此框架中实现了经典的连续贪心算法：客户端并行执行局部小步长优化以逼近局部解，随后将局部更新聚合至中央服务器。为应对海量客户端，聚合操作仅对采样子集执行。此外，聚合仅在连续多轮并行局部优化之间间歇进行，从而显著降低通信开销。我们证明，即使在采用上述成本控制策略的情况下，所提出的联邦算法仍能保证获得高质量的近似解。最后，我们展示了如何将联邦框架应用于解决最大覆盖和设施选址等经典离散子模优化问题。