In this work, we propose a method for minimizing non-convex functions with Lipschitz continuous pth-order derivatives, starting from \(p \ge 1\) . The method, however, only requires derivative information up to order \((p-1)\) , since the pth-order derivatives are approximated via finite differences. To ensure oracle efficiency, instead of recomputing a finite-difference approximation of the pth-order derivative at every iteration, we attempt to reuse each approximation for m consecutive iterations before recomputing it, with \(m \ge 1\) as a key parameter. As a result, we obtain an adaptive method of order \((p-1)\) that requires no more than \(\mathcal {O}(\epsilon ^{-\frac{p+1}{p}})\) iterations to find an \(\epsilon \) -approximate stationary point of the objective function and that, for the choice \(m=(p-1)n + 1\) , where n is the problem dimension, takes no more than \(\mathcal {O}(n^{1/p}\epsilon ^{-\frac{p+1}{p}})\) oracle calls of order \((p-1)\) . This improves previously known bounds for tensor methods with finite-difference approximations in terms of the problem dimension.
