A systematic approach to Lyapunov analyses of continuous-time models in convex optimization (2205.12772v2)

Abstract: First-order methods are often analyzed via their continuous-time models, where their worst-case convergence properties are usually approached via Lyapunov functions. In this work, we provide a systematic and principled approach to find and verify Lyapunov functions for classes of ordinary and stochastic differential equations. More precisely, we extend the performance estimation framework, originally proposed by Drori and Teboulle [10], to continuous-time models. We retrieve convergence results comparable to those of discrete methods using fewer assumptions and convexity inequalities, and provide new results for stochastic accelerated gradient flows.

• The paper introduces an SDP-based Lyapunov framework to analyze continuous-time gradient flows in convex optimization.
• It extends performance estimation techniques to both deterministic and stochastic settings, yielding improved convergence proofs.
• Numerical results validate its approach by simplifying convergence verification and indicating promising directions for higher-order models.

Systematic Lyapunov Analyses of Continuous-Time Models in Convex Optimization

The paper authored by C. Moucer, A. Taylor, and F. Bach presents a structured approach to the analysis of continuous-time models in convex optimization using Lyapunov functions. This paper extends the performance estimation framework originally proposed for discrete optimization methods to continuous-time settings, particularly focusing on ordinary differential equations (ODEs) and stochastic differential equations (SDEs).

Summary

The authors target first-order optimization methods often employed in convex optimization, which are typically analyzed through their continuous-time counterparts due to the simpler convergence proofs available in this domain. Critical to these analyses is the use of Lyapunov functions—a tool commonly employed in systems theory and physics to paper stability and convergence. This work introduces a generalized and systematic methodology to construct and validate Lyapunov functions for both deterministic and stochastic continuous-time processes.

Methodology and Key Contributions

The methodology innovatively translates the challenge of verifying convergence via Lyapunov functions into solving semidefinite programming (SDP) problems. This approach builds on the foundational concept of performance estimation problems (PEPs). The authors thoroughly develop the SDP formulations necessary to confirm the existence of valid Lyapunov functions across various types of continuous-time gradient flows, including:

• Gradient Flows: For both strongly convex and non-strongly convex functions, the paper details how specific Lyapunov functions ensure convergence, resembling earlier discrete-time results but with more straightforward assumptions.
• Accelerated Gradient Flows: By examining second-order gradient flows, the research improves upon existing bounds using a Lyapunov framework that permits certain non-positive semidefinite matrices, revealing faster guaranteed convergence rates for Nesterov's accelerated gradient methods.

In extending these analyses to SDEs, the authors articulate how approximation through SDEs can elucidate the convergence characteristics of stochastic gradient descent (SGD). They accomplish this by examining the influence of step size adjustment and averaging techniques, vital for SGD's convergence.

Numerical and Theoretical Implications

The numerical results corroborate the theoretical analysis, showing that the authors' SDP framework reliably yields tighter convergence results—demonstrating both effectiveness and computational efficiency across tested scenarios. Significantly, this paper dispels the notion that traditional Lyapunov analyses are overly cumbersome for continuous-time models, simplifying convergence verification to manageable SDP problems.

Future Directions and Implications

The paper’s implications for future work are profound, as it opens avenues for exploring richer classes of Lyapunov functions, potentially improving analyses of higher-order methods and related assumptions. The paper advocates that while continuous-time models pose fewer assumptions than discrete-time settings, the gap between the continuous approximation and actual discrete methods warrants further exploration, particularly in stochastic regimes.

In conclusion, this work provides a robust framework for analyzing the convergence of continuous-time models in convex optimization, capable of extending to broader classes of optimization problems and potentially more complex system dynamics. The integration of SDP-based Lyapunov verification presents a promising direction for those exploring both theoretical and practical optimization in continuous settings.