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We propose a new first-order method for minimizing nonconvex functions withLipschitz continuous gradients and H\"older continuous Hessians. The proposedalgorithm is a heavy-ball method equipped with two particular restartmechanisms. 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 theH\"older exponent and constant, respectively. Our algorithm is$\nu$-independent and thus universal; it automatically achieves the abovecomplexity bound with the optimal $\nu \in [0, 1]$ without knowledge of$H_{\nu}$. In addition, the algorithm does not require other problem-dependentparameters as input, including the gradient's Lipschitz constant or the targetaccuracy $\epsilon$. Numerical results illustrate that the proposed method ispromising.

Universal heavy-ball method for nonconvex optimization under H\"older continuous Hessians