{"id":"a17e0035-2bab-4e76-894b-c6257b6e39d6","arxiv_id":"1908.03878","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Bregman forward-backward splitting is shown to converge for monotone inclusions in reflexive Banach spaces, with sharper rates in the convex minimization case.","lead":"This paper proves that a Bregman-distance version of the forward-backward splitting algorithm converges for sums of monotone operators in reflexive Banach spaces, a setting where even Euclidean convergence was previously open. It also derives faster convergence rates for convex minimization problems, which could make this method more useful in optimization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.8's focusing condition is, via Proposition 2.5, equivalent to the desired cluster-point inclusion W(xn)⊂zer(A+B); the general convergence theorem is a reduction rather than an unconditional proof.","rationale":"I read the paper in good faith. The derivations in Propositions 2.5 and 2.6 and the applications in Section 3 are coherent, and the special cases are genuinely proven, so the mathematical soundness is not in question. My concern is about the load-bearing status of Definition 2.7: Proposition 2.5 already establishes every antecedent of the focusing condition, making the condition equivalent to the inclusion of weak cluster points in zer(A+B). Thus Theorem 2.8 does not itself prove the hard part of convergence in the general monotone inclusion setting; it reduces convergence to an equivalent asymptotic property that must be verified separately in each application. This matches the reader's observation that the focusing condition is a significant nontrivial structural assumption, though I sharpen it further by noting the near-tautological relationship with the conclusion. The reader's weakest assumption was condition (1.1); I see (1.1) as explicit and at least checkable through the sufficient conditions of Proposition 2.1, whereas focusing is as hard as the convergence conclusion itself. The conditional verdict remains appropriate: the paper is a valuable framework with correct proofs, but the headline theorem is a reduction more than an unconditional convergence guarantee. No change to the reader's verdict is needed.","tokens_in":22529,"tokens_out":12208,"duration_ms":110758,"concrete_test":"Analytical check: rewrite Definition 2.7 by replacing its four antecedents with the conclusions of Proposition 2.5(i)–(iv), which hold whenever Algorithm 2.4's hypotheses are satisfied. If the antecedents are entailed, the focusing condition simplifies to W(xn)⊂zer(A+B), and Theorem 2.8 should be restated with that inclusion in place of [c]. Then verify whether Theorem 2.8 supplies any independent argument for that inclusion; if it does not, the central claim is conditional on the very conclusion it purports to prove. A complementary check: take a concrete instance satisfying Algorithm 2.4 and Theorem 2.8[a],[b],[d] for which inclusion in zer(A+B) is not yet known, and see whether Theorem 2.8 alone certifies convergence without an additional cluster-point analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2.7 defines focusing by a conditional whose antecedent lists exactly four properties: (Dfn(z,xn)) converges; the sum of ⟨xn+1−z, γn−1(∇fn(xn)−∇fn(xn+1))−Bxn+Bz⟩ is finite; the sum of (1−δ2)⟨xn−z,Bxn−Bz⟩ is finite; and the sum of (1−κγn/α)Dfn(xn+1,xn) is finite. Proposition 2.5(i)–(iv) proves precisely these four properties for every orbit generated under the hypotheses of Algorithm 2.4. Hence, for every admissible orbit, the focusing condition is logically equivalent to the inclusion W(xn)⊂zer(A+B). Theorem 2.8 then cites [c] (focusing) together with [a], [b], and [d] to conclude weak convergence; the hard analytic step, proving that every weak cluster point is a zero of A+B, is not established by the theorem but is imported through [c]. The corollaries do discharge this step individually via Lemma 3.1 and other arguments, so the paper is internally coherent; however, the general statement in Theorem 2.8 and the abstract's claim of establishing convergence for the operator-splitting scheme are weaker than they appear: the general result is a reduction of convergence to an equivalent asymptotic condition rather than a proof of that condition. The novel inequality (1.1) is also a substantial global assumption, but it is at least explicit and checkable through Proposition 2.1; focusing is checkable only by repeating the convergence proof in each application.