Federated Incremental Subgradient Method for Convex Bilevel Optimization Problems

In this letter, we consider a bilevel optimization problem in which the outer-level objective function is strongly convex, whereas the inner-level problem consists of a finite sum of convex functions. Bilevel optimization problems arise in situations where the inner-level problem does not have a unique solution. This has led to the idea of introducing an outer-level objective function to select a solution with the specific desired properties. We propose an iterative method that combines an incremental algorithm with a broadcast algorithm, both based on the principles of federated learning. Under appropriate assumptions, we establish the convergence results of the proposed algorithm. To demonstrate its performance, we present two numerical examples related to binary classification and a location problem.