Tight analyses of first-order methods with error feedback

2025-06-05

AI Summary

In distributed learning, gradient compression alleviates communication bottlenecks but often degrades convergence. Existing error-feedback (EF/EF21) methods lack tight theoretical analyses, hindering precise characterization of their fundamental performance limits. This paper constructs the first *optimal* Lyapunov functions for both EF and EF21, enabling derivation of *matching upper and lower bounds* on their convergence rates under convex smooth settings. These bounds facilitate a rigorous, apples-to-apples comparison among three key algorithms: EF, EF21, and compressed gradient descent. The resulting convergence rates constitute the tightest known theoretical guarantees to date. Crucially, they reveal—for the first time—the intrinsic impact of the error-feedback mechanism on convergence speed and its fundamental performance ceiling. This work thus establishes a solid theoretical foundation for designing and selecting communication-efficient compression strategies in distributed optimization.

Technology Category

• Machine Learning: Learning on the Edge & Model Compression
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Communication between agents often constitutes a major computational bottleneck in distributed learning. One of the most common mitigation strategies is to compress the information exchanged, thereby reducing communication overhead. To counteract the degradation in convergence associated with compressed communication, error feedback schemes -- most notably $mathrm{EF}$ and $mathrm{EF}^{21}$ -- were introduced. In this work, we provide a tight analysis of both of these methods. Specifically, we find the Lyapunov function that yields the best possible convergence rate for each method -- with matching lower bounds. This principled approach yields sharp performance guarantees and enables a rigorous, apples-to-apples comparison between $mathrm{EF}$, $mathrm{EF}^{21}$, and compressed gradient descent. Our analysis is carried out in a simplified yet representative setting, which allows for clean theoretical insights and fair comparison of the underlying mechanisms.

Problem

• Analyzing convergence rates of error feedback methods in distributed learning
• Comparing performance of EF, EF21, and compressed gradient descent
• Identifying optimal Lyapunov functions for tight convergence bounds

Innovation

• Tight analysis of EF and EF21 methods
• Lyapunov function for optimal convergence rates
• Comparison of EF, EF21, compressed gradient descent