Federated learning is a setting where agents, each with access to their own data source, combine models from local data to create a global model. If agents are drawing their data from different distributions, though, federated learning might produce a biased global model that is not optimal for each agent. This means that agents face a fundamental question: should they choose the global model or their local model? We show how this situation can be naturally analyzed through the framework of coalitional game theory. We propose the following game: there are heterogeneous players with different model parameters governing their data distribution and different amounts of data they have noisily drawn from their own distribution. Each players goal is to obtain a model with minimal expected mean squared error (MSE) on their own distribution. They have a choice of fitting a model based solely on their own data, or combining their learned parameters with those of some subset of the other players. Combining models reduces the variance component of their error through access to more data, but increases the bias because of the heterogeneity of distributions. Here, we derive exact expected MSE values for problems in linear regression and mean estimation. We then analyze the resulting game in the framework of hedonic game theory; we study how players might divide into coalitions, where each set of players within a coalition jointly construct model(s). We analyze three methods of federation, modeling differing degrees of customization. In uniform federation, the agents collectively produce a single model. In coarse-grained federation, each agent can weight the global model together with their local model. In fine-grained federation, each agent can flexibly combine models from all other agents in the federation. For each method, we analyze the stable partitions of players into coalitions.
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