Convergence Analysis of Asynchronous Federated Learning with Gradient Compression for Non-Convex Optimization

In practical federated learning (FL), the large communication overhead between clients and the server is often a significant bottleneck. Gradient compression methods can effectively reduce this overhead, while error feedback (EF) restores model accuracy. Moreover, due to device heterogeneity, synchronous FL often suffers from stragglers and inefficiency-issues that asynchronous FL effectively alleviates. However, in asynchronous FL settings-which inherently face three major challenges: asynchronous delay, data heterogeneity, and flexible client participation-the complex interactions among these system/statistical constraints and compression/EF mechanisms remain poorly understood theoretically. In this paper, we fill this gap through a comprehensive convergence study that adequately decouples and unravels these complex interactions across various FL frameworks. We first consider a basic asynchronous FL framework AsynFL, and establish an improved convergence analysis that relies on fewer assumptions and yields a superior convergence rate than prior studies. We then extend our study to a compressed version, AsynFLC, and derive sufficient conditions for its convergence, indicating the nonlinear interaction between asynchronous delay and compression rate. Our analysis further demonstrates how asynchronous delay and data heterogeneity jointly exacerbate compression-induced errors, thereby hindering convergence. Furthermore, we study the convergence of AsynFLC-EF, the framework that further integrates EF. We prove that EF can effectively reduce the variance of gradient estimation under the aforementioned challenges, enabling AsynFLC-EF to match the convergence rate of AsynFL. We also show that the impact of asynchronous delay and flexible participation on EF is limited to slowing down the higher-order convergence term. Experimental results substantiate our analytical findings very well.