AI 中文总结
研究使用不精确二阶预言机的凸优化问题，提出加速不精确牛顿外梯度法，给出不同条件下找到ε-解的复杂度，且每次迭代时间与矩阵乘法相当，还建立匹配下界证明方法最优性。
AI 中文摘要
本文提出一种名为加速不精确牛顿外梯度法（AINE）的新型二阶方法用于凸优化，该方法使用δ-不精确海森矩阵。当海森矩阵是L₂-利普希茨连续时，AINE能在不精确二阶预言机（ISO）复杂度为\(\mathcal{O}((\delta/\epsilon)^{1/2}+(L_2/\epsilon)^{2/7})\)时找到一个ε-解；当三阶导数是L₃-利普希茨连续时，复杂度为\(\mathcal{O}((\delta/\epsilon)^{1/2}+(L_3/\epsilon)^{1/5})\)。此外，还建立了两种情况的匹配预言机复杂度下界，证明了方法的最优性。
英文摘要
In this paper, we present a novel second-order method called Accelerated Inexact Newton Extragradient (AINE) for convex optimization using $δ$-inexact Hessians. We show that AINE can find an $ε$-solution in the inexact second-order oracle (ISO) complexity of $\mathcal{O}( (δ/ε)^{1/2} + (L_2/ε)^{2/7} )$ when the Hessian is $L_2$-Lipschitz continuous, and a better complexity of $\mathcal{O}( (δ/ε)^{1/2} + (L_3/ε)^{1/5} )$ when the third-order derivative is $L_3$-Lipschitz continuous. Notably, each iteration of our method can be conducted in the same running time as matrix multiplication up to logarithmic factors. In addition, we also establish matching oracle complexity lower bounds for both setups, demonstrating the optimality of our methods.
