Optimal algorithms for smooth and strongly convex distributed optimization in networks

Theory group, Microsoft Research, Redmond, United Stales, In this paper, wc determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings: centralized and decentralized communications over a network. For centralized (i.e. master/slave) algorithms, we show that distributing Nesterov's accelerated gradient descent is optimal and achieves a precision e > 0 in time 0(KJ(1 + Ar)ln(l/e)), where Kg is the condition number of the (global) function to optimize, A is the diameter of the network, and r (resp. 1) is the time needed to communicate values between two neighbors (resp. perform local computations). For decentralized algorithms based on gossip, we provide the first optimal algorithm, called the multi-step dual accelerated (MSDA) method, that achieves a precision e > 0 in time 0 0 7 ( 1 + where KJ is the condition number of the local functions and 7 is the (normalized) eigengap of the gossip matrix used for communication between nodes. We then verify the efficiency of MSDA against state-of-theart methods for two problems: least-squares regression and classification by logistic regression.