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# Byzantine Machine Learning Made Easy By Resilient Averaging of Momentums

*Proceedings of the 39th International Conference on Machine Learning*, PMLR 162:6246-6283, 2022.

#### Abstract

Byzantine resilience emerged as a prominent topic within the distributed machine learning community. Essentially, the goal is to enhance distributed optimization algorithms, such as distributed SGD, in a way that guarantees convergence despite the presence of some misbehaving (a.k.a.,

*Byzantine*) workers. Although a myriad of techniques addressing the problem have been proposed, the field arguably rests on fragile foundations. These techniques are hard to prove correct and rely on assumptions that are (a) quite unrealistic, i.e., often violated in practice, and (b) heterogeneous, i.e., making it difficult to compare approaches. We present*RESAM (RESilient Averaging of Momentums)*, a unified framework that makes it simple to establish optimal Byzantine resilience, relying only on standard machine learning assumptions. Our framework is mainly composed of two operators:*resilient averaging*at the server and*distributed momentum*at the workers. We prove a general theorem stating the convergence of distributed SGD under RESAM. Interestingly, demonstrating and comparing the convergence of many existing techniques become direct corollaries of our theorem, without resorting to stringent assumptions. We also present an empirical evaluation of the practical relevance of RESAM.