Abstract
This paper studies the problem of distributed stochastic optimization in an adversarial setting where, out of the machines which allegedly compute stochastic gradients every iteration, an -fraction are Byzantine, and can behave arbitrarily and adversarially. Our main result is a variant of stochastic gradient descent (SGD) which finds -approximate minimizers of convex functions in $T = \tilde{O}\big( \frac{1}{\varepsilon^2 m} + \frac{\alpha^2}{\varepsilon^2} \big)$ iterations. In contrast, traditional mini-batch SGD needs 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.