Byzantine Stochastic Gradient Descent

This paper studies the problem of distributed stochastic optimization in an adversarial setting where, out of the m machines which allegedly compute stochastic gradients every iteration, an $α$-fraction are Byzantine, and can behave arbitrarily and adversarially. Our main result is a variant of stochastic gradient descent (SGD) which finds ε-approximate minimizers of convex functions in T = Õ( 1/ε² m + α²/ε² ) iterations. In contrast, traditional mini-batch SGD needs T = O( 1/ε² m ) iterations, but cannot tolerate Byzantine failures. Further, we provide a lower bound showing that, up to logarithmic factors, our algorithm is information-theoretically optimal both in terms of sampling complexity and time complexity.