Near-Optimal Second-Order Convex Finite-Sum Optimization under Individual Smoothness
Xinliang ZhangLesi ChenZonghan LiJingzhao Zhang
Nov 2026
Artificial IntelligenceData Science
Abstract
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 $\tilde{\Theta}(n + n^{6/7} \epsilon^{-2/7})$ 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 $\Omega(n+ n^{5/7} \epsilon^{-2/7})$ 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 $\tilde{\mathcal{O}}(n + n^{5/7} \epsilon^{-2/7})$ by applying the optimal Monteiro-Svaiter acceleration, matching our lower bound up to only logarithmic factors.
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