The Communication Complexity of Optimization
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We consider the communication complexity of a number of distributed optimization problems. We start with the problem of solving a linear system. Suppose there is a coordinator together with s servers P_1, ..., P_s, the i-th of which holds a subset A^(i) x = b^(i) of n_i constraints of a linear system in d variables, and the coordinator would like to output x ∈R^d for which A^(i) x = b^(i) for i = 1, ..., s. We assume each coefficient of each constraint is specified using L bits. We first resolve the randomized and deterministic communication complexity in the point-to-point model of communication, showing it is Θ̃(d^2L + sd) and Θ̃(sd^2L), respectively. We obtain similar results for the blackboard model. When there is no solution to the linear system, a natural alternative is to find the solution minimizing the ℓ_p loss. While this problem has been studied, we give improved upper or lower bounds for every value of p > 1. One takeaway message is that sampling and sketching techniques, which are commonly used in earlier work on distributed optimization, are neither optimal in the dependence on d nor on the dependence on the approximation ϵ, thus motivating new techniques from optimization to solve these problems. Towards this end, we consider the communication complexity of optimization tasks which generalize linear systems. For linear programming, we first resolve the communication complexity when d is constant, showing it is Θ̃(sL) in the point-to-point model. For general d and in the point-to-point model, we show an Õ(sd^3 L) upper bound and an Ω̃(d^2 L + sd) lower bound. We also show if one perturbs the coefficients randomly by numbers as small as 2^-Θ(L), then the upper bound is Õ(sd^2 L) + poly(dL).
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