Lower Bounds and Optimal Algorithms for Personalized Federated Learning

TL;DR

This paper establishes the first lower bounds and develops optimal algorithms for personalized federated learning, significantly advancing theoretical understanding and practical methods in the field.

Contribution

It introduces the first lower bounds for personalized federated learning and proposes optimal algorithms matching these bounds in various regimes.

Findings

Proved lower bounds for communication and oracle complexity.

Designed optimal accelerated algorithms for personalized federated learning.

Demonstrated practical superiority through numerical experiments.

Abstract

In this work, we consider the optimization formulation of personalized federated learning recently introduced by Hanzely and Richt\'arik (2020) which was shown to give an alternative explanation to the workings of local {\tt SGD} methods. Our first contribution is establishing the first lower bounds for this formulation, for both the communication complexity and the local oracle complexity. Our second contribution is the design of several optimal methods matching these lower bounds in almost all regimes. These are the first provably optimal methods for personalized federated learning. Our optimal methods include an accelerated variant of {\tt FedProx}, and an accelerated variance-reduced version of {\tt FedAvg}/Local {\tt SGD}. We demonstrate the practical superiority of our methods through extensive numerical experiments.