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A systematic approach to Lyapunov analyses of continuous-time models in convex optimization

Céline Moucer, Adrien Taylor, Francis Bach

TL;DR

The paper develops a systematic, SDP-based framework to certify convergence of continuous-time gradient-based models in convex optimization via Lyapunov functions, extending the performance estimation paradigm to both ODEs and SDEs. By formulating Lyapunov decrease as small LMIs, it enables automated worst-case rate computation for deterministic gradient flows and accelerated flows, as well as stochastic approximations of SGD with step-size schedules and averaging. It yields known optimal rates for strongly convex and convex settings, analyzes time dilation and non-autonomous dynamics, and reveals that averaging may not preserve acceleration in stochastic contexts, necessitating careful step-size design. The approach provides practical, tight certificates and highlights pathways for extending Lyapunov-based analysis to broader continuous-time models and higher-order stochastic dynamics, with numerical tools for reproducing results.

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.

A systematic approach to Lyapunov analyses of continuous-time models in convex optimization

TL;DR

The paper develops a systematic, SDP-based framework to certify convergence of continuous-time gradient-based models in convex optimization via Lyapunov functions, extending the performance estimation paradigm to both ODEs and SDEs. By formulating Lyapunov decrease as small LMIs, it enables automated worst-case rate computation for deterministic gradient flows and accelerated flows, as well as stochastic approximations of SGD with step-size schedules and averaging. It yields known optimal rates for strongly convex and convex settings, analyzes time dilation and non-autonomous dynamics, and reveals that averaging may not preserve acceleration in stochastic contexts, necessitating careful step-size design. The approach provides practical, tight certificates and highlights pathways for extending Lyapunov-based analysis to broader continuous-time models and higher-order stochastic dynamics, with numerical tools for reproducing results.

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.
Paper Structure (33 sections, 15 theorems, 71 equations, 2 figures)

This paper contains 33 sections, 15 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lyapunov functions
  3. Prior works
  4. Contributions and organization
  5. Assumptions
  6. A principled approach to Lyapunov functions for gradient flows
  7. The gradient flow
  8. Minimizing strongly convex functions
  9. Minimizing convex functions
  10. Accelerated gradient flows
  11. Minimizing strongly convex functions
  12. Minimizing convex functions
  13. Higher-order convergence and time dilation
  14. A non-autonomous first-order gradient flow
  15. A non-autonomous second-order gradient flow
  16. ...and 18 more sections

Key Result

Theorem 2.1

\newlabelconv_gf_mu0 Let $a,c, \tau \geqslant 0$ and $\mu > 0$. The following assertions are equivalent:

Figures (2)

  • Figure 1: Comparison between numerical values for $\tau$ obtained by solving the LMI \ref{['LMI']} and the reference established in the literature scieur2017integration, for trajectories $X_t$ generated by gradient flow \ref{['eq:gf_init']} originating from a $\mu$-strongly convex function.
  • Figure 2: Comparison between the worst-case guarantee obtained numerically with PEP, and its references, for the Polyak damped oscillator \ref{['str_agf']} originating from $\mu$-strongly convex functions, and for quadratic Lyapunov functions \ref{['lyapunov_order2']}.

Theorems & Definitions (36)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proof 1
  • Theorem 2.5
  • Proof 2
  • Corollary 2.6
  • Proof 3
  • Theorem 2.7
  • ...and 26 more