Simple Minimax Optimal Byzantine Robust Algorithm for Nonconvex Objectives with Uniform Gradient Heterogeneity

In this study, we consider nonconvex federated learning problems with the existence of Byzantine workers. We propose a new simple Byzantine robust algorithm called Momentum Screening. The algorithm is adaptive to the Byzantine fraction, i.e., all its hyperparameters do not depend on the number of Byzantine workers. We show that our method achieves the best optimization error of $O(\delta^2\zeta_\mathrm{max}^2)$ for nonconvex smooth local objectives satisfying $\zeta_\mathrm{max}$-uniform gradient heterogeneity condition under $\delta$-Byzantine fraction, which can be better than the best known error rate of $O(\delta\zeta_\mathrm{mean}^2)$ for local objectives satisfying $\zeta_\mathrm{mean}$-mean heterogeneity condition when $\delta \leq (\zeta_\mathrm{max}/\zeta_\mathrm{mean})^2$. Furthermore, we derive an algorithm independent lower bound for local objectives satisfying $\zeta_\mathrm{max}$-uniform gradient heterogeneity condition and show the minimax optimality of our proposed method on this class. In numerical experiments, we validate the superiority of our method over the existing robust aggregation algorithms and verify our theoretical results.