Technical Report: A Totally Asynchronous Nesterov's Accelerated Gradient Method for Convex Optimization

Abstract

We present a totally asynchronous algorithm for convex optimization that is based on a novel generalization of Nesterov's accelerated gradient method. This algorithm is developed for fast convergence under "total asynchrony," i.e., allowing arbitrarily long delays between agents' computations and communications without assuming any form of delay bound. These conditions may arise, for example, due to jamming by adversaries. Our framework is block-based, in the sense that each agent is only responsible for computing updates to (and communicating the values of) a small subset of the network-level decision variables. In our main result, we present bounds on the algorithm's parameters that guarantee linear convergence to an optimizer. Then, we quantify the relationship between (i) the total number of computations and communications executed by the agents and (ii) the agents' collective distance to an optimum. Numerical simulations show that this algorithm requires 28% fewer iterations than the heavy ball algorithm and 61% fewer iterations than gradient descent under total asynchrony.