From Interpretation to Correction: A Decentralized Optimization Framework for Exact Convergence in Federated Learning

This work introduces a novel decentralized framework to interpret federated learning (FL) and, consequently, correct the biases introduced by arbitrary client participation and data heterogeneity, which are two typical traits in practical FL. Specifically, we first reformulate the core processes of FedAvg - client participation, local updating, and model aggregation - as stochastic matrix multiplications. This reformulation allows us to interpret FedAvg as a decentralized algorithm. Leveraging the decentralized optimization framework, we are able to provide a concise analysis to quantify the impact of arbitrary client participation and data heterogeneity on FedAvg's convergence point. This insight motivates the development of Federated Optimization with Exact Convergence via Push-pull Strategy (FOCUS), a novel algorithm inspired by the decentralized algorithm that eliminates these biases and achieves exact convergence without requiring the bounded heterogeneity assumption. Furthermore, we theoretically prove that FOCUS exhibits linear convergence (exponential decay) for both strongly convex and non-convex functions satisfying the Polyak-Lojasiewicz condition, regardless of the arbitrary nature of client participation.