0 if data is heterogeneous. Then we further divide the vanishing error ε_T into three parts caused by Byzantine, stochasticity and insufficient iteration, and analyse their complexities (in the worst case), respectively. To prove the lower bound is tight, we design a stochastic Nesterov accelerated gradient method for Byzantine strongly convex problems and a recursive regularization method for Byzantine convex problems. We establish the oracle complexities of our methods, matching the corresponding lower bounds, implying the lower bound is tight and the proposed methods are optimal." />

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