  First-order optimization methods for nonconvex functions with Lipschitz
continuous gradients and Hessian have been studied intensively in both machine
learning and optimization. State-of-the-art methods 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 method. However,
these algorithms employ additional mechanisms, such as restart schemes and
negative curvature exploitation, which complicate the algorithms' behavior and
make it challenging to apply them to more advanced settings (e.g., stochastic
optimization). To realize a simpler algorithm, we investigate the heavy-ball
differential equation, a continuous-time analogy of the AGD and heavy-ball
methods; we prove that the dynamics attains an $\varepsilon$-stationary point
within $O(\varepsilon^{-{7/4}})$ time. We also show that a vanilla heavy-ball
algorithm, obtained by discretizing the dynamics, achieves the complexity of
$O(\varepsilon^{-{7/4}})$ under an additional assumption.