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies forward-backward splitting with Bregman distances for finding zeros of the sum of two maximally monotone operators in reflexive Banach spaces. The iteration is (1.3), with Bregman kernels f_n that may vary with n, and the main structural hypothesis is the two-point condition (1.1) on the single-valued operator B. The central result, Theorem 2.8, asserts weak convergence of bounded orbits whose weak cluster points lie in int dom f, provided the algorithm is 'focusing' in the sense of Definition 2.7 and either the solution set is a singleton or the kernels satisfy an asymptotic gradient-strictness condition. The paper then shows that several known frameworks (Bregman proximal point, variable-metric forward-backward splitting, the Renaud-Cohen algorithm, and Nguyen's minimization algorithm) are recovered, and in the minimization setting proves monotone decrease of the objective, summability of the objective error, and the rate o(1/n) for the objective gap.","tokens_in":22903,"tokens_out":18430,"duration_ms":190203,"significance":"The paper contains a genuinely useful unification. The sufficient conditions for condition (1.1) in Proposition 2.1 cover cocoercivity, Lipschitz/angle-bounded operators, strong monotonicity, and descent-type inequalities, and the minimization results in Theorem 3.9 improve known rates from O(1/n) to o(1/n) while also giving summability of the objective error. The concrete corollaries and examples are carefully developed and do recover or extend existing methods, including new Euclidean-space results for monotone inclusions that are not minimization problems. The proofs are long but generally well structured, and the verification of assumptions in each corollary is explicit. The main weakness is that the general convergence theorem is stated in terms of a 'focusing' hypothesis that, because of Proposition 2.5, is logically equivalent to the desired conclusion that all weak cluster points are zeros; the unconditional content therefore sits in the corollaries rather than in Theorem 2.8 itself.","major_comments":[{"comment":"Proposition 2.5 proves that, for every sequence generated by Algorithm 2.4 and every z in S, the four antecedent conditions displayed in (2.36) all hold. Consequently, for any admissible orbit, the 'focusing' condition is equivalent to the inclusion W(x_n) subset of zer(A+B). The proof of Theorem 2.8 then obtains the central inclusion W(x_n) subset of S in one line by invoking hypothesis [c], rather than by proving it. This makes Theorem 2.8 a formal reduction rather than an unconditional convergence theorem, and the abstract's claim that the paper establishes the convergence of (1.3) is stronger than what the general theorem actually provides. The corollaries do discharge the focusing condition individually, so the paper is internally coherent, but the main theorem and the abstract should be reframed, for instance by stating explicitly that the new analytic work for the general inclusion problem is the verification of focusing in the corollaries, or by replacing [c] with a sufficient condition that is not equivalent to the conclusion.","section":"Definition 2.7 and Theorem 2.8"},{"comment":"Condition (1.1) is the main novel assumption, and Proposition 2.1 shows that it captures several useful known conditions. However, the paper does not prove that (1.1), together with the other hypotheses of Algorithm 2.4, implies W(x_n) subset of zer(A+B). The summability results of Proposition 2.5 are derived from (1.1), but none of them yields the cluster-point inclusion without the separate focusing assumption. Since focusing is equivalent to that inclusion, the general convergence result is conditional in a way that is not transparent from the abstract. The authors should either prove a general sufficient condition for focusing from (1.1), or explicitly state in the abstract and in Theorem 2.8 that the general monotone-inclusion convergence statement assumes the cluster-point property and that the unconditional results are those established in the subsequent corollaries and examples.","section":"Problem 1.1, condition (1.1), and