A Tight Theory of Error Feedback Algorithms in Distributed Optimization

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.

Training modern machine learning models often requires many computers to work together, but these computers can spend a large amount of time communicating updates to each other. To reduce this cost, researchers often compress the messages being sent, at the risk of making training slower or less reliable. Error feedback is a simple idea for repairing this: each computer keeps track of what was lost during compression and tries to send that missing information later. In this paper, we study two widely used error-feedback methods and ask how fast they can reach a solution in the worst case. We identify the best step sizes and the right mathematical measures of progress for these methods, showing when distributed training behaves just like the simpler single-computer case. We also show that one classic version can fail when different workers have genuinely different objectives, clarifying an important limitation. These results give researchers and practitioners clearer guidance on when communication-saving training methods are reliable, how to tune them, and where existing theory still needs to be improved.