  Federated Learning is a collaborative training framework that leverages
heterogeneous data distributed across a vast number of clients. Since it is
practically infeasible to request and process all clients during the
aggregation step, partial participation must be supported. In this setting, the
communication between the server and clients poses a major bottleneck. To
reduce communication loads, there are two main approaches: compression and
local steps. Recent work by Mishchenko et al. [2022] introduced the new
ProxSkip method, which achieves an accelerated rate using the local steps
technique. Follow-up works successfully combined local steps acceleration with
partial participation [Grudzie\'n et al., 2023, Condat et al. 2023] and
gradient compression [Condat et al. [2022]. In this paper, we finally present a
complete method for Federated Learning that incorporates all necessary
ingredients: Local Training, Compression, and Partial Participation. We obtain
state-of-the-art convergence guarantees in the considered setting. Moreover, we
analyze the general sampling framework for partial participation and derive an
importance sampling scheme, which leads to even better performance. We
experimentally demonstrate the advantages of the proposed method in practice.
