On Centralized and Distributed Mirror Descent: Convergence Analysis Using Quadratic Constraints
This work provides a novel analysis framework for optimization algorithms, which is incremental but offers improved convergence guarantees for researchers and practitioners in machine learning and optimization.
The authors tackled the problem of analyzing convergence rates for centralized and distributed mirror descent (MD) by developing a semi-definite programming (SDP) framework using quadratic constraints and Lyapunov stability, resulting in certified exponential rates for strongly convex cases and O(1/k) rates for convex settings, with numerical experiments showing superior rates compared to distributed gradient descent.
Mirror descent (MD) is a powerful first-order optimization technique that subsumes several optimization algorithms including gradient descent (GD). In this work, we develop a semi-definite programming (SDP) framework to analyze the convergence rate of MD in centralized and distributed settings under both strongly convex and non-strongly convex assumptions. We view MD with a dynamical system lens and leverage quadratic constraints (QCs) to provide explicit convergence rates based on Lyapunov stability. For centralized MD under strongly convex assumption, we develop a SDP that certifies exponential convergence rates. We prove that the SDP always has a feasible solution that recovers the optimal GD rate as a special case. We complement our analysis by providing the $O(1/k)$ convergence rate for convex problems. Next, we analyze the convergence of distributed MD and characterize the rate using SDP. To the best of our knowledge, the numerical rate of distributed MD has not been previously reported in the literature. We further prove an $O(1/k)$ convergence rate for distributed MD in the convex setting. Our numerical experiments on strongly convex problems indicate that our framework certifies superior convergence rates compared to the existing rates for distributed GD.