Tight analyses of first-order methods with error feedback
This work offers incremental improvements in theoretical understanding for researchers in distributed optimization, focusing on single-agent settings for clean insights.
The paper tackled the problem of communication bottlenecks in distributed learning by analyzing error feedback methods (EF and EF21) for compressed communication, providing tight convergence rates with matching lower bounds to enable rigorous comparisons.
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.