Superfast Second-Order Methods for Unconstrained Convex Optimization

Abstract In this paper, we present new second-order methods with convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This faster than existing lower bound for type of schemes (Agarwal and Hazan in Proceedings 31st conference on learning theory, PMLR, pp. 774–792, 2018; Arjevani Shiff Math Program 178(1–2):327–360, 2019), which k^{-7/2} \right) 7 / 2 . Our progress can be explained by a finer specification problem class. The main idea approach consists implementation third-order scheme from Nesterov (Math 186:157–183, 2021) using oracle. At each our method, solve nontrivial auxiliary linearly convergent based relative non-degeneracy condition (Bauschke et al. Oper Res 42:330–348, 2016; Lu SIOPT 28(1):333–354, 2018). During process, Hessian objective function computed once, gradient \ln {1 \over \epsilon }\right) ln 1 ? times, $$\epsilon desired accuracy solution problem.