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arxiv: 2604.21443 · v1 · submitted 2026-04-23 · 🧮 math.OC

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Mini-Batch Stochastic Halpern Algorithm for Nonexpansive Fixed point Problems

Hideaki Iiduka
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Pith reviewed 2026-05-09 21:37 UTC · model grok-4.3

classification 🧮 math.OC
keywords nonexpansive mappingfixed point problemHalpern algorithmmini-batch stochasticmean square convergencestochastic approximationconvergence rateoptimization
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The pith

The mini-batch stochastic Halpern algorithm converges in mean square to the closest fixed point of a nonexpansive mapping.

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

The Halpern algorithm approximates the fixed point of a nonexpansive mapping that is closest to the starting point. For large-scale problems the full mapping evaluation is too costly, so this paper replaces each evaluation with an average over a small random mini-batch drawn from an unbiased stochastic oracle. With step sizes that shrink to zero and batch sizes that grow, the sequence of iterates converges in mean square to the desired fixed point. The rate of convergence depends on how fast the step sizes diminish.

Core claim

The paper introduces the mini-batch stochastic Halpern algorithm, which at each step computes a mini-batch estimate of the nonexpansive mapping and applies the Halpern update using diminishing step sizes and increasing batch sizes. It proves that the iterates converge in mean square to the projection of the initial point onto the fixed point set, and provides a convergence rate analysis showing dependence on the step-size schedule.

What carries the argument

The mini-batch stochastic Halpern iteration, which uses an unbiased stochastic oracle to approximate the nonexpansive mapping by averaging over a batch of size growing with iteration number, combined with step sizes alpha_n tending to zero.

If this is right

  • The algorithm becomes usable on large-scale problems where full mapping evaluations would be prohibitive.
  • Convergence holds in the mean-square sense, so the expected squared distance to the target fixed point tends to zero.
  • Convergence speed can be tuned by the choice of the diminishing step-size sequence.
  • The method keeps the original Halpern guarantee of selecting the fixed point closest to the initial point.

Where Pith is reading between the lines

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

  • The same mini-batch stochastic replacement could be applied to other fixed-point schemes such as the Mann iteration.
  • The technique may transfer to stochastic variational inequalities and equilibrium problems that rely on nonexpansive operators.
  • Implementations face a trade-off between per-step cost savings from small batches and the eventual need for larger batches to control variance.

Load-bearing premise

The stochastic oracle must be unbiased so that its expectation recovers the true nonexpansive mapping, and the step sizes and batch sizes must decay and grow at rates that drive the accumulated error to zero.

What would settle it

For a known nonexpansive mapping with an explicit fixed point set, such as a linear operator in low-dimensional Euclidean space, run the algorithm and check whether the expected squared distance of the iterates to the closest fixed point fails to approach zero.

read the original abstract

The Halpern algorithm is a powerful fixed point approximation method for finding the closest point in the fixed point set of a nonexpansive mapping to the initial point. However, in practice, it is not necessarily true that this algorithm can be applied to large-scale fixed point problems, since the computation of the nonexpansive mapping is expensive. In this paper, we present mini-batch stochastic Halpern algorithm to resolve the issue caused by the computational difficulty of the mapping. We preform a convergence analysis demonstrating that the algorithm with diminishing step sizes and increasing batch sizes converges in mean square to the closest point in the fixed point set to the initial point. We also perform a convergence rate analysis demonstrating that convergence speed of the algorithm depends on the settings of the diminishing step sizes.

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

2 major / 2 minor

Summary. The manuscript proposes a mini-batch stochastic variant of the Halpern iteration for approximating the fixed point of a nonexpansive mapping that is closest to the initial point. The algorithm replaces exact evaluations of the mapping with unbiased mini-batch stochastic oracles. The authors claim to prove mean-square convergence to the desired fixed point when the step-size sequence diminishes and the batch-size sequence increases, together with a rate analysis that depends on the specific choice of diminishing steps.

