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REVIEW 3 major objections 4 minor 39 references

A Double Inertial Forward-Backward Splitting Algorithm With Applications to Regression and Classification Problems

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper's central claim is that Algorithm 1*, a forward-backward splitting method with two inertial extrapolation terms, converges weakly to a solution of the monotone inclusion problem $0\in(A+B)z$.

desk verdict The algorithm is a minor variant of existing double-inertial methods, but the convergence proof rests on a false norm identity, so the central theorem is unproven. read the letter →

arxiv 2505.03794 v1 pith:TSHSEJ5I submitted 2025-05-01 cs.LG math.OC

İrfan Işik , Ibrahim Karahan , Okan Erkaymaz This is my paper · ORCID
classification cs.LGmath.OC MSC 47H0547J2565K0590C25
keywords doubleinertialforward-backwardsplittingmonotoneinclusionproblemweakconvergenceco-coerciveoperatormaximalregressionclassification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to prove that a forward-backward splitting method carrying two inertial extrapolation terms, Algorithm 1*, converges weakly to a zero of $A+B$ in a real Hilbert space. The intended payoff is practical: momentum terms are a standard way to speed up iterative solvers, and the paper wants to show that a second momentum term can be added without losing the weak-convergence guarantee. The authors also run regression and classification experiments, comparing their update with several existing algorithms and reporting lower errors and shorter run times on their test tasks. The result is an extension claim in the monotone-inclusion literature: a new member of the family of inertial splitting methods, with a convergence proof and supporting numerics.

What carries the argument

The central object is the forward-backward operator $T:=J_\lambda^B(I-\lambda A)$, where $J_\lambda^B=(I+\lambda B)^{-1}$ is the resolvent of $B$. By the cited theorem in [8], $T$ is $\kappa$-averaged with $\kappa=2\alpha/(4\alpha-\lambda)$, which supplies the contractive inequality used throughout. Algorithm 1* is a relaxed inertial iteration over $T$: the three-term momentum $w_k=p_k+\vartheta(p_k-p_{k-1})+\delta(p_{k-1}-p_{k-2})$ is followed by $p_{k+1}=(1-E_k)w_k+E_kTw_k$. The proof is carried by the Lyapunov function $\Gamma_k=\|p_k-p^*\|^2-\vartheta\|p_{k-1}-p^*\|^2-\delta\|p_{k-2}-p^*\|^2+E_1(1-\kappa)/\kappa(1-|\delta|-\vartheta)\|p_k-p_{k-1}\|^2$; showing $\Gamma_{k+1}\le\Gamma_k$ up to a negative term is what yields the limit of successive differences, and demiclosedness of $T$ then converts that into weak convergence.

What would settle it

One could settle the theorem by expanding Eq. (18) symbolically for a concrete Hilbert space and checking whether the printed identity holds; if it does not, the proof of Lemma 3.2 collapses. A second check is to search for an admissible triple $(\vartheta,\delta,E_1)$ under Assumption B for, say, $\alpha=1$, $\lambda=1$; if none exists, the theorem is vacuous. A third, behavioral check is to run Algorithm 1* on a simple two-dimensional problem with a known solution and verify whether $\|p_k-p_{k-1}\|$ and $\|w_k-Tw_k\|$ actually tend to zero for parameters inside the admissible region.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 3.3: if $A$ is $\alpha$-co-coercive, $B$ is maximal monotone, $(A+B)^{-1}(0)$ is nonempty, and the parameters satisfy Assumption B, then the sequence $\{p_k\}$ defined by $w_k=p_k+\vartheta(p_k-p_{k-1})+\delta(p_{k-1}-p_{k-2})$ and $p_{k+1}=(1-E_k)w_k+E_kJ_\lambda^B(w_k-\lambda Aw_k)$ converges weakly to a point in $(A+B)^{-1}(0)$. The proof recasts the iteration as a relaxed fixed-point method for the averaged operator $T=J_\lambda^B(I-\lambda A)$, builds a Lyapunov function over three consecutive iterates, shows it is nonincreasing, forces $\|p_k-p_{k-1}\|\to0$, and uses demiclosedness to conclude that every weak cluster point is a solution. The numerical sections then claim that, on sine-curve regression and Iris classification tasks, Algorithm 1* is faster and more accurate than the compared methods in most settings.

