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We consider the distributed statistical learning problem in a high-dimensional adversarial scenario. At each iteration, $m$ worker machines compute stochastic gradients and send them to a master machine. However, an $\alpha$-fraction of $m$ worker machines, called Byzantine machines, may act adversarially and send faulty gradients. To guard against faulty information sharing, we develop a distributed robust learning algorithm based on Nesterov's dual averaging. This algorithms is provably robust against Byzantine machines whenever $\alpha\in[0, 1/2)$. For smooth convex functions, we show that running the proposed algorithm for $T$ iterations achieves a statistical error bound $\tilde{O}\big(1/\sqrt{mT}+\alpha/\sqrt{T}\big)$. This result holds for a large class of normed spaces and it matches the known statistical error bound for Byzantine stochastic gradient in the Euclidean space setting. A key feature of the algorithm is that the dimension dependence of the bound scales with the dual norm of the gradient; in particular, for probability simplex, we show that it depends logarithmically on the problem dimension $d$. Such a weak dependence on the dimension is desirable in high-dimensional statistical learning and it has been known to hold for the classical mirror descent but it appears to be new for the Byzantine gradient scenario. 

We study a distributed policy evaluation problem in which a group of agents with jointly observed states and private local actions and rewards collaborate to learn the value function of a given policy via local computation and communication. This problem arises in various large-scale multi-agent systems, including power grids, intelligent transportation systems, wireless sensor networks, and multi-agent robotics. We develop and analyze a new distributed temporal-difference learning algorithm that minimizes the mean-square projected Bellman error. Our approach is based on a stochastic primal-dual method and we improve the best-known convergence rate from $O(1/\sqrt{T})$ to $O(1/T)$, where $T$ is the total number of iterations. Our analysis explicitly takes into account the Markovian nature of the sampling and addresses a broader class of problems than the commonly-used i.i.d. sampling scenario.