On Centralized and Distributed Mirror Descent: Convergence Analysis Using Quadratic Constraints

Mirror descent (MD) is a powerful first-order optimization technique that subsumes several algorithms including gradient (GD). In this work, we leverage quadratic constraints and Lyapunov functions to analyze the stability characterize convergence rate of MD algorithm as well its distributed variant using semidefinite programming (SDP). For both algorithms, consider strongly convex nonstrongly assumptions. centralized problems, construct an SDP certifies exponential rates derive closed-form feasible solution recovers optimal GD special case. We complement our analysis by providing explicit $O(1/k)$ for problems. Next, numerically whose dimensions are independent network size. To best knowledge, numerical has not been previously reported in literature. further prove setting. Our experiments on problems indicate framework superior compared existing GD.