Tight analyses of first-order methods with error feedback

TL;DR

This paper provides a precise theoretical analysis of error feedback methods in distributed learning, establishing optimal convergence rates and enabling fair comparison with other compression techniques.

Contribution

It introduces tight convergence bounds for EF and EF^{21} methods, supported by matching lower bounds, in a simplified single-agent setting.

Findings

Derived the best possible Lyapunov functions for EF and EF^{21}

Established matching lower bounds for convergence rates

Provided a clear comparison between EF, EF^{21}, and compressed gradient descent

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 the simplified single-agent setting, which allows for clean theoretical insights and fair comparison of the underlying mechanisms.