Design of Gradient Push Algorithm in Time-Varying Directed Graph

Abstract

This paper proposes a novel distributed mechanism, named gradient push algorithm (GPA), to improve the shortage in convex function optimization, such as limitations of adjacent matrix and fixed topology. The algorithm is raised based on gradient algorithm and perturbed push sum protocol in dynamic topology of wireless sensor network (WSN). First, we develop the update rule of GPA, and bring 4 lemmas to reveal its convergence attribution. Second, in order to prove convergence of the lemma, we analyze the algorithm quantitatively, whose result shows that perturbed push sum protocol (PSP) converged to the optimal solution at rate of $O\left( {{{\ln t} \mathord{\left/ {\vphantom {{\ln t} {\sqrt t }}} \right. \kern-0pt} {\sqrt t }}} \right)$. Finally, we check its convergence through simulation. Both theory analysis and simulation results show that, the new algorithm avoids the limitation of adjacent matrix, and will convergent to the same optimal solution, which meets the need of dynamic topology in WSN.