Linear Convergence of Distributed Mirror Descent with Integral Feedback for Strongly Convex Problems
This work provides a theoretical guarantee of exponential convergence for a distributed optimization algorithm, which is important for researchers and practitioners in distributed machine learning and control.
This paper examines a continuous-time decentralized mirror descent algorithm for distributed optimization of strongly convex global objective functions. It proves that the algorithm, which uses integral feedback for consensus, achieves local exponential convergence, improving upon prior asymptotic convergence results.
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.