Linear Convergence of Distributed Mirror Descent with Integral Feedback for Strongly Convex Problems

Distributed optimization often requires finding the minimum of a global objective function written as a sum of local functions. A group of agents work collectively to minimize the global function. We study a continuous-time decentralized mirror descent algorithm that uses purely local gradient information to converge to the global optimal solution. The algorithm enforces consensus among agents using the idea of integral feedback. Recently, the asymptotic convergence of this algorithm was studied for when the global function is strongly convex but local functions are convex. Using control theoretical tools, in this work, we prove (theoretically) that the algorithm indeed achieves local exponential convergence. We also provide a numerical experiment on a real data-set as a validation of the convergence speed of our algorithm.