Load-bearing premise

The load-bearing premise is that the coupled inequalities in Assumption B admit at least one admissible choice of the parameters and that the algebraic expansion in Eq. (18) is correct; if the parameter region is empty or that identity is wrong, the boundedness proof and the convergence theorem lose their foundation.

Editorial extensions

If this is right

  • If the theorem holds, practitioners can add two inertial extrapolation terms to forward-backward splitting for monotone inclusions and still retain a weak-convergence guarantee.
  • The relaxation parameters $E_k$ can be tuned inside $(E_1,E_2)$ without leaving the convergence regime, giving a family of algorithms rather than a single fixed update.
  • For convex minimization with $B=\partial f$ and $A=\nabla g$, the scheme becomes a two-momentum proximal-gradient method, so the result applies to regression and classification objectives of the kind tested in the paper.
  • The convergence is weak, not strong: on infinite-dimensional problems the iterates may fail to converge in norm while still approaching the solution set in the weak topology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof leans on $\delta\le0$ through $|\delta|=-\delta$; a natural extension to test is whether a symmetric $\delta\ge0$ admits the same Lyapunov argument, which would show whether the momentum sign restriction is an artifact of the proof.
  • The numerical experiments embed the splitting step inside an extreme learning machine; a direct test on sparse regression, where the resolvent becomes a soft-threshold map and the solution set is nontrivial, would show whether the double-inertial acceleration transfers to the standard Lasso setting.
  • Because Assumption B is a system of coupled inequalities, the quickest check of the theorem's substance is to try to exhibit an admissible triple $(\vartheta,\delta,E_1)$ for a concrete pair $(\alpha,\lambda)$; if no such triple exists, the convergence result is vacuous.
  • The expansion in Eq. (18) appears to drop cross terms that a correct expansion of a squared three-term combination would contain; whether the Lyapunov proof survives a corrected identity is the decisive check on the paper's main theorem.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a forward-backward splitting algorithm with two inertial parameters (Algorithm 1*) for finding a zero of A+B, where A is co-coercive and B is maximal monotone in a real Hilbert space. The central theoretical claim is that, under Assumptions A and B, the generated sequence converges weakly to a solution of the monotone inclusion problem. The paper also reports numerical experiments applying the algorithm to regression and classification tasks and compares it with FISTA, VIFBA, and two other algorithms. The main proof is carried out in Lemma 3.2 (boundedness) and Theorem 3.3 (weak convergence), relying on an averagedness property of the forward-backward operator and on a Lyapunov-type quantity Gamma_k.

Significance. If the weak-convergence theorem were correct, the contribution would be a modest but useful addition to the inertial splitting literature: another double-inertial forward-backward method with a convergence guarantee under stated parameter restrictions. The paper appropriately invokes a standard external result on the averagedness of the forward-backward operator, so I do not see a circularity problem. The numerical experiments cover several baselines, but the theoretical core is not sound as written: the key norm identity in Eq. (18) is false, and the feasibility of the coupled parameter conditions in Assumption B(iii) is never demonstrated. These issues undermine the main claim.

