A conditional gradient algorithm for distributed online optimization in networks

Abstract

This paper addresses a network of computing nodes aiming to solve an online convex optimisation problem in a distributed manner, that is, by means of the local estimation and communication, without any central coordinator. An online distributed conditional gradient algorithm based on the conditional gradient is developed, which can effectively tackle the problem of high time complexity of the distributed online optimisation. The proposed algorithm allows the global objective function to be decomposed into the sum of the local objective functions, and nodes collectively minimise the sum of local time-varying objective functions while the communication pattern among nodes is captured as a connected undirected graph. By adding a regularisation term to the local objective function of each node, the proposed algorithm constructs a new time-varying objective function. The proposed algorithm also utilises the local linear optimisation oracle to replace the projection operation such that the regret bound of the algorithm can be effectively improved. By introducing the nominal regret and the global regret, the convergence properties of the proposed algorithm are also theoretically analysed. It is shown that, if the objective function of each agent is strongly convex and smooth, these two types of regrets grow sublinearly with the order of O(log T ), where T is the time horizon. Numerical experiments also demonstrate the advantages of the proposed algorithm over existing distributed optimisation algorithms.

A distributed parallel optimization algorithm via alternating direction method of multipliers

“…Each agent updates its state with the local subgradient information and states collected from neighbours. The subgradient-based methods have been developed for various scenarios [12][13][14][15][16][17][18][19]. In the dual-based algorithms, alternating direction method of multipliers (ADMM) is a well-known method for large-scale systems.…”

A trust‐based resilient consensus algorithm for distributed optimization considering node and edge attacks

“…16 For distributed online optimization problems, a distributed algorithm was designed based on the conditional gradient without any central coordinator. 17 Furthermore, to deal with the distributed optimization problems hybrid linear constraints, a discrete-time algorithm based on matrix and graph theories was developed, and the effectiveness was verified by numerical examples. 18 Based on the state impulsive dynamical theory, a distributed hybrid impulsive algorithm was proposed to solve the large-scale nonlinear optimization problems with linear convergence rate.…”

“…Moreover, a fully distributed algorithm was used for solving grid optimization problem with three types of load, 15 and series of distributed algorithms were proposed to solve the stochastic big‐data problems with expected convergence rate 16 . For distributed online optimization problems, a distributed algorithm was designed based on the conditional gradient without any central coordinator 17 . Furthermore, to deal with the distributed optimization problems hybrid linear constraints, a discrete‐time algorithm based on matrix and graph theories was developed, and the effectiveness was verified by numerical examples 18 .…”