Uncertainty Principle for Communication Compression in Distributed and Federated Learning and the Search for an Optimal Compressor

In order to mitigate the high communication cost in distributed and federated learning, various vector compression schemes, such as quantization, sparsification and dithering, have become very popular. In designing a compression method, one aims to communicate as few bits as possible, which minimizes the cost per communication round, while at the same time attempting to impart as little distortion (variance) to the communicated messages as possible, which minimizes the adverse effect of the compression on the overall number of communication rounds. However, intuitively, these two goals are fundamentally in conflict: the more compression we allow, the more distorted the messages become. We formalize this intuition and prove an {\em uncertainty principle} for randomized compression operators, thus quantifying this limitation mathematically, and {\em effectively providing asymptotically tight lower bounds on what might be achievable with communication compression}. Motivated by these developments, we call for the search for the optimal compression operator. In an attempt to take a first step in this direction, we consider an unbiased compression method inspired by the Kashin representation of vectors, which we call {\em Kashin compression (KC)}. In contrast to all previously proposed compression mechanisms, KC enjoys a {\em dimension independent} variance bound for which we derive an explicit formula even in the regime when only a few bits need to be communicate per each vector entry.