Near-Optimal Second-Order Convex Finite-Sum Optimization under Individual Smoothness

We study the stochastic second-order oracle complexity of solving smooth convex finite-sum optimization with $n$ components. Very recently, Tsedenov & Kovalev (2026) showed that the minimax optimal complexity under averaged smoothness and convexity is $\tildeΘ(n + n^{6/7} ε^{-2/7})$ by establishing nearly matching upper and lower bounds. In this paper, we show that individual smoothness and convexity provably enable further acceleration, where the lower complexity bound is $Ω(n+ n^{5/7} ε^{-2/7})$ under this stronger assumption. Then, by establishing a novel second-order self-bounding inequality, we design a new Taylor-corrected lazy Newton method, which can lead to a better complexity of $\tilde{\mathcal{O}}(n + n^{5/7} ε^{-2/7})$ by applying the optimal Monteiro-Svaiter acceleration, matching our lower bound up to only logarithmic factors.

Source: arXiv:2610.42040v1 - http://arxiv.org/abs/2610.42040v1 PDF: https://arxiv.org/pdf/2610.42040v1 Original Link: http://arxiv.org/abs/2610.42040v1