A Damped Newton Method Achieves Global $\mathcal O \left(\frac{1}{k^2}\right)$ and Local Quadratic Convergence Rate

Slavomir Hanzely, Dmitry Kamzolov, Dmitry Pasechnyuk, Alexander Gasnikov, Peter Richtárik, Martin Takáč

Keywords: Cubic Newton method, Damped Newton method, fast global convergence, convex optimization

Abstract: In this paper, we present the first stepsize schedule for Newton method resulting in fast global and local convergence guarantees. In particular, we a) prove an $\mathcal O \left( 1/{k^2} \right)$ global rate, which matches the state-of-the-art global rate of cubically regularized Newton method of Polyak and Nesterov (2006) and of regularized Newton method of Mishchenko (2021), and the later variant of Doikov and Nesterov (2021), b) prove a local quadratic rate, which matches the best-known local rate of second-order methods, and c) our stepsize formula is simple, explicit, and does not require solving any subproblem. Our convergence proofs hold under affine-invariant assumptions closely related to the notion of self-concordance. Finally, our method has competitive performance when compared to existing baselines which share the same fast global convergence guarantees.