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This research advances the theoretical efficiency of convex optimization by establishing tight complexity bounds for algorithms utilizing lazy second-order oracles, which query Hessians less frequently than gradients. By matching a novel lower bound with an improved upper bound, the work provides fundamental insights into the computational limits of scalable machine learning training methods that rely on Hessian information.
Near-Optimal Convex Optimization with Lazy Second-Order Oracles
arXiv:2610.03222v1 Announce Type: cross 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 $\Omega(m+ m^{1/7} \epsilon^{-2/7})$ on the number of total iterations to find an $\epsilon$-solution using a novel block zero-chain construction. Then we propose a novel method that achieves a new upper bound of $\tilde{\mathcal{O}}(m+ m^{1/7} \epsilon^{-2/7})$, which significantly improves the prior one (Chen, Liu, Luo, and Zhang, COLT 2026) of $\tilde{\mathcal{O}}(m+ m^{13/21} \epsilon^{-2/7})$ and is tight up to logarithmic factors.
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 $\Omega(m+ m^{1/7} \epsilon^{-2/7})$ on the number of total iterations to find an $\epsilon$-solution using a novel block zero-chain construction. Then we propose a novel method that achieves a new upper bound of $\tilde{\mathcal{O}}(m+ m^{1/7} \epsilon^{-2/7})$, which significantly improves the prior one (Chen, Liu, Luo, and Zhang, COLT 2026) of $\tilde{\mathcal{O}}(m+ m^{13/21} \epsilon^{-2/7})$ and is tight up to logarithmic factors.
Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2610.03222 [math.OC]
(or arXiv:2610.03222v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2610.03222 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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