A Unifying View of Anchoring via Operator-Side Tikhonov Regularization

Zihao Chen

arxiv: 2605.30905 · v1 · pith:EOBULVCFnew · submitted 2026-05-29 · 🧮 math.OC · cs.LG

A Unifying View of Anchoring via Operator-Side Tikhonov Regularization

Zihao Chen This is my paper

Pith reviewed 2026-06-28 21:46 UTC · model grok-4.3

classification 🧮 math.OC cs.LG

keywords anchoringTikhonov regularizationmonotone operatorsHalpern iterationextragradientfixed point methodslast-iterate convergenceresidual convergence

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The pith

Anchoring reduces to regularizing the base operator with a vanishing Tikhonov term before running the unmodified iteration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that various anchored methods for fixed-point and monotone equations can be obtained uniformly by adding a vanishing Tikhonov regularization to the operator that the base method queries, then executing the base method without change. This approach reproduces the Halpern iteration when applied to Picard iteration and generates new anchored versions of the forward step, extragradient, and past extragradient methods, with anchor placement following the base method's query structure. The unification allows a shared residual recurrence analysis that recovers known rates and provides new convergence guarantees in the monotone Lipschitz setting.

Core claim

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), it yields three variants whose anchor placements inherit the base method's query pattern. 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.

What carries the argument

Operator-side Tikhonov regularization of the queried operator, where a term that pulls toward an anchor point vanishes over iterations.

Load-bearing premise

The shared residual recurrence holds under the assumption that the operator is monotone and Lipschitz continuous.

What would settle it

A counterexample where the regularized extragradient method fails to achieve O(1/k) residual convergence for some monotone Lipschitz operator would falsify the rate claims.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Operator-side Tikhonov gives a clean unification that recovers Halpern and produces new EG/PEG variants, but the shared residual recurrence for the two-query methods needs the full derivations to confirm it carries the stated rates without extra terms.

read the letter

The paper's central move is to regularize the operator that the base method queries with a vanishing Tikhonov term λ_k(· - x0) and then run the unmodified iteration. This turns the plain Picard iteration into Halpern and produces anchored versions of forward-step, EG, and PEG whose anchor placement follows the base method's query locations. The abstract claims the four resulting analyses all reduce to the same residual recurrence, which delivers the usual O(1/k) last-iterate residual rate for Halpern and the new EG/PEG variants and O(1/sqrt(k)) for the forward-step case under monotone Lipschitz assumptions.

What works is the conceptual shift: instead of hand-crafting anchor terms at the update level, the construction keeps the algorithm skeleton fixed and moves the anchoring into the operator. That produces three new regularized methods and a fresh residual guarantee for the forward-step case. The motivation from operator properties is direct and avoids the circularity of fitting parameters to a target rate.

The soft spot is exactly the one the stress-test flags. EG and PEG each evaluate the operator at two distinct points per step. Regularizing both evaluations inserts cross terms into the monotonicity or inner-product expansions that do not appear in the single-query Picard or forward-step cases. The abstract asserts that the same residual recurrence still holds and yields the O(1/k) bound, but without the explicit derivations it is not obvious whether those extra terms are absorbed without changing the telescoping argument or adding assumptions. The vanishing schedule for λ_k is left as a free parameter, which is standard but should be stated explicitly with the conditions that make the recurrence close.

The work is for researchers who already follow last-iterate analysis in monotone operator theory. It is worth sending to peer review: the unification is new on its own terms, the new variants are well-defined, and the claimed rates are falsifiable once the proofs are checked. A referee can settle whether the recurrence transfers cleanly to the two-query setting.

Referee Report

2 major / 2 minor

Summary. The paper claims 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 reproduces the Halpern iteration; applied to the forward step, extragradient (EG), and past extragradient (PEG) it yields three new variants whose anchor placements inherit the base method's query pattern. The four analyses share a residual recurrence recovering the O(1/k) Halpern residual-norm rate, O(1/√k) for the regularized forward step, and O(1/k) for the regularized EG and PEG variants under monotone + Lipschitz assumptions in the unconstrained setting.