Theorem 2.8"}],"minor_comments":[{"comment":"The four lines in the antecedent of the implication are separated by commas; since the implication concerns their conjunction, please make this explicit, for example by writing 'if all of the following four conditions hold' before the display.","section":"Definition 2.7, display (2.36)"},{"comment":"The step from convergence of (Delta_n) to convergence of (D_{f_n}(z,x_n)) uses the fact that the term delta_1 gamma_n <x_n - z, x*_n + Bz> is summable, which follows from the summability of <x_{n+1}-z, x*_{n+1}+Bz> and the boundedness of (gamma_n). This is implicit; adding one sentence would make the argument fully transparent.","section":"Proposition 2.5, proof after (2.26)"},{"comment":"The strong convergence claim in Example 3.11 is stated without proof or reference. If it is intended as a new result, please provide the argument or identify the theorem from which it follows; if it is an illustration, make that clear.","section":"Example 3.11"},{"comment":"The sentence 'the convergence of such an iterative process has not yet been established, even in finite-dimensional spaces with a single function f_n = f and constant parameters' should be qualified relative to the Renaud-Cohen algorithm (1.7), which treats a constant strongly convex kernel in Hilbert space; otherwise the novelty claim is easy to misread as stronger than intended.","section":"Introduction, Section 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically solid in its concrete applications, and I do not see an algebraic error in the main chains of inequalities. The central issue is presentation and framing: Theorem 2.8's focusing hypothesis is equivalent to the cluster-point property that the paper is supposed to establish, so the general theorem is a reduction. This can be fixed in revision by restructuring the main theorem, adjusting the abstract, and making the conditional nature of the general result explicit. No citation or scope concerns; the paper fits a mathematical optimization journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: the abstract sells Theorem 2.8 as establishing convergence of Bregman forward-backward splitting for inclusions, but the general theorem is weaker than that. The focusing condition (Definition 2.7) is, via Proposition 2.5, logically equivalent to the desired cluster-point inclusion W(xn) ⊂ zer(A+B). So the theorem reduces convergence to exactly the hard analytic step, rather than proving it. That said, the paper is still worth serious attention, because the corollaries genuinely discharge that step case by case, and the minimization results are new even in Euclidean spaces.\n\nWhat the paper does well: it assembles a clean framework around condition (1.1), which unifies several scattered assumptions (cocoercivity, the Renaud–Cohen condition, the Bregman descent lemma, etc.). Proposition 2.5 is the technical core, and I checked the key inequality chain — it is coherent. The finite-dimensional corollary (3.6) and the minimization rates o(1/n) with summable errors are concrete advances. Example 2.9 gives a real problem outside the reach of all four prior frameworks, which is the right way to justify a new condition.\n\nThe soft spots: the focusing condition is the main one. It is not a checkable assumption in the usual sense; verifying it in each application essentially means repeating a convergence proof via Lemma 3.1. If a reader only looks at Theorem 2.8 and the abstract, they will overestimate what is proven for the general inclusion problem. A second, minor concern is that condition (1.1) is a global two-point inequality involving constants that are not constructive. The paper shows it covers known cases, but it does not offer much guidance for genuinely new operators, so practical verification could be a barrier. The paper is also long and technical, but that is justified by the machinery.\n\nBottom line: this is a real and useful paper for researchers working on Bregman splitting and monotone operator theory in Banach spaces. The corollaries and rate results stand on their own, and the proofs appear sound. I would bring it to a reading group and cite the minimization part.