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arXiv · 2610.03222

Near-Optimal Convex Optimization with Lazy Second-Order Oracles

Xinliang Zhang·Lesi Chen·Chengchang Liu·Jingzhao Zhang
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Abstract

This paper studies the complexity of convex optimization using lazy second-order oracles (Doikov, Chayti, and Jaggi, ICML 2023), where an algorithm queries gradients every iteration and Hessians once per $m$ iterations. Under this setting, we show a lower bound of $Ω(m+ m^{1/7} ε^{-2/7})$ on the number of total iterations to find an $ε$-solution using a novel block zero-chain construction. Then we propose a novel method that achieves a new upper bound of $\tilde{\mathcal{O}}(m+ m^{1/7} ε^{-2/7})$, which significantly improves the prior one (Chen, Liu, Luo, and Zhang, COLT 2026) of $\tilde{\mathcal{O}}(m+ m^{13/21} ε^{-2/7})$ and is tight up to logarithmic factors.

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Xinliang Zhang, Lesi Chen, Chengchang Liu, Jingzhao Zhang. 2026-10-02. Near-Optimal Convex Optimization with Lazy Second-Order Oracles. https://arxiv.org/abs/2610.03222

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