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Acceleration via Perturbations on Low-resolution Ordinary Differential Equations

Xudong Li, Lei Shi, Mingqi Song

TL;DR

This work introduces a generalized perturbed ODE with gradient and gradient-correction perturbations to analyze accelerated optimization for mu-strongly convex functions. By establishing a Lyapunov-based framework in both continuous and discrete time, it reveals that a careful balance between the perturbations can preserve or enhance convergence rates, while the gradient-correction term alone may slow convergence in some setups. The authors develop implicit and symplectic Euler discretizations, derive explicit rate conditions on the perturbation parameters, and propose accelerated algorithms that extend known schemes. Numerical experiments on quadratic and logistic-regression problems corroborate the theory, showing reduced oscillations and improved speed when perturbations are properly coordinated. The results offer practical guidelines for designing perturbed dynamics and discretizations to achieve faster optimization performance in strongly convex settings.

Abstract

Recently, the high-resolution ordinary differential equation (ODE) framework, which retains higher-order terms, has been proposed to analyze gradient-based optimization algorithms. Through this framework, the term $\nabla^2 f(X_t)\dot{X_t}$, known as the gradient-correction term, was found to be essential for reducing oscillations and accelerating the convergence rate of function values. Despite the importance of this term, simply adding it to the low-resolution ODE may sometimes lead to a slower convergence rate. To fully understand this phenomenon, we propose a generalized perturbed ODE and analyze the role of the gradient and gradient-correction perturbation terms under both continuous-time and discrete-time settings. We demonstrate that while the gradient-correction perturbation is essential for obtaining accelerations, it can hinder the convergence rate of function values in certain cases. However, this adverse effect can be mitigated by involving an additional gradient perturbation term. Moreover, by conducting a comprehensive analysis, we derive proper choices of perturbation parameters. Numerical experiments are also provided to validate our theoretical findings.

Acceleration via Perturbations on Low-resolution Ordinary Differential Equations

TL;DR

This work introduces a generalized perturbed ODE with gradient and gradient-correction perturbations to analyze accelerated optimization for mu-strongly convex functions. By establishing a Lyapunov-based framework in both continuous and discrete time, it reveals that a careful balance between the perturbations can preserve or enhance convergence rates, while the gradient-correction term alone may slow convergence in some setups. The authors develop implicit and symplectic Euler discretizations, derive explicit rate conditions on the perturbation parameters, and propose accelerated algorithms that extend known schemes. Numerical experiments on quadratic and logistic-regression problems corroborate the theory, showing reduced oscillations and improved speed when perturbations are properly coordinated. The results offer practical guidelines for designing perturbed dynamics and discretizations to achieve faster optimization performance in strongly convex settings.

Abstract

Recently, the high-resolution ordinary differential equation (ODE) framework, which retains higher-order terms, has been proposed to analyze gradient-based optimization algorithms. Through this framework, the term , known as the gradient-correction term, was found to be essential for reducing oscillations and accelerating the convergence rate of function values. Despite the importance of this term, simply adding it to the low-resolution ODE may sometimes lead to a slower convergence rate. To fully understand this phenomenon, we propose a generalized perturbed ODE and analyze the role of the gradient and gradient-correction perturbation terms under both continuous-time and discrete-time settings. We demonstrate that while the gradient-correction perturbation is essential for obtaining accelerations, it can hinder the convergence rate of function values in certain cases. However, this adverse effect can be mitigated by involving an additional gradient perturbation term. Moreover, by conducting a comprehensive analysis, we derive proper choices of perturbation parameters. Numerical experiments are also provided to validate our theoretical findings.

Paper Structure

This paper contains 10 sections, 6 theorems, 118 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Perturbed ODE for strongly convex functions
  3. Optimization algorithms obtained by discretizing (general perturbed ODE)
  4. Optimization algorithms obtained by the implicit discretization
  5. Optimization algorithms obtained by symplectic discretizations
  6. Numerical experiments
  7. Numerical experiments on quadratic function minimization problem
  8. Numerical experiments on $\ell_2$-regularized logistic regression problem
  9. Conclusion
  10. Analysis for the modified symplectic Euler discretization \ref{['eq: iteration of modified symplectic discretization']}

Key Result

Theorem 1

Suppose that $f\in{\cal C}^2$ is $\mu$-strongly convex. Then, the following inequality holds If the non-negative perturbation parameters $\Delta_1, \Delta_2$ satisfy then it holds that Besides, if $\Delta_1=0$, $\Delta_2>0$ and $f$ is $L$-smooth, then the following estimation holds

Figures (5)

  • Figure 1: An illustration of four stages of a horizontal damped spring oscillator described by \ref{['equation of motion']} with $f(X)=\frac{1}{2}\mathcal{K}X^2$.
  • Figure 2: Numerical comparisons of scheme \ref{['eq: iteration of direct symplectic discretization']} with different $(\widehat{\Delta}_1, \widehat{\Delta}_2)$ on solving problem \ref{['eq: quadratic function for numerical experiments']}.
  • Figure 3: Numerical comparisons of scheme \ref{['eq: iteration of direct symplectic discretization']} with different $(\widehat{\Delta}_1, \widehat{\Delta}_2)$ on solving $\ell_2$-regularized logistic regression \ref{['eq: logistic regression']} with dataset CINA.
  • Figure 4: Numerical comparisons of scheme \ref{['eq: iteration of direct symplectic discretization']} with different $(\widehat{\Delta}_1, \widehat{\Delta}_2)$ on solving $\ell_2$-regularized logistic regression \ref{['eq: logistic regression']} with dataset a9a
  • Figure 5: Numerical comparisons of scheme \ref{['eq: iteration of direct symplectic discretization']} with different $(\widehat{\Delta}_1, \widehat{\Delta}_2)$ on solving $\ell_2$-regularized logistic regression \ref{['eq: logistic regression']} with dataset ijcnn1.

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary 1
  • ...and 2 more