  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, Sun and
Shahrampour (2020) studied the asymptotic convergence of this algorithm for
when the global function is strongly convex but local functions are convex.
Using control theory tools, in this work, we prove 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.
