# A Unifying View of Anchoring via Operator-Side Tikhonov Regularization

arXiv: https://arxiv.org/abs/2605.30905 · HTML: https://mlsanity.com/paper/2605.30905/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 38 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 17 of 17)

- [Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k2) Rate on Squared Gradient Norm](https://mlsanity.com/paper/2102.07922/) (2021)
- [A First Order Method for Solving Convex Bi-Level Optimization Problems](https://mlsanity.com/paper/1702.03999/) (2017)
- [Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2106.02326/) (2021)
- [Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities](https://mlsanity.com/paper/2002.08872/) (2020)
- [Extragradient Method: O(1/K) Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity](https://mlsanity.com/paper/2110.04261/) (2021)
- [Last-Iterate Convergence of Optimistic Gradient Method for Monotone Variational Inequalities](https://mlsanity.com/paper/2205.08446/) (2022)
- [Exact Optimal Accelerated Complexity for Fixed-Point Iterations](https://mlsanity.com/paper/2201.11413/) (2022)
- [Halpern-Type Accelerated and Splitting Algorithms For Monotone Inclusions](https://mlsanity.com/paper/2110.08150/) (2021)
- [Optimal error bounds for nonexpansive fixed-point iterations in normed spaces](https://mlsanity.com/paper/2108.10969/) (2021)
- [Fast Optimistic Gradient Descent Ascent (OGDA) method in continuous and discrete time](https://mlsanity.com/paper/2203.10947/) (2022)
- [On modified Halpern and Tikhonov-Mann iterations](https://mlsanity.com/paper/2203.11003/) (2022)
- [An Improved Last-Iterate Convergence Rate for Anchored Gradient Descent Ascent](https://mlsanity.com/paper/2604.03782/) (2026)
- [Last-Iterate Convergence of Anchored Gradient Descent](https://mlsanity.com/paper/2604.12235/) (2026)
- [Accelerated Extragradient-Type Methods -- Part 2: Generalization and Sublinear Convergence Rates under Co-Hypomonotonicity](https://mlsanity.com/paper/2501.04585/) (2025)
- [Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals](https://mlsanity.com/paper/2406.00852/) (2024)
- [Extra-Gradient Method with Flexible Anchoring: Strong Convergence and Fast Residual Decay](https://mlsanity.com/paper/2410.14369/) (2024)
- [Stochastic Moving Anchor Algorithms and a Popov's Scheme with Moving Anchor](https://mlsanity.com/paper/2506.07290/) (2025)

## Abstract (arXiv)

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/√k) for the regularized forward step, and giving O(1/k) for the regularized EG and PEG variants in the unconstrained monotone Lipschitz setting.