Due to the high communication overhead when training machine learning models in a distributed environment, modern algorithms invariably rely on lossy communication compression. However, when untreated, the errors caused by compression propagate, and can lead to severely unstable behavior, including exponential divergence. Almost a decade ago, Seide et al [2014] proposed an error feedback (EF) mechanism, which we refer to as EF14, as an immensely effective heuristic for mitigating this issue. However, despite steady algorithmic and theoretical advances in the EF field in the last decade, our understanding is far from complete. In this work we address one of the most pressing issues. In particular, in the canonical nonconvex setting, all known variants of EF rely on very large batch sizes to converge, which can be prohibitive in practice. We propose a surprisingly simple fix which removes this issue both theoretically, and in practice: the application of Polyak's momentum to the latest incarnation of EF due to Richt\'{a}rik et al. [2021] known as EF21. Our algorithm, for which we coin the name EF21-SGDM, improves the communication and sample complexities of previous error feedback algorithms under standard smoothness and bounded variance assumptions, and does not require any further strong assumptions such as bounded gradient dissimilarity. Moreover, we propose a double momentum version of our method that improves the complexities even further. Our proof seems to be novel even when compression is removed from the method, and as such, our proof technique is of independent interest in the study of nonconvex stochastic optimization enriched with Polyak's momentum.

翻译：由于分布式环境下训练机器学习模型时存在较高的通信开销，现代算法通常依赖有损通信压缩。然而，若不加以处理，由压缩引起的误差会传播，并可能导致严重的不稳定行为，包括指数发散。近十年前，Seide等人[2014]提出了一种称为EF14的误差反馈机制，作为缓解该问题极为有效的启发式方法。然而，尽管过去十年间EF领域在算法与理论上稳步推进，我们的理解仍远未完善。本文旨在解决其中一个最紧迫的问题：在典型的非凸设置中，所有已知的EF变体均需依赖极大批量大小才能收敛，而这在实践中可能难以承受。我们提出了一种惊人简单的修正方案，可从理论与实践中消除此问题：将Polyak动量应用于Richtárik等人[2021]提出的最新EF变体（即EF21）。我们将该算法命名为EF21-SGDM，其在标准光滑性和有界方差假设下（无需有界梯度差异性等强假设），改善了此前误差反馈算法的通信复杂度与样本复杂度。此外，我们提出该方法的双动量版本，进一步提升了复杂度。即使从方法中移除压缩环节，我们的证明仍具新颖性：因此，我们的证明技术在对融合Polyak动量的非凸随机优化研究中具有独立价值。