A Quantised Push‐Sum Distributed Adaptive Momentum Algorithm for Optimisation Over Directed Networks

ABSTRACT In this paper, we investigate a distributed constrained optimisation problem over directed networks. The agents in the networks conduct local computations and communications, endeavouring to collaboratively minimise the aggregation of all locally known convex cost functions subject to a global constraint set. However, since the agents are constantly transmitting information, most existing algorithms for this problem are prone to communication burdens, especially in large‐scale networks under a limited communication bandwidth. Problems of this nature emerge in a number of applications, mostly evident in distributed classification tasks, distributed image restoration, distributed compressive sensing etc. To solve these kinds of problems, we propose an effective quantised push‐sum distributed adaptive momentum (QPS‐DADAM) algorithm. On the one hand, the QPS‐DADAM algorithm employs the random quantiser to reduce the communication overhead and avoid the channel blockage. On the other hand, the QPS‐DADAM algorithm incorporates the adaptive momentum method into the push‐sum protocol to further accelerate the convergence over directed networks. Rigorous theoretical analyses are provided to illustrate that the QPS‐DADAM algorithm converges sublinearly to the optimal solution. In addition, numerical simulations further demonstrate the efficacy of the QPS‐DADAM algorithm and the correctness of the theoretical discoveries.