Regularized Overestimated Newton

We propose Regularized Overestimated Newton (RON), a Newton-type method with low per-iteration cost and strong global and local convergence guarantees for smooth convex optimization. RON interpolates between gradient descent and globally regularized Newton, with behavior determined by the largest Hessian overestimation error. Globally, when the optimality gap of the objective is large, RON achieves an accelerated $O(n^{-2})$ convergence rate; when small, its rate becomes $O(n^{-1})$. Locally, RON converges superlinearly and linearly when the overestimation is exact and inexact, respectively, toward possibly non-isolated minima under the local Quadratic Growth (QG) condition. The linear rate is governed by an improved effective condition number depending on the overestimation error. Leveraging a recent randomized rank-$k$ Hessian approximation algorithm, we obtain a practical variant with $O(\text{dim}\cdot k^2)$ cost per iteration. When the Hessian rank is uniformly below $k$, RON achieves a per-iteration cost comparable to that of first-order methods while retaining the superior convergence rates even in degenerate local landscapes. We validate our theoretical findings through experiments on entropic optimal transport and inverse problems.