#### Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates

##### Dong Yin, Yudong Chen, Kannan Ramchandran, Peter Bartlett

In large-scale distributed learning, security issues have become increasingly important. Particularly in a decentralized environment, some computing units may behave abnormally, or even exhibit Byzantine failures---arbitrary and potentially adversarial behavior. In this paper, we develop distributed learning algorithms that are provably robust against such failures, with a focus on achieving optimal statistical performance. A main result of this work is a sharp analysis of two robust distributed gradient descent algorithms based on median and trimmed mean operations, respectively. We prove statistical error rates for three kinds of population loss functions: strongly convex, non-strongly convex, and smooth non-convex. In particular, these algorithms are shown to achieve order-optimal statistical error rates for strongly convex losses. To achieve better communication efficiency, we further propose a median-based distributed algorithm that is provably robust, and uses only one communication round. For strongly convex quadratic loss, we show that this algorithm achieves the same optimal error rate as the robust distributed gradient descent algorithms.

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