Acceleration via Perturbations on Low-resolution Ordinary Differential Equations

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.