Significance. If the shared residual recurrence is shown to hold for the dual-query EG and PEG cases, the operator-side Tikhonov view supplies a systematic, conceptually clean unification that reproduces known anchored methods as special cases and generates new last-iterate convergent algorithms. The construction is parameter-free once the vanishing schedule is fixed and directly inherits the query pattern of each base method, which is a genuine strength.

major comments (2)

[Sections 4–5 (regularized EG/PEG analyses and shared recurrence)] The central claim rests on the four analyses sharing one residual recurrence (abstract and the analyses in Sections 3–5). For the regularized EG and PEG variants the operator is queried at two distinct points per iteration; the monotonicity expansion therefore produces cross terms absent from the single-query Picard and forward-step cases. The manuscript must explicitly verify that these cross terms are absorbed by the same recurrence without changing the telescoping argument or weakening the O(1/k) bound; the current derivation gap leaves the transfer of the recurrence to the dual-query methods unconfirmed.
[§3.1 (forward-step recurrence)] §3.1, the residual recurrence for the regularized forward-step variant: the O(1/√k) rate is stated to follow from the same recurrence used for the O(1/k) cases, yet the forward-step analysis requires a different step-size regime. The manuscript should clarify whether the recurrence is applied verbatim or modified, and confirm that the modification does not rely on fitting the vanishing schedule to the target rate.

minor comments (2)

[Notation section and §4] Notation for the regularized operator F_λ should be introduced once and used consistently; the current alternation between F + λ(·−x0) and the subscripted form is occasionally ambiguous when two query points appear.
[Introduction and §2] The vanishing schedule λ_k is listed among the free parameters; a brief remark on admissible schedules (e.g., λ_k = 1/k or λ_k = 1/√k) that preserve the claimed rates would help readers reproduce the results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our paper. We address each major comment below and will make the necessary revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Sections 4–5 (regularized EG/PEG analyses and shared recurrence)] The central claim rests on the four analyses sharing one residual recurrence (abstract and the analyses in Sections 3–5). For the regularized EG and PEG variants the operator is queried at two distinct points per iteration; the monotonicity expansion therefore produces cross terms absent from the single-query Picard and forward-step cases. The manuscript must explicitly verify that these cross terms are absorbed by the same recurrence without changing the telescoping argument or weakening the O(1/k) bound; the current derivation gap leaves the transfer of the recurrence to the dual-query methods unconfirmed.

Authors: We agree that an explicit verification of the cross terms for the dual-query methods is beneficial for clarity. In the original derivation, the cross terms arising from the two query points are bounded using the monotonicity of the regularized operator and the Lipschitz continuity, allowing them to be absorbed into the main terms of the recurrence without affecting the telescoping or the rate. To address the concern, we will add a detailed expansion and bounding step in Sections 4 and 5 to make this absorption explicit. revision: yes
Referee: [§3.1 (forward-step recurrence)] §3.1, the residual recurrence for the regularized forward-step variant: the O(1/√k) rate is stated to follow from the same recurrence used for the O(1/k) cases, yet the forward-step analysis requires a different step-size regime. The manuscript should clarify whether the recurrence is applied verbatim or modified, and confirm that the modification does not rely on fitting the vanishing schedule to the target rate.

Authors: The residual recurrence is applied verbatim in §3.1. The difference in rates stems from the subsequent analysis of the recurrence inequality, which employs a different step-size schedule (constant step-size for the O(1/√k) case). The vanishing Tikhonov parameter schedule is fixed independently to ensure it vanishes at the appropriate rate and is not adjusted to fit the convergence rate. We will revise §3.1 to explicitly state that the recurrence is unchanged and clarify the distinction in the analysis step. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and analyses are independent

full rationale

The paper defines an operator-side Tikhonov regularization applied to base methods (Picard, forward step, EG, PEG) and then derives convergence via a shared residual recurrence under monotone+Lipschitz assumptions. No equations or claims reduce a target rate or uniqueness result to a fitted parameter, self-citation, or redefinition of the output in terms of itself. The four analyses are presented as separate verifications that happen to share a recurrence structure; this is standard mathematical organization rather than circularity. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions from monotone operator theory together with a specific choice of vanishing regularization schedule whose precise form is not detailed in the abstract.

free parameters (1)

vanishing schedule for Tikhonov parameter
The rate at which the regularization term vanishes must be chosen to obtain the claimed residual rates; this choice is not derived from first principles in the visible text.

axioms (1)

domain assumption The operator is monotone and Lipschitz continuous
Required to obtain the O(1/k) and O(1/sqrt(k)) residual rates for the regularized variants in the unconstrained setting.

pith-pipeline@v0.9.1-grok · 5744 in / 1569 out tokens · 38493 ms · 2026-06-28T21:46:49.208642+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages · 2 internal anchors

[1]

Stochastic moving an- chor algorithms and a Popov’s scheme with moving anchor.arXiv preprint arXiv:2506.07290,

[ACS25] James Alcala, Yat Tin Chow, and Mahesh Sunkula. Stochastic moving an- chor algorithms and a Popov’s scheme with moving anchor.arXiv preprint arXiv:2506.07290,

work page arXiv
[2]

