  In this work, we quantitatively calibrate the performance of global and local
models in federated learning through a multi-criterion optimization-based
framework, which we cast as a constrained program. The objective of a device is
its local objective, which it seeks to minimize while satisfying nonlinear
constraints that quantify the proximity between the local and the global model.
By considering the Lagrangian relaxation of this problem, we develop a novel
primal-dual method called Federated Learning Beyond Consensus (\texttt{FedBC}).
Theoretically, we establish that \texttt{FedBC} converges to a first-order
stationary point at rates that matches the state of the art, up to an
additional error term that depends on a tolerance parameter introduced to
scalarize the multi-criterion formulation. Finally, we demonstrate that
\texttt{FedBC} balances the global and local model test accuracy metrics across
a suite of datasets (Synthetic, MNIST, CIFAR-10, Shakespeare), achieving
competitive performance with state-of-the-art.