Significance. If the convergence theorem holds with explicit, verifiable conditions on the step and batch sequences, the work supplies a practical stochastic extension of a classical fixed-point method that is relevant for large-scale problems in optimization and variational inequalities. The mean-square convergence result follows standard stochastic-approximation arguments and the rate analysis could guide parameter selection, but the contribution would be incremental rather than transformative unless the rates improve upon existing stochastic Halpern or Krasnosel'skii-Mann analyses.

major comments (2)
  1. [Abstract] The abstract states that convergence holds 'with diminishing step sizes and increasing batch sizes' but supplies neither the explicit schedules (e.g., α_k = 1/k^β or batch size b_k = k^γ) nor the precise summability conditions required for the mean-square error to vanish. Without these, the central claim cannot be verified from the given statement.
  2. [Abstract] The convergence-rate analysis is said to depend on the settings of the diminishing step sizes, yet no explicit rate (e.g., O(1/k) or O(1/sqrt(k))) is stated in the abstract and no comparison is made to the deterministic Halpern rate or to existing stochastic fixed-point methods. This weakens the practical utility of the rate result.
minor comments (2)
  1. [Abstract] The abstract contains a typographical error: 'preform' should be 'perform'.
  2. [Title] The title uses 'fixed point' as two words; consistency with the abstract's usage of 'fixed point' is fine, but the hyphenated form 'fixed-point' would be preferable in the title for standard mathematical English.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and the recommendation of minor revision. The suggestions regarding the abstract are constructive, and we will revise the manuscript to improve the precision and utility of the abstract while preserving its high-level nature.

read point-by-point responses

  1. Referee: [Abstract] The abstract states that convergence holds 'with diminishing step sizes and increasing batch sizes' but supplies neither the explicit schedules (e.g., α_k = 1/k^β or batch size b_k = k^γ) nor the precise summability conditions required for the mean-square error to vanish. Without these, the central claim cannot be verified from the given statement.

    Authors: We agree that the abstract would benefit from greater specificity to allow direct verification. The convergence theorem in the manuscript provides explicit conditions on the sequences, including summability requirements such as ∑ α_k = ∞ and ∑ α_k²/b_k < ∞ to ensure the mean-square error vanishes. In the revised version we will update the abstract to state representative schedules (for instance α_k = 1/k and b_k = k) together with the key summability conditions drawn from the theorem. revision: yes

  2. Referee: [Abstract] The convergence-rate analysis is said to depend on the settings of the diminishing step sizes, yet no explicit rate (e.g., O(1/k) or O(1/sqrt(k))) is stated in the abstract and no comparison is made to the deterministic Halpern rate or to existing stochastic fixed-point methods. This weakens the practical utility of the rate result.

    Authors: We acknowledge that the abstract does not quote an explicit rate or include comparisons. The rate analysis in Section 4 derives explicit orders that depend on the choice of diminishing step sizes, recovering the same leading term as the deterministic Halpern iteration when the batch size grows appropriately. We will revise the abstract to state a concrete rate (for example O(1/k) under α_k ∼ 1/k) and add a brief comparison noting that the stochastic mini-batch version attains the deterministic rate order while reducing per-iteration cost. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper describes a standard stochastic extension of the Halpern iteration for nonexpansive mappings, using an unbiased mini-batch oracle together with diminishing steps and growing batch sizes to prove mean-square convergence to the nearest fixed point. No equations or parameters are defined in terms of the target convergence result, no fitted quantities are relabeled as predictions, and no load-bearing steps reduce to self-citations or ansatzes imported from the authors' prior work. The analysis is self-contained against external benchmarks of stochastic approximation theory and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; the ledger therefore records only the assumptions explicitly named or implied by the abstract. The central claim rests on the nonexpansive property and on the existence of a stochastic oracle whose expectation recovers the full mapping, plus unspecified conditions on step and batch sizes.

free parameters (2)
  • diminishing step-size sequence
    The abstract states that convergence holds for 'diminishing step sizes' but gives neither the functional form nor any concrete sequence.
  • increasing batch-size sequence
    The abstract requires 'increasing batch sizes' without specifying the growth rate or schedule.
axioms (2)
  • domain assumption The mapping T is nonexpansive, i.e., ||T(x) - T(y)|| ≤ ||x - y|| for all x, y.
    Standard assumption for the Halpern iteration; invoked implicitly by the problem statement.
  • domain assumption A stochastic mini-batch oracle exists whose conditional expectation equals the full mapping T.
    Required for the stochastic version to be unbiased; not stated explicitly but necessary for the mean-square analysis.

pith-pipeline@v0.9.0 · 5423 in / 1505 out tokens · 47377 ms · 2026-05-09T21:37:16.900061+00:00 · methodology

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Reference graph

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