Polynomial-time implementation of higher-order regularized Newton methods

Develop an algorithm for implementing Nesterov’s higher-order regularized Newton method for orders d greater than or equal to 4 in a way that improves upon the classical second-order Newton method.

Background

Nesterov’s higher-order regularized Newton method minimizes a dth-order Taylor approximation with a degree-(d+1) regularization term selected to preserve convexity. The paper notes that the method is implementable in polynomial time for d=3, but that no corresponding algorithm is known for orders d≥4 that offers an improvement over ordinary Newton’s method. Thus, the unresolved issue concerns whether higher-order convergence can be coupled with an algorithmically efficient implementation beyond the cubic case.

References

To the best of our knowledge, no algorithm is known that can implement this method for $d\geq4$ in a way that improves upon Newton's method.

— Further analysis and extension of the higher-order Newton method of Ahmadi, Chaudhry, and Zhang  (2609.01001 - Voert et al., 1 Sep 2026) in Section 1, subsection “Previous research”