Examples of slow convergence for adaptive regularization optimization methods are not isolated
<|reference_start|>Examples of slow convergence for adaptive regularization optimization methods are not isolated: The adaptive regularization algorithm for unconstrained nonconvex optimization was shown in Nesterov and Polyak (2006) and Cartis, Gould and Toint (2011) to require, under standard assumptions, at most $\mathcal{O}(\epsilon^{3/(3-q)})$ evaluations of the objective function and its derivatives of degrees one and two to produce an $\epsilon$-approximate critical point of order $q\in\{1,2\}$. This bound was shown to be sharp for $q \in\{1,2\}$. This note revisits these results and shows that the example for which slow convergence is exhibited is not isolated, but that this behaviour occurs for a subset of univariate functions of nonzero measure.<|reference_end|>
@article{toint2024examples,
    title={Examples of slow convergence for adaptive regularization optimization
  methods are not isolated},
    author={Philippe L. Toint},
    journal={arXiv preprint arXiv:2409.16047},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.16047},
    primaryClass={math.OC cs.CC}
}