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.

翻译：通信成本是分布式学习与一阶优化中的主要瓶颈。缓解该问题的常见方法是对智能体间交换的梯度信息进行压缩，然而此类压缩通常会削弱基于梯度的方法的收敛保证。误差反馈机制为此提供了一种简单且计算代价低廉的补救方案，但已有众多变体被提出，其相对性能仍未被充分理解。本文针对文献中两种主要的误差反馈算法——经典误差反馈方法（EF）与误差反馈21（EF21），通过识别最优步长选择并构建针对每种方法的最优Lyapunov函数，提供了紧致的收敛分析。所得结果独立于智能体数量，并恢复了单智能体场景下已知的最优保证。