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In this thesis, we propose new theoretical frameworks for the analysis ofstochastic and distributed methods with error compensation and local updates.Using these frameworks, we develop more than 20 new optimization methods,including the first linearly converging Error-Compensated SGD and the firstlinearly converging Local-SGD for arbitrarily heterogeneous local functions.Moreover, the thesis contains several new distributed methods with unbiasedcompression for distributed non-convex optimization problems. The derivedcomplexity results for these methods outperform the previous best-known resultsfor the considered problems. Finally, we propose a new scalable decentralizedfault-tolerant distributed method, and under reasonable assumptions, we derivethe iteration complexity bounds for this method that match the ones ofcentralized Local-SGD.

Distributed and Stochastic Optimization Methods with Gradient Compression and Local Steps