Coalitional FL: Coalition Formation and Selection in Federated Learning with Heterogeneous Data

The model accuracy achieved by federated learning (FL) depends significantly on devices' data distributions. To improve the model accuracy of FL with heterogeneous data distributions on devices, existing works propose some device sampling methods for the central server, but face the problem that the selected devices may still have unbalanced data. In this paper, we propose a novel coalitional FL framework for FL with heterogeneous data. Specifically, devices can cooperate and form device coalitions to reduce the data unbalancedness, and we formulate devices' interactions as a coalition formation game. Then the server selects an optimal subset of device coalitions to improve the model accuracy. Analyzing the coalition formation and selection framework is challenging since the relationship between model accuracy and data heterogeneity is not clear, and devices' coalition formation decisions and the server's coalition selection strategy are coupled in a highly non-trivial manner. We first derive a novel theoretical characterization of the relationship between model accuracy loss and data heterogeneity which follows an inverse function. With the novel theoretical relationship, we analyze devices' coalition formation game. We characterize the conditions under which the Nash stable partition exists, and propose an accelerated algorithm for devices to reach the Nash stable partition. For the server's device coalition selection problem, we show that the model accuracy loss depends on both data heterogeneity and the number of data samples of device coalitions in a non-monotonous way, and we propose a low-complexity algorithm for the server to select device coalitions efficiently. We conduct extensive simulations and show that our proposed coalition formation and selection framework reduces the data heterogeneity of selected device coalitions by up to
$58.6\%$
and increases the model accuracy by up to
$6.8\%$
compared with four existing benchmarks.