🤖 AI Summary
Existing anchoring algorithms suffer from a lack of unified perspective due to their reliance on method-specific anchor constructions, hindering systematic understanding and generalization. This work proposes an operator-side Tikhonov regularization framework that unifies diverse anchoring methods by simply incorporating a vanishing regularizer into the base operator and running the original algorithm unchanged. The framework reveals the intrinsic commonality among methods such as Halpern iteration and extrapolated anchored gradient schemes, and naturally yields new variants. Theoretically, it recovers Halpern iteration with an $O(1/k)$ residual convergence rate, establishes a novel $O(1/\sqrt{k})$ guarantee for forward-step methods, and—under unconstrained monotone Lipschitz settings—achieves, for the first time, an $O(1/k)$ convergence rate for both the extragradient (EG) and past extragradient (PEG) methods.
📝 Abstract
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.