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We propose and study a new class of gradient communication mechanisms for communication-efficient training -- three point compressors (3PC) -- as well as efficient distributed nonconvex optimization algorithms that can take advantage of them. Unlike most established approaches, which rely on a static compressor choice (e.g., Top-$K$), our class allows the compressors to {\em evolve} throughout the training process, with the aim of improving the theoretical communication complexity and practical efficiency of the underlying methods. We show that our general approach can recover the recently proposed state-of-the-art error feedback mechanism EF21 (Richtárik et al., 2021) and its theoretical properties as a special case, but also leads to a number of new efficient methods. Notably, our approach allows us to improve upon the state of the art in the algorithmic and theoretical foundations of the {\em lazy aggregation} literature (Chen et al., 2018). As a by-product that may be of independent interest, we provide a new and fundamental link between the lazy aggregation and error feedback literature. A special feature of our work is that we do not require the compressors to be unbiased.