Semidefinite-Programming-Based Network Code for Ring-Topological Decentralized Federated Learning

This paper examines the decentralized stochastic gradient descent algorithm for federated learning over a wireless ring network, where each device connects to its 2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> adjacent devices, termed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>-tier coverage. Given this topology, the consensus coefficients, or mixing matrix, can be optimized via semidefinite programming (SDP). By employing network coding, the learning topology can be densified without additional communication costs, introducing linear constraints to the uncoded problem. The joint design of the mixing matrix and coding parameters is also formulated as an SDP problem, allowing for efficient determination. Numerical results for linear regression and image classification (using MNIST and CIFAR-10 datasets) demonstrate that our SDP-based network coding approach significantly accelerates convergence in decentralized federated learning under a variation of the pathological non-IID data distribution.