Extra-gradient method with flexi- ble anchoring: Strong convergence and fast residual decay.arXiv preprint arXiv:2410.14369,

[BC24] Radu Ioan Bot ¸ and Enis Chenchene. Extra-gradient method with flexi- ble anchoring: Strong convergence and fast residual decay.arXiv preprint arXiv:2410.14369,

work page arXiv
[3]

[BN24] Radu Ioan Bot ¸ and Dang-Khoa Nguyen. Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals.arXiv preprint arXiv:2406.00852,

work page arXiv
[4]

Last-Iterate Convergence of Anchored Gradient Descent

[CZ26] Yang Cai and Weiqiang Zheng. Last-iterate convergence of anchored gradient descent.arXiv preprint arXiv:2604.12235,

work page internal anchor Pith review Pith/arXiv arXiv
[5]

[SST+26] Anja Surina, Arun Suggala, George Tsoukalas, Anton Kovsharov, Sergey Shi- robokov, Francisco J. R. Ruiz, Pushmeet Kohli, and Swarat Chaudhuri. An improved last-iterate convergence rate for anchored gradient descent ascent. arXiv preprint arXiv:2604.03782,

work page internal anchor Pith review Pith/arXiv arXiv
[6]

Halpern-type accelerated and splitting algorithms for monotone inclusions

[TDL21] Quoc Tran-Dinh and Yang Luo. Halpern-type accelerated and splitting algo- rithms for monotone inclusions.arXiv preprint arXiv:2110.08150,

work page arXiv
[7]

Tran-Dinh and N

[TDNT25] Quoc Tran-Dinh and Nghia Nguyen-Trung. Accelerated extragradient-type methods – part 2: Generalization and sublinear convergence rates under co- hypomonotonicity.arXiv preprint arXiv:2501.04585,

work page arXiv
[8]

Regularization and iterative methods for monotone variational inequalities.Fixed Point Theory and Applications, 2010:765206,

36 [XX10] Xiubin Xu and Hong-Kun Xu. Regularization and iterative methods for monotone variational inequalities.Fixed Point Theory and Applications, 2010:765206,

2010

[1] [1]

Stochastic moving an- chor algorithms and a Popov’s scheme with moving anchor.arXiv preprint arXiv:2506.07290,

[ACS25] James Alcala, Yat Tin Chow, and Mahesh Sunkula. Stochastic moving an- chor algorithms and a Popov’s scheme with moving anchor.arXiv preprint arXiv:2506.07290,

work page arXiv

[2] [2]

Extra-gradient method with flexi- ble anchoring: Strong convergence and fast residual decay.arXiv preprint arXiv:2410.14369,

[BC24] Radu Ioan Bot ¸ and Enis Chenchene. Extra-gradient method with flexi- ble anchoring: Strong convergence and fast residual decay.arXiv preprint arXiv:2410.14369,

work page arXiv

[3] [3]

[BN24] Radu Ioan Bot ¸ and Dang-Khoa Nguyen. Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals.arXiv preprint arXiv:2406.00852,

work page arXiv

[4] [4]

Last-Iterate Convergence of Anchored Gradient Descent

[CZ26] Yang Cai and Weiqiang Zheng. Last-iterate convergence of anchored gradient descent.arXiv preprint arXiv:2604.12235,

work page internal anchor Pith review Pith/arXiv arXiv

[5] [5]

[SST+26] Anja Surina, Arun Suggala, George Tsoukalas, Anton Kovsharov, Sergey Shi- robokov, Francisco J. R. Ruiz, Pushmeet Kohli, and Swarat Chaudhuri. An improved last-iterate convergence rate for anchored gradient descent ascent. arXiv preprint arXiv:2604.03782,

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

Halpern-type accelerated and splitting algorithms for monotone inclusions

[TDL21] Quoc Tran-Dinh and Yang Luo. Halpern-type accelerated and splitting algo- rithms for monotone inclusions.arXiv preprint arXiv:2110.08150,

work page arXiv

[7] [7]

Tran-Dinh and N

[TDNT25] Quoc Tran-Dinh and Nghia Nguyen-Trung. Accelerated extragradient-type methods – part 2: Generalization and sublinear convergence rates under co- hypomonotonicity.arXiv preprint arXiv:2501.04585,

work page arXiv

[8] [8]

Regularization and iterative methods for monotone variational inequalities.Fixed Point Theory and Applications, 2010:765206,

36 [XX10] Xiubin Xu and Hong-Kun Xu. Regularization and iterative methods for monotone variational inequalities.Fixed Point Theory and Applications, 2010:765206,

2010