\n\nFor peer review: yes, send it to a careful referee who knows the Legendre function literature. The referee should ask the authors to state plainly that Theorem 2.8 is a reduction, and to make the role of the focusing condition explicit in the abstract or introduction. That is a minor revision, not a rejection.","headline":"The general convergence theorem is a reduction: the 'focusing' condition essentially assumes the key inclusion, but the corollaries and new minimization rates do the real work.","tokens_in":23377,"tokens_out":2419,"would_cite":true,"duration_ms":26800,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","47J25","90C25","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that Bregman forward-backward splitting converges weakly to a zero of A+B in reflexive Banach spaces under a new condition on B.","keywords":["Bregman distance","forward-backward splitting","monotone inclusion","reflexive Banach space","weak convergence","Legendre function","convergence rate","variable metric"],"falsifier":"A concrete test: take $X=\\mathbb{R}^2$, $A$ the normal cone of a cone, $B$ a monotone but non-cocoercive operator, and $f=\\frac12\\|\\cdot\\|^2$; scan a grid over $x,y\\in C$ and $z\\in S$ and numerically evaluate the difference between the two sides of (1.1). If any triple gives a positive remainder, the theory's main assumption is violated. In the minimization setting, run Algorithm 3.8 on a convex problem satisfying (3.21) and check whether the objective error decays faster than $1/n$; a plateau at order $1/n$ would contradict the claimed $o(1/n)$ rate.","tokens_in":22313,"feed_emoji":"📐","tokens_out":9409,"duration_ms":97133,"temperature":0.7,"pith_summary":"This paper proves that the Bregman forward-backward splitting algorithm converges for monotone inclusions of the form $0\\in Ax+Bx$ in reflexive Banach spaces, even when the Bregman kernel changes from iteration to iteration. Convergence of this algorithm was previously known for minimization problems, and the paper's result for general monotone inclusions is new even in Euclidean spaces. The proof rests on a single inequality, condition (1.1), imposed on the pair $(A,B)$ and the Bregman kernel, which is shown to unify older assumptions such as cocoercivity, the descent lemma, strong monotonicity, and angle-boundedness. In the convex minimization case, the method produces monotonically decreasing objective values with summable errors and an $o(1/n)$ rate.","feed_headline":"Bregman forward-backward now converges for monotone inclusions","feed_subtitle":"Proof covers reflexive Banach spaces, recovers several older methods, and adds an o(1/n) minimization rate.","key_machinery":"The load-bearing mechanism is the new inequality (1.1) coupled with a quasi-Fejér monotonicity estimate for the sequence of Bregman energies. For every $x,y\\in C$, $z\\in S$, and selections $y^*\\in Ay$, $z^*\\in Az$, condition (1.1) requires $\\langle y-x,By-Bz\\rangle\\leq \\kappa D_f(x,y)+\\langle y-z,\\delta_1(y^*-z^*)+\\delta_2(By-Bz)\\rangle$; the proof inserts this inequality into a four-point identity for Bregman distances (Proposition 2.3(ii) of [6]) to obtain the telescoping bound $\\Delta_{n+1}\\leq (1+\\eta_n)\\Delta_n-\\theta_n$, where $\\theta_n$ is a sum of nonnegative terms. Summability of $\\theta_n$ then forces the desired limits, and the focusing condition converts weak cluster points into zeros of $A+B$.","core_discovery":"The central finding is that the iteration $x_{n+1}=(\\nabla f_n+\\gamma_n A)^{-1}(\\nabla f_n(x_n)-\\gamma_n B x_n)$, with $A,B$ maximally monotone and $f_n$ a sequence of Legendre functions whose Bregman distances are compatible, converges weakly to a point in $S=\\operatorname{zer}(A+B)$ whenever the orbit is bounded, its weak cluster points lie in $\\operatorname{int}\\operatorname{dom} f$, and a focusing condition holds (Theorem 2.8). The argument controls the Bregman energy $\\Delta_n=D_{f_n}(z,x_n)+\\delta_1\\gamma_n\\langle x_n-z,x_n^*+Bz\\rangle$ through a quasi-Fejér inequality, and condition (1.1) is the place where the coupling between $A$, $B$, and the Bregman kernel enters. The paper shows that condition (1.1) covers most previously used assumptions, so the theorem recovers the Bregman proximal point algorithm, variable-metric forward-backward splitting, and the auxiliary-problem splitting method of [20] as special cases, while also handling examples that none of them can. For convex minimization, the objective error is summable and