Heavy-ball Differential Equation Achieves $O(\varepsilon^{-7/4})$ Convergence for Nonconvex Functions

TL;DR

This paper demonstrates that the heavy-ball differential equation, a continuous-time model of optimization algorithms, can reach an $ ext{ε}$-stationary point in $O( ext{ε}^{-7/4})$ time for nonconvex functions, simplifying convergence analysis.

Contribution

The paper establishes that the heavy-ball differential equation achieves the optimal $O( ext{ε}^{-7/4})$ convergence rate without additional mechanisms, providing a simpler approach for nonconvex optimization.

Findings

Heavy-ball differential equation attains $O(ε^{-7/4})$ convergence.

Simplifies understanding of heavy-ball method dynamics.

Provides a continuous-time perspective on optimization complexity.

Abstract

First-order optimization methods for nonconvex functions with Lipschitz continuous gradient and Hessian have been extensively studied. State-of-the-art methods for finding an $\varepsilon$-stationary point within $O(\varepsilon^{-{7/4}})$ or $\tilde{O}(\varepsilon^{-{7/4}})$ gradient evaluations are based on Nesterov's accelerated gradient descent (AGD) or Polyak's heavy-ball (HB) method. However, these algorithms employ additional mechanisms, such as restart schemes and negative curvature exploitation, which complicate their behavior and make it challenging to apply them to more advanced settings (e.g., stochastic optimization). As a first step in investigating whether a simple algorithm with $O(\varepsilon^{-{7/4}})$ complexity can be constructed without such additional mechanisms, we study the HB differential equation, a continuous-time analogue of the AGD and HB methods. We prove that its dynamics attain an $\varepsilon$-stationary point within $O(\varepsilon^{-{7/4}})$ time.