Toward a Systematic Understanding and Interactive Search of Lyapunov-Style Proofs in Optimization

Abstract

Lyapunov-style convergence proofs, which establish a nonincreasing sequence to provide a quantitative convergence rate for an algorithm, are popular and often considered desirable in first-order optimization. However, existing approaches to finding such Lyapunov functions rely on hand-designed templates or prior insight on the proof structure, and do not certify that the resulting Lyapunov-style analysis provides the sharpest convergence bound. In this work, we introduce a systematic framework for converting a tight, analytic convergence proof of an optimization algorithm, often found via computer assistance, into an equivalent proof based on Lyapunov functions. We implement a concrete procedure that combines a performance estimation problem (PEP) toolbox with elementary linear algebra, and show that it captures a number of prior Lyapunov analyses within a single Jupyter notebook. Based on our implementation, a user can straightforwardly test our systematic and interactive procedure on their own optimization algorithm of interest to search for a tight Lyapunov-style proof via code, without the need to comprehend the implementation details. We extend the application of our framework and discover four novel analytic Lyapunov-style proofs, where notably, one of them identifies a new exact optimal proximal algorithm for strongly monotone inclusion problems.