{"ID":2872978,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.07580","arxiv_id":"2509.07580","title":"On Global Rates for Regularization Methods based on Secant Derivative Approximations","abstract":"An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the $p$th-order tensor, based on lower-order derivatives. Between each recalculation of the $p$th-order derivative approximation, a high-order secant equation can be used to update the $p$th-order tensor as proposed in (Welzel 2024) or the approximation can be kept constant in a lazy manner. When refreshing the $p$th-order tensor approximation after $m$ steps, an exact evaluation of the tensor or a finite difference approximation can be used with an explicit discretization stepsize. For all the newly adaptive regularization variants, we prove an $\\mathcal{O}\\left( \\max[ ε_1^{-(p+1)/p}, \\, ε_2^{(-p+1)/(p-1)} ] \\right)$ bound on the number of iterations needed to reach an $(ε_1, \\, ε_2)$ second-order stationary points. Discussions on the number of oracle calls for each introduced variant are also provided. When $p=2$, we obtain a second-order method that uses quasi-Newton approximations with an $\\mathcal{O}\\left(\\max[ε_1^{-3/2}, \\, \\, ε_2^{-3}]\\right)$ iteration bound to achieve approximate second-order stationarity.","short_abstract":"An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the $p$th-order tensor, based on lower-order derivatives. Between each recalculation of the $p$th-order derivative approximation, a high-order secant equation can be used to update the $p$th-order t...","url_abs":"https://arxiv.org/abs/2509.07580","url_pdf":"https://arxiv.org/pdf/2509.07580v1","authors":"[\"Coralia Cartis\",\"Sadok Jerad\"]","published":"2025-09-09T10:43:35Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.NA\"]","methods":"[]","has_code":false}