FROST—Fast row-stochastic optimization with uncoordinated step-sizes

Abstract

In this paper, we discuss distributed optimization over directed graphs, where doubly stochastic weights cannot be constructed. Most of the existing algorithms overcome this issue by applying push-sum consensus, which utilizes column-stochastic weights. The formulation of column-stochastic weights requires each agent to know (at least) its out-degree, which may be impractical in, for example, broadcast-based communication protocols. In contrast, we describe FROST (Fast Row-stochastic-Optimization with uncoordinated STep-sizes), an optimization algorithm applicable to directed graphs that does not require the knowledge of out-degrees, the implementation of which is straightforward as each agent locally assigns weights to the incoming information and locally chooses a suitable step-size. We show that FROST converges linearly to the optimal solution for smooth and strongly convex functions given that the largest step-size is positive and sufficiently small.