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We establish a novel $\Omega(m+ m^{1/7} \epsilon^{-2/7})$ lower bound for convex optimization with lazy second-order oracles, proving that existing algorithms are suboptimal. A new method is then proposed, achieving a near-optimal $\tilde{\mathcal{O}}(m+ m^{1/7} \epsilon^{-2/7})$ iteration complexity, which significantly improves upon prior $\tilde{\mathcal{O}}(m+ m^{13/21} \epsilon^{-2/7})$ bounds. This work tightens the theoretical understanding of optimization under limited Hessian queries.
Lazy second-order optimization is far more efficient than previously thought, with a new bound achieving near-optimality.
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.
