Robust Distributed Optimization With Randomly Corrupted Gradients

In this paper, we propose a first-order distributed optimization algorithm that is provably robust to Byzantine failures-arbitrary and potentially adversarial behavior, where all the participating agents are prone to failure. We model each agent's state over time as a two-state Markov chain that indicates Byzantine or trustworthy behaviors at different time instants. We set no restrictions on the maximum number of Byzantine agents at any given time. We design our method based on three layers of defense: 1) Temporal gradient averaging, 2) robust aggregation, and 3) gradient normalization. We study two settings for stochastic optimization, namely Sample Average Approximation and Stochastic Approximation, and prove that for strongly convex and smooth non-convex cost functions, our algorithm achieves order-optimal statistical error and convergence rates.