major comments (3)
  1. [Section 3, Lemma 3.2, Eq. (18)] The displayed expansion of ||w_k - p^*||^2 is algebraically false. With x = p_k - p^*, y = p_{k-1} - p^*, z = p_{k-2} - p^*, the correct identity is ||(1+ϑ)x - (ϑ-δ)y - δz||^2 = (1+ϑ)||x||^2 - (ϑ-δ)||y||^2 - δ||z||^2 + (1+ϑ)(ϑ-δ)||x-y||^2 + δ(1+ϑ)||x-z||^2 - δ(ϑ-δ)||y-z||^2, whereas the manuscript writes δ(1-ϑ)||p_k - p_{k-2}||^2 in place of δ(1+ϑ)||p_k - p_{k-2}||^2. A concrete counterexample is H = R, p^* = 0, p_{k-1} = p_{k-2} = 0, p_k = 1, ϑ = 0.1, δ = -0.1, for which the left-hand side equals 1.21 and the manuscript's right-hand side equals 1.23. Since Eq. (18) is used to construct Gamma_k and to derive the monotonicity inequality (24), the boundedness proof of Lemma 3.2 and the weak-convergence conclusion of Theorem 3.3 are unsupported.
  2. [Section 3, Lemma 3.2, Eqs. (29)-(32)] The proof asserts that from (30) the limit of Gamma_k + c_1||p_{k-1} - p_{k-2}||^2 is zero. Monotone decrease combined with nonnegativity of this sequence yields only existence of a finite limit; the conclusion ||p_{k-1} - p_{k-2}|| -> 0 follows instead by summing the telescoping inequality and using convergence of the series of c_2||p_{k-1} - p_{k-2}||^2. As written, the argument is incorrect, although this particular step is repairable.
  3. [Section 3, Assumption B(iii)] The manuscript never demonstrates that the parameter restrictions in Assumption B(iii) are jointly feasible. The displayed conditions couple ϑ, δ, E_1, and κ through multiple strict inequalities, and no admissible numerical assignment is provided. If this set of parameters is empty, Theorem 3.3 is vacuous. This issue is independent of the algebraic error in Eq. (18) and needs to be settled explicitly.
minor comments (4)
  1. [Section 3, Eq. (22)] In the displayed chain of inequalities leading to (22), the expression '-|δ| ||p_{k-1} - p_{k-2}||^2 ϑ^2 ||p_k - p_{k-1}||^2' is missing a plus sign; it should read '-|δ| ||p_{k-1} - p_{k-2}||^2 + ϑ^2 ||p_k - p_{k-1}||^2'.
  2. [Section 4, Tables 3 and 4] For the sigmoid activation function, FISTA achieves lower MSE, lower RMSE, and higher R^2 than Algorithm 1* in both Table 3 and Table 4, so the abstract's and conclusion's blanket claim that the proposed algorithm yields superior outcomes is not supported by the reported experiments.
  3. [Introduction, Eq. (2)] The displayed forward-backward iteration in Eq. (2) uses the resolvent of A on the term involving B; for problem (1), with B maximal monotone and A co-coercive, the standard forward-backward iteration is x_{n+1} = (I + λ_n B)^{-1}(I - λ_n A)x_n. Please correct the expression or the attribution.
  4. [Throughout] The manuscript contains several typographical and linguistic issues, including 'was was first suggested', 'successfull', and 'efficient of the proposed algorithm'; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence proof rests on an external averagedness theorem, and no fitted parameter is relabeled as a prediction.

