Convergence-Rate-Matching Discretization of Accelerated Optimization Flows Through Opportunistic State-Triggered Control

A recent body of exciting work seeks to shed light on the behavior of accelerated methods in optimization via high-resolution differential equations. These differential equations are continuous counterparts of the discrete-time optimization algorithms, and their convergence properties can be characterized using the powerful tools provided by classical Lyapunov stability analysis. An outstanding question of pivotal importance is how to discretize these continuous flows while maintaining their convergence rates. This paper provides a novel approach through the idea of opportunistic state-triggered control. We take advantage of the Lyapunov functions employed to characterize the rate of convergence of high-resolution differential equations to design variable-stepsize forward-Euler discretizations that preserve the Lyapunov decay of the original dynamics. The philosophy of our approach is not limited to forward-Euler discretizations and may be combined with other integration schemes.