统一了多种锚定方法的构造原理，用Tikhonov正则化实现更清晰的算法设计。
A Unifying View of Anchoring via Operator-Side Tikhonov Regularization
- 通过在算子上添加衰减的Tikhonov正则项，统一构建锚定机制。
- 在无约束单调利普希茨条件下，收敛速度达O(1/k)或O(1/√k)。
- 适用于研究迭代算法收敛性与结构设计的学者，尤其关注稳定更新的场景。
锚定不动点与单调方程方法（如Halpern迭代、额外锚定梯度等）通过引入趋近参考点的衰减拉力，获得末次迭代保证。现有锚定变体虽具优异收敛性，但锚点位置在更新层面常依赖算法特定且概念模糊。本文证明：锚定可统一为单一算子侧构造——对基方法查询的算子施加衰减Tikhonov正则项，再运行未经修改的基方法。应用于Picard迭代，复现Halpern迭代；应用于前向步、外梯度（EG）、过去外梯度（PEG，又称Popov法），生成三种新变体，其锚点位置继承基方法的查询模式。前向步实例给出新的残差收敛保证，而EG与PEG实例产生新正则化版本。四类分析共享一个残差递推关系，恢复出Halpern的$O(1/k)$残差范数收敛率，对正则化前向步为$O(1/\\/sqrt{k})$，对正则化EG与PEG在无约束单调利普希茨设定下为$O(1/k)$。
原文摘要 · Abstract (English)
Anchored fixed point and monotone equation methods, including Halpern iteration, extra anchored gradient, and their relatives, add a vanishing pull toward a reference point to obtain last-iterate guarantees. Existing anchored variants often achieve sharp last-iterate guarantees, but from the update-level perspective the placement of the anchor can be algorithm-specific and conceptually opaque. We show that anchoring admits a single operator-side construction: regularize the operator queried by the base method with a vanishing Tikhonov term, then run the unmodified base method. Applied to the Picard iteration, this recipe reproduces the Halpern iteration; applied to the forward step, extragradient (EG), and past extragradient (PEG, also known as Popov's method), it yields three variants whose anchor placements inherit the base method's query pattern. The forward-step instantiation gives a new residual convergence guarantee, while the EG and PEG instantiations give new regularized variants. The four analyses share a residual recurrence, recovering the $O(1/k)$ Halpern residual-norm convergence rate, giving $O(1/\sqrt{k})$ for the regularized forward step, and giving $O(1/k)$ for the regularized EG and PEG variants in the unconstrained monotone Lipschitz setting.
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