A Tight Theory of Error Feedback Algorithms in Distributed Optimization

2026-05-29

AI Summary

This work addresses the degraded convergence in distributed first-order optimization caused by gradient compression and the lack of tight theoretical comparisons among existing error feedback algorithms (EF/EF21). To resolve these issues, the authors develop a unified convergence analysis framework. By constructing optimal Lyapunov functions tailored to EF and EF21 and identifying their respective optimal step sizes, they establish the first tight convergence guarantees that are independent of the number of nodes and match the optimal convergence rate of single-agent methods. This analysis not only clarifies the distinct convergence behaviors of EF and EF21 under arbitrary network scales but also characterizes their optimal rates without relying on specific communication graph structures, thereby significantly enhancing both the clarity and practical utility of the theoretical understanding.

Abstract

Communication costs are a major bottleneck in distributed learning and first-order optimization. A common approach to alleviate this issue is to compress the gradient information exchanged between agents. However, such compression typically degrades the convergence guarantees of gradient-based methods. Error feedback mechanisms provide a simple and computationally cheap remedy for this issue, but numerous variants have been proposed, and their relative performance remains poorly understood. This paper provides tight convergence analyses for two of the main error-feedback algorithms from the literature, the classic Error Feedback method (EF) and Error Feedback 21 (EF21), by identifying optimal step-size choices and constructing optimal Lyapunov functions tailored to each method. The results hold independently of the number of agents and recover the known best guarantees possible in the single-agent regime.

Problem

• distributed optimization
• gradient compression
• error feedback
• convergence analysis
• communication cost

Innovation

• error feedback
• distributed optimization
• gradient compression
• Lyapunov analysis
• tight convergence