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arxiv: 2203.04869 · v3 · submitted 2022-03-09 · 🧮 math.OC

From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems

Quoc Tran-Dinh This is my paper

Pith reviewed 2026-05-24 12:11 UTC · model grok-4.3

classification 🧮 math.OC
keywords Halpern iterationNesterov accelerationfixed point iterationroot finding problemsmonotone inclusionsextra-anchored gradient methodsconvergence ratesco-coercive operators
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The pith

Halpern's fixed-point iteration is equivalent to Nesterov's accelerated scheme via a simple transformation for solving co-coercive equations.

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

The paper establishes an equivalence between Halpern's fixed-point iteration and a Nesterov-accelerated form for root-finding problems with co-coercive operators. This equivalence provides a straightforward convergence proof for the accelerated scheme and derives a new convergence rate for Halpern's method. It is then used to create accelerated versions of extra-anchored gradient methods for monotone inclusions that require only monotonicity and Lipschitz continuity rather than co-coerciveness. Numerical examples confirm that observed rates align with the theoretical predictions.

Core claim

Halpern's fixed-point iteration scheme for solving a co-coercive equation is equivalent to Nesterov's accelerated interpretation through a simple transformation. This equivalence leads to a straightforward convergence proof for Nesterov's accelerated scheme and, as a consequence, a new convergence rate for Halpern's fixed-point iteration. The same approach produces new Nesterov's accelerated variants for extra-anchored gradient and past-extra anchored gradient methods to solve monotone inclusions under the weaker assumption of monotonicity and Lipschitz continuity.

What carries the argument

The simple transformation establishing equivalence between Halpern's fixed-point iteration and Nesterov's accelerated scheme.

If this is right

  • Halpern's fixed-point iteration obtains a new convergence rate.
  • New accelerated variants of extra-anchored gradient methods are derived for monotone inclusions.
  • The accelerated past-extra anchored gradient method uses two past-iterate correction terms.
  • The formulation can guide development of new accelerated methods for minimax problems without requiring co-coerciveness.
  • Theoretical convergence rates are validated by numerical experiments matching up to a constant factor.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence suggests that acceleration techniques can be transferred between different classes of iterative methods for root-finding.
  • Similar transformations might be applied to other fixed-point schemes to uncover hidden acceleration structures.
  • Extending this to continuous-time dynamical systems could reveal new accelerated flows for solving inclusions.
  • These variants may improve practical performance in applications involving monotone operators where co-coerciveness fails.

Load-bearing premise

The operator is co-coercive for the original gradient scheme or monotone and Lipschitz continuous for the new variants.

What would settle it

A numerical counterexample where the transformed Nesterov scheme fails to converge at the predicted rate for a co-coercive operator would disprove the equivalence.

Figures

Figures reproduced from arXiv: 2203.04869 by Quoc Tran-Dinh.

Figure 1
Figure 1. Figure 1: The convergence behavior of the two Nesterov’s accelerated variants on two problem in￾stances. Left: Case 1 with (n, p) = (500, 1000), and Right: Case 2 with (n, p) = (1000, 1000) [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The performance of Nesterov’s accelerated variants of EAG and PEAG on (64). Left: Case 1, and Right: Case 2 [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
read the original abstract

We derive an equivalent form of Halpern's fixed-point iteration scheme for solving a co-coercive equation (also called a root-finding problem), which can be viewed as a Nesterov's accelerated interpretation. We show that one method is equivalent to another via a simple transformation, leading to a straightforward convergence proof for Nesterov's accelerated scheme. Alternatively, we directly establish convergence rates of Nesterov's accelerated variant, and as a consequence, we obtain a new convergence rate of Halpern's fixed-point iteration. Next, we apply our results to different methods to solve monotone inclusions, where our convergence guarantees are applied. Since the gradient/forward scheme requires the co-coerciveness of the underlying operator, we derive new Nesterov's accelerated variants for both recent extra-anchored gradient and past-extra anchored gradient methods in the literature. These variants alleviate the co-coerciveness condition by only assuming the monotonicity and Lipschitz continuity of the underlying operator. Interestingly, our new Nesterov's accelerated interpretation of the past-extra anchored gradient method involves two past-iterate correction terms. This formulation is expected to guide us developing new Nesterov's accelerated methods for minimax problems and their continuous views without co-coericiveness. We test our theoretical results on two numerical examples, where the actual convergence rates match well the theoretical ones up to a constant factor.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes an equivalence between Halpern's fixed-point iteration and a Nesterov-accelerated scheme for solving co-coercive root-finding problems (equations) via a simple transformation. This equivalence yields a straightforward convergence proof for the accelerated form and, as a consequence, a new convergence rate for Halpern iteration. The same framework is applied to monotone inclusions to derive new Nesterov-accelerated variants of extra-anchored gradient and past-extra-anchored gradient methods; these variants require only monotonicity and Lipschitz continuity of the operator rather than co-coerciveness. The theoretical rates are illustrated on two numerical examples, where observed convergence matches the predicted rates up to a constant factor.

Significance. If the algebraic equivalence and the resulting rates hold under the stated operator assumptions, the work supplies a direct bridge between classical fixed-point iterations and accelerated schemes, permitting transfer of convergence analyses and the construction of accelerated methods for a broader class of monotone inclusions. The explicit transformation and the two-correction-term formulation for the past-extra-anchored variant are concrete strengths that could guide further development for minimax problems. The paper credits the algebraic nature of the equivalence and supplies numerical corroboration rather than relying on it as primary evidence.

minor comments (3)
  1. [Abstract] Abstract, line 12: 'co-coericiveness' is a typographical error and should read 'co-coerciveness'.
  2. [Abstract] Abstract, line 10: the phrase 'past-extra anchored gradient' should be hyphenated consistently as 'past-extra-anchored gradient' to match the later usage in the text.
  3. [§2 or §3] The manuscript would benefit from an explicit statement, early in §2 or §3, of the precise transformation that maps one iteration to the other (including the auxiliary sequence definitions), so that the equivalence can be verified without reconstructing it from the convergence proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript, the accurate description of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central derivation is an algebraic equivalence between Halpern iteration and a Nesterov-accelerated form obtained by direct transformation of the iterates. Convergence rates follow from standard analysis under the stated operator assumptions (co-coercivity or monotonicity+Lipschitz). New variants for extra-anchored methods are constructed explicitly from the equivalence and do not reduce to fitted inputs or prior self-citations. Numerical examples serve only as corroboration. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of monotone and co-coercive operators from convex analysis; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption The operator is co-coercive or monotone and Lipschitz continuous.
    Invoked when the paper distinguishes the gradient scheme (needs co-coerciveness) from the new variants (only monotonicity + Lipschitz).
  • standard math Fixed-point iterations converge under the stated operator conditions.
    Background result from monotone operator theory used to obtain the rates.

pith-pipeline@v0.9.0 · 5770 in / 1308 out tokens · 18469 ms · 2026-05-24T12:11:57.143207+00:00 · methodology

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Forward citations

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