decays as $o(1/n)$, with a companion summability result for the Bregman distances between successive iterates (Theorem 3.9).","pith_inferences":["Condition (1.1) is global, but Proposition 2.1 shows it is implied by simpler structural assumptions such as cocoercivity, strong monotonicity plus Lipschitzness, or a Bregman descent inequality; a practical check would be to test these simpler sufficient conditions first, since (1.1) itself is hard to verify directly.","The $o(1/n)$ rate and summability of Bregman displacements suggest the method is first-order optimal in the same sense as gradient descent; whether an accelerated variant with $O(1/n^2)$ objective error exists in Bregman geometry is a natural open question.","The focusing condition is the least transparent assumption; Corollary 3.6 shows that in finite-dimensional spaces it can be replaced by simpler hypotheses, so a plausible conjecture is that focusing is automatic for essentially strictly convex Legendre kernels with open conjugate domain.","Changing the Bregman kernel each iteration opens a design axis: one could choose $f_n$ adaptively to improve conditioning or to make the prox of $A$ easy, and the theorem suggests such adaptivity costs only a mild multiplicative drift condition $D_{f_{n+1}}\\le(1+\\eta_n)D_{f_n}$."],"forward_implications":["Convergence of Bregman forward-backward splitting now holds for the full monotone inclusion problem $0\\in Ax+Bx$ in reflexive Banach spaces, not just for minimization, and the result is new even in Euclidean spaces.","Existing algorithms—the Bregman monotone proximal point method, variable-metric forward-backward splitting, and the auxiliary-problem splitting method of [20]—are recovered as special cases of one theorem, and new instances are constructed that none of those frameworks can handle.","In convex minimization the method yields a monotonically decreasing objective sequence, summable objective errors, and the rate $(\\phi+\\psi)(x_n)-\\min(\\phi+\\psi)=o(1/n)$, together with $\\sum_n n(D_{f_n}(x_{n+1},x_n)+D_{f_n}(x_n,x_{n+1}))<+\\infty$.","Variational inequalities with non-cocoercive operators can be solved outside Hilbert spaces by choosing convenient Bregman kernels."],"supporting_citations":[{"why":"Supplies the Bregman distance calculus and the proximal-point algorithm that the new scheme extends; its Proposition 2.3(ii) drives the four-point identity in the main proof.","marker":"[6]"},{"why":"Variable-metric forward-backward splitting in Hilbert spaces, recovered as a special case in Corollary 3.3.","marker":"[15]"},{"why":"Earlier Bregman forward-backward splitting for convex minimization, generalized in Theorem 3.9 with sharper rates.","marker":"[18]"},{"why":"Auxiliary-problem splitting method whose key condition appears as Proposition 2.1(iv) and whose convergence theorem is recovered.","marker":"[20]"},{"why":"Standard Hilbert-space forward-backward theory and technical lemmas, including Lemma 5.31, used throughout the proof.","marker":"[7]"},{"why":"Legendre function theory used to control boundedness and weak cluster points via Bregman distances.","marker":"[5]"},{"why":"Euclidean Bregman descent lemma and finite-dimensional convergence rates that Theorem 3.9 improves.","marker":"[3]"},{"why":"Bregman approximation methods and Lemma 3.1, used to identify cluster points as zeros of $A+B$.","marker":"[13]"},{"why":"Classical forward-backward method with cocoercive $B$, recovered as a special case of the new framework.","marker":"[17]"}],"fun_headline_variants":["Bregman splitting converges for monotone inclusions","Beyond minimization: Bregman forward-backward provably converges","Variable Bregman distances give sharper splitting convergence","From minimization to inclusions: Bregman splitting unifies proofs","Reflexive Banach spaces: Bregman splitting convergence proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the new inequality (1.1): it must hold for all points in the domain and all solutions, with constants $\\delta_1,\\delta_2,\\kappa$; if it does not, the energy decrease that drives the proof collapses.","fun_headline_variants_meta":{"raw":{"variants":["Bregman splitting converges for monotone inclusions","Beyond minimization: Bregman forward-backward provably converges","Variable Bregman distances give sharper splitting convergence","From minimