锚定额外近端方法：单调包含问题的最优高阶方法
Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems
- Google Research（谷歌研究院）
- Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出锚定额外近端框架，结合锚定外推与相对误差非精确近端更新，构造高阶方法，在切向残差准则下达到最优预言机复杂度，并给出匹配下界。
AI中文摘要：
我们研究了在切向残差准则下，寻找由光滑单值单调算子与极大单调集值算子之和构成的复合单调包含问题的近似解的确定性预言机复杂度。我们引入了锚定额外近端（AEP）框架，该框架将锚定外推步骤与满足相对误差条件的非精确锚定近端更新相结合。该框架在一阶设置中恢复了复合快速外梯度方法，并通过将隐式更新中的算子替换为其在外推点的泰勒近似，自然地产生了二阶及更高阶的扩展。对于每个$p\geq 2$，假设单值算子的$(p-1)$阶导数是Lipschitz连续的，我们将此构造与二分线搜索相结合，得到一种$p$阶方法，该方法在$\widetilde{O}(\varepsilon^{-2/(3p-1)})$次预言机调用内找到切向残差至多为$\varepsilon$的点。这改进了所有先前$p$阶方法的上界：特别是，它改进了先前已知的最佳$\widetilde{O}(\varepsilon^{-1/p})$切向残差复杂度，以及在较弱的对偶间隙准则下高阶混合近端外梯度方法的经典$O(\varepsilon^{-2/(p+1)})$界。我们通过一个最坏情况下的下界$\Omega(\varepsilon^{-2/(3p-1)})$来补充这一结果，该下界适用于$p$阶预言机模型中的任何确定性算法，而不限制算法采用张量步骤或任何其他规定的更新结构。因此，对于所有$p\geq 2$，所提出的方法在$\varepsilon$上达到了最优依赖关系，直至对数因子。
英文摘要：
We study the deterministic oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximally monotone set-valued operator, under the tangent-residual criterion. We introduce the Anchored Extra-Proximal (AEP) framework, which combines an anchored extrapolation step with an inexact anchored proximal update satisfying a relative-error condition. The framework recovers the composite Fast Extragradient method in the first-order setting and yields natural second- and higher-order extensions by replacing the operator in the implicit update with its Taylor approximation at the extrapolated point. For every $p\geq 2$, assuming that the $(p-1)$th derivative of the single-valued operator is Lipschitz continuous, we combine this construction with a bisection line search to obtain a $p$th-order method that finds a point with tangent residual at most $\varepsilon$ in $\widetilde{O}(\varepsilon^{-2/(3p-1)})$ oracle calls. This improves all prior upper bounds for $p$th-order methods: in particular, it improves the previous best-known $\widetilde{O}(\varepsilon^{-1/p})$ tangent-residual complexity as well as the classical $O(\varepsilon^{-2/(p+1)})$ bound of higher-order hybrid proximal extragradient methods under the weaker duality-gap criterion. We complement this result with a worst-case lower bound of $Ω(\varepsilon^{-2/(3p-1)})$ for every deterministic algorithm in the $p$th-order oracle model, without restricting the algorithm to tensor steps or any other prescribed update structure. Thus, the proposed method attains the optimal dependence on $\varepsilon$, up to logarithmic factors, for all $p\geq2$.
