  We propose a new first-order method for minimizing nonconvex functions with
Lipschitz continuous gradients and H\"older continuous Hessians. The proposed
algorithm is a heavy-ball method equipped with two particular restart
mechanisms. It finds a solution where the gradient norm is less than $\epsilon$
in $O(H_{\nu}^{\frac{1}{2 + 2 \nu}} \epsilon^{- \frac{4 + 3 \nu}{2 + 2 \nu}})$
function and gradient evaluations, where $\nu \in [0, 1]$ and $H_{\nu}$ are the
H\"older exponent and constant, respectively. Our algorithm is
$\nu$-independent and thus universal; it automatically achieves the above
complexity bound with the optimal $\nu \in [0, 1]$ without knowledge of
$H_{\nu}$. In addition, the algorithm does not require other problem-dependent
parameters as input, including the gradient's Lipschitz constant or the target
accuracy $\epsilon$. Numerical results illustrate that the proposed method is
promising.
