The adaptive regularization algorithm for unconstrained nonconvex optimization was shown in [7, 20] 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 in [5, 6] for \(q=1\) and in [11] for arbitrary \(q \in \{1,2\}\) . 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.
