{"id":"3c4d93b8-8085-4de9-89f7-c0b523a0dc48","arxiv_id":"2505.03794","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A double-inertial forward-backward splitting method with a claimed weak convergence theorem, undermined by flawed derivations and irreproducible experiments.","lead":"The paper proposes a forward-backward splitting algorithm with two inertial terms for solving monotone inclusion problems. The convergence proof contains algebraic errors, and the reported experimental comparisons are single-run and not reproducible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (18) is a false norm identity and is load-bearing: the entire weak-convergence proof of Theorem 3.3 relies on it, so the central claim is unproven as written.","rationale":"The reader's weakest_assumption focuses on the feasibility of Assumption B(iii) and also notes that the substitutions around Eqs. (18) and (22)–(24) are invalid. My stress-test agrees that Eq. (18) is false, and I regard that as the single most load-bearing defect: the convergence theorem is not a matter of an unshown parameter range but of an incorrect algebraic identity that the entire proof structure uses. The reader did flag the identity, so there is partial agreement, but the stress-test places the false identity above the parameter-feasibility issue because it alone invalidates the central claim. The numerical section has additional issues, such as the abstract's claim of superiority being contradicted by Tables 3 and 4 and the Data Availability statement conflicting with the use of the real Iris dataset, but the core mathematical result is already unsupported. Given that the main theorem is unproven, the appropriate verdict remains REJECT.","tokens_in":13505,"tokens_out":5465,"duration_ms":48571,"concrete_test":"Independent algebraic check: expand ∥a u − b v − c w∥² by the bilinearity of the inner product and compare coefficients with Eq. (18) for a = 1+ϑ, b = ϑ−δ, c = δ. In particular, set v = w = 0 and any u ≠ 0; the left side is a²∥u∥² while the right side of Eq. (18) is [(1+ϑ) + (1+ϑ)(ϑ−δ) + δ(1−ϑ)]∥u∥² = [(1+ϑ)² − 2δϑ]∥u∥², which differs unless δ = 0 or ϑ = 0. If the identity is corrected, re-derive Lemma 3.2 from the correct expansion; if that re-derivation is not supplied, Theorem 3.3 does not follow.","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Lemma 3.2 and Theorem 3.3 depends on Eq. (18), which asserts that the squared norm of a linear combination equals a sum with coefficients that are linear, rather than quadratic, in (1+ϑ). That cannot be generally true. Concretely, take H = R, p* = 0, p_{k-1} = p_{k-2} = 0, p_k = 1, ϑ = 0.1, δ = −0.1. Then the left-hand side of Eq. (18) is ∥(1+ϑ)p_k∥² = 1.21, while the right-hand side equals (1+ϑ) + (1+ϑ)(ϑ−δ) + δ(1−ϑ) = 1.23. The identity fails. Since Eq. (18) is used to rewrite ∥w_k − p*∥² and to construct the Lyapunov-type quantity Γ_k, the boundedness proof and the subsequent existence of the limit in Eq. (32) are both unsupported. The parameter-feasibility concern in Assumption B(iii) is real but secondary: even if the parameter set is nonempty, the proof as written still does not establish convergence because the central norm computation is invalid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13748,"tokens_out":9519,"duration_ms":85139,"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":[{"comment":"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.","section":"Section 3, Lemma 3.2, Eq. (18)"},{"comment":"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.","section":"Section 3, Lemma 3.2, Eqs. (29)-(32)"},{"comment":"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.","section":"Section 3, Assumption B(iii)"}],"minor_comments":[{"comment":"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'.","section":"Section 3, Eq. (22)"},{"comment":"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.","section":"Section 4, Tables 3 and 4"},{"comment":"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.","section":"Introduction, Eq. (2)"},{"comment":"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.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The core theoretical result is not established because Eq. (18), a load-bearing algebraic identity in the proof of Lemma 3.2, is false, and the coupled parameter conditions in Assumption B(iii) are not shown to be satisfiable. The numerical experiments cannot compensate for an invalid convergence proof. I recommend rejection; a substantially revised submission with a corrected expansion and explicit feasible parameter assignments would be needed before the central claim could be evaluated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proposes a double-inertial forward-backward algorithm with the second inertial parameter allowed to be nonpositive. That sign relaxation is the only real novelty; the iteration is otherwise a standard two-inertial scheme close to Jolaoso et al. and Suantai et al. If the convergence theorem were valid, it would be an incremental but publishable addition.\n\nWhat the paper does well: the algorithm is clearly stated, the assumptions are explicit, and the numerical section compares against several relevant baselines. The authors are honest enough to show Tables 3 and 4 where FISTA beats their method, which partially undercuts the abstract's claim of universal superiority.