Federated K-means clustering via dual decomposition-based distributed optimization

The use of distributed optimization in machine learning can be motivated either by the resulting preservation of privacy or the increase in computational efficiency. On the one hand, training data might be stored across multiple devices. Training a global model within a network where each node only has access to its confidential data requires the use of distributed algorithms. Even if the data is not confidential, sharing it might be prohibitive due to bandwidth limitations. On the other hand, the ever-increasing amount of available data leads to large-scale machine learning problems. By splitting the training process across multiple nodes its efficiency can be significantly increased. This paper aims to demonstrate how dual decomposition can be applied for distributed training of K -means clustering problems. After an overview of distributed and federated machine learning, the mixed-integer quadratically constrained programming-based formulation of the K -means clustering training problem is presented. The training can be performed in a distributed manner by splitting the data across different nodes and linking these nodes through consensus constraints. Finally, the performance of the subgradient method, the bundle trust method, and the quasi-Newton dual ascent algorithm are evaluated on a set of benchmark problems. The main benefit stemming from the formulation of the clustering problem as a mixed-integer program and from the use of dual decomposition within a federated learning framework, apart from the preservation of privacy, is the computation of a lower bound of the objective of the overall clustering problem. In this way, the worst-case distance of any found solution to the global optimum can be easily assessed. While the mixed-integer programming-based formulation of the clustering problems suffers from weak integer relaxations, the presented approach can potentially be used to enable an efficient solution in the future, both in a central and distributed setting.