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Due to the communication bottleneck in distributed and decentralized federated learning applications, algorithms using compressed communication have attracted significant attention. The Error Feedback (EF) is a widely-studied compression framework for convergence with biased compressors such as top-k sparsification. Although various improvements have been obtained in recent years, the theoretical guarantee for EF-type framework is still limited. Previous works either 1) rely on strong assumptions such as bounded gradient/dissimilarity assumptions, thus can not deal with arbitrary data heterogeneity and also slow the convergence speed, or 2) can not enjoy linear speedup in the number of clients. In this work, we propose a new EFSkip framework which removes the strong assumptions to allow arbitrary data heterogeneity and enjoys linear speedup for significantly improving upon previous results. In particular, EFSkip achieves a substantially lower computational complexity compared to the previous EF21, i.e., EFSkip enjoys the linear speedup in the number of clients (reducing the result linearly using more clients). We also show that EFSkip enjoys linear speedup and achieves faster convergence for nonconvex problems satisfying Polyak-Lojasiewicz (PL) condition. We believe that the new EFSkip framework will have a large impact on the communication- and computation-efficient distributed and decentralized federated learning.