full rationale

The paper's central theoretical claim (Theorem 3.3) is not circular. Algorithm 1* is defined directly, and the proof imports the κ-averagedness of J_B^λ(I−λA) from [8, Theorem 7], an external result by Bot, Csetnek, and Meier, not by the present authors. This is genuine external support. The algorithm parameters (ϑ, δ, E_k) are constrained by Assumption B before the conclusion and are not fitted to data; the proof then attempts to construct a Lyapunov-type quantity Γ_k and shows monotonicity under algebraic inequalities. No equation is asserted to be equivalent to the target result by definition, and no fitted quantity is renamed as a prediction. The self-citations in the reference list (e.g., [1–3,31] by co-author Karahan) are background citations for inertial and extragradient ideas and are not load-bearing in Lemma 3.2 or Theorem 3.3. Two serious correctness concerns exist, but they are not circularity. First, Eq. (18) is a false norm identity: the right-hand side has coefficients that are linear in (1+ϑ) rather than quadratic, so the squared norm of a linear combination is not expanded correctly. This identity is load-bearing for the boundedness proof and for the existence of the limit in Eq. (32), so the proof as written does not establish the theorem. Second, Assumption B(iii) asserts the joint feasibility of several coupled parameter inequalities without exhibiting an admissible tuple, leaving open the possibility that the admissible parameter set is empty. These concerns undermine the mathematical validity of the derivation, but they do not constitute a reduction of the claimed result to its own inputs. Under the circularity criteria, which require exhibiting a definitional equivalence, a fitted-input-renamed-as-prediction, or a load-bearing self-citation chain, the finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central claim depends on standard monotone operator assumptions plus an asserted but unverified feasibility condition on the algorithm parameters. The experiments introduce additional free choices (activation function, hidden nodes, iterations) that are not reported.

free parameters (4)
  • ϑ
    First inertial parameter; must satisfy Assumption B(ii). No value is reported in the experiments.
  • δ
    Second inertial parameter; restricted to δ ≤ 0 and further constrained by Assumption B(iii). No value is reported in the experiments.
  • λ
    Step size for the resolvent; chosen in (0, 2α) in Step 1. No value is reported.
  • E_k bounds (E1, E2)
    Relaxation sequence is required to lie in (E1, E2) with 0 < E1 < E2 < 1. The bounds E1 and E2 are free parameters; no values are reported.
assumptions (3)
  • domain assumption A is α-cocoercive and B is maximal monotone on a real Hilbert space; (A+B)^{-1}(0) is nonempty.
    Assumption A defines the problem class the algorithm is claimed to solve.
  • standard math T = J^B_λ(I−λA) is κ-averaged with κ = 2α/(4α−λ).
    Cited from Reference [8, Theorem 7]; used in Lemma 3.2 to apply quasi-nonexpansiveness and demiclosedness.
  • ad hoc to paper The parameter restrictions in Assumption B are jointly feasible for a nonempty range of values.
    The paper asserts the existence of admissible ϑ, δ, E1, E2 but does not demonstrate a concrete feasible assignment; if the admissible set is empty the convergence theorem is vacuous.

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Pith. "Pith review of A Double Inertial Forward-Backward Splitting Algorithm With Applications to Regression and Classification Problems." pith.science (2026). https://pith.science/paper/TSHSEJ5I

@misc{pith2026250503794,
  author       = {Pith},
  title        = {Pith review of: A Double Inertial Forward-Backward Splitting Algorithm With Applications to Regression and Classification Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TSHSEJ5I}},
  note         = {Machine review of arXiv:2505.03794}
}
read the original abstract

This paper presents an improved forward-backward splitting algorithm with two inertial parameters. It aims to find a point in the real Hilbert space at which the sum of a co-coercive operator and a maximal monotone operator vanishes. Under standard assumptions, our proposed algorithm demonstrates weak convergence. We present numerous experimental results to demonstrate the behavior of the developed algorithm by comparing it with existing algorithms in the literature for regression and data classification problems. Furthermore, these implementations suggest our proposed algorithm yields superior outcomes when benchmarked against other relevant algorithms in existing literature.

Figures

Figures reproduced from arXiv: 2505.03794 by the authors.

Figure 1
Figure 1. Regression analysis using linear activation function for random 10 sine values [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Regression analysis using linear activation function for random 100 sine [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Regression analysis using sigmoid activation function for random 10 sine [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: : Regression analysis using sigmoid activation function for random 100 sine [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Confusion matrix for Algorithm 1* Precision Recall F1-Score Support 0 (Setosa) 1.00 1.00 1.00 44 1 (Versicolor) 1.00 0.97 0.99 39 2 (Virginica) 0.97 1.00 0.99 37 Accuracy 0.99 Total Support 120 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

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