FedBHG: Bayesian Coalition Stability in Federated Learning

At a federated learning checkpoint, clients must choose between local and shared continuation before either alternative is trained. We study this decision through posterior gradient-estimation risk under a Gaussian hierarchy. With known scalar variances, clients whose sampling variance is below the between-client variance remain local; the rest pool using inverse-total-variance weights. Under the specified summary disclosure, we prove that this partition almost surely admits no coalition that weakly reduces every member's risk and strictly reduces at least one. For at least three pooling-eligible clients, these are the unique fixed positive linear weights that all members accept over staying local for every disclosed summary configuration. When every covariance direction favors the same local-or-shared choice, the construction extends to general covariances; when directions favor conflicting choices, every partition can be unstable. Under shared quadratic curvature, the posterior-risk comparison also determines the expected-loss comparison between the corresponding Newton updates. FedBHG estimates the variances at a checkpoint and continues with ordinary training. Synthetic experiments isolate failures from route errors, directional conflict and aggregation weights. Across 20 runs in four selected task/clusterer settings with fixed client partitions, FedBHG beats both universal continuations in accuracy in 19 and loss in 17. Four fresh-partition observations improve accuracy in all four and loss in three. These experiments support checkpoint-based selective continuation in the tested settings, with the Bayesian routing rule's incremental benefit over simpler routes varying by task. The theory concerns oracle posterior-risk stability; the neural experiments evaluate predictive transfer under fitted routing and ordinary training.