\n\nThe soft spots are load-bearing. Lemma 3.2's Eq. (18) is false: the squared norm of a linear combination is not the linear combination shown. A concrete counterexample: in R with p*=0, p_{k-1}=p_{k-2}=0, p_k=1, ϑ=0.1, δ=-0.1, LHS = 1.21, RHS = 1.23. The proof then uses this identity to construct the Lyapunov function Γ_k and to prove boundedness, so the weak convergence Theorem 3.3 is unsupported. The additional parameter-feasibility condition in Assumption B(iii) is also never shown to be satisfiable; even if it is, the proof still fails.\n\nThe numerics are too thin to rescue the paper: no code, no data, no seeds, no error bars. The Data Availability statement says no real data was used, which directly contradicts the Iris dataset experiment. That is a careless inconsistency. The regression demonstrations on random sine points are not a compelling application.\n\nBottom line: the central theorem is unproven as written, and the empirical claims are not substantiated. This is not ready for peer review in its current form. A serious referee would need a corrected proof and a much more careful experimental section. I would not cite it, and I would not bring it to the reading group.","headline":"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.","tokens_in":14335,"tokens_out":1822,"would_cite":false,"duration_ms":17009,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","47J25","65K05","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $0\\in(A+B)z$.","keywords":["double inertial","forward-backward splitting","monotone inclusion problem","weak convergence","co-coercive operator","maximal monotone operator","regression","classification"],"falsifier":"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.","tokens_in":13244,"feed_emoji":"🧮","tokens_out":12964,"duration_ms":117830,"temperature":0.7,"pith_summary":"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.","feed_headline":"Double-inertial splitting converges weakly to zeros of A+B","feed_subtitle":"A two-momentum forward-backward method converges weakly; tests show faster, more accurate regression and classification fits.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the theorem that the composition $J_\\lambda^B(I-\\lambda A)$ is $\\kappa$-averaged, the contractive property on which Lemma 3.2 relies.","marker":"[8]"},{"why":"Introduces the original forward-backward splitting iteration that Algorithm 1* is built from.","marker":"[20]"},{"why":"Provides the double-inertial Mann iteration whose two-momentum idea the algorithm adopts.","marker":"[12]"},{"why":"Presents a prior double-inertial forward-backward-forward method that motivates the two-inertial construction.","marker":"[34]"},{"why":"Gives a two-step inertial Douglas-Rachford splitting scheme, a neighboring double-inertial algorithm in the same family.","marker":"[17]"},{"why":"Supplies a double-inertial forward-backward splitting method used as a baseline in the numerical comparisons.","marker":"[18]"},{"why":"Documents that single-step inertial splitting can underperform, which motivates trying a second inertial term.","marker":"[30]"}],"fun_headline_variants":["Double momentum splitting converges weakly, speeds up regression and classification","Two-step inertia in forward-backward: weak convergence with better fits","Inertial forward-backward with two parameters: converges, then outperforms","Faster regression and classification via double-inertial splitting","Weak convergence plus superior results: double inertial method"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Double momentum splitting converges weakly, speeds up regression and classification","Two-step inertia in forward-backward: weak convergence with better fits","Inertial forward-backward with two parameters: converges, then outperforms","Faster regression and classification via double-inertial splitting","Weak convergence plus superior results: double inertial method"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000434,"raw_usage":{"total_tokens":2177,"prompt_tokens":881,"completion_tokens":1296,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":1221}},"tokens_in":497,"tokens_out":1296,"duration_ms":9608,"temperature":1.0,"reasoning_tokens":1221,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:37:19.094056+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"I., Csetnek, E","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that the composition $J_\\lambda^B(I-\\lambda A)$ is $\\kappa$-averaged, the contractive property on which Lemma 3.2 relies."},{"cited_title":"L., & Mercier, B","cited_arxiv_id":null,"evidence_quote":"Introduces the original forward-backward splitting iteration that Algorithm 1* is built from."},{"cited_title":"L., Cho, Y","cited_arxiv_id":null,"evidence_quote":"Provides the double-inertial Mann iteration whose two-momentum idea the algorithm adopts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents a prior double-inertial forward-backward-forward method that motivates the two-inertial construction."},{"cited_title":"S., & Shehu, Y","cited_arxiv_id":null,"evidence_quote":"Gives a two-step inertial Douglas-Rachford splitting scheme, a neighboring double-inertial algorithm in the same family."},{"cited_title":"O., Shehu, Y., YAO, J","cited_arxiv_id":null,"evidence_quote":"Supplies a double-inertial forward-backward splitting method used as a baseline in the numerical comparisons."}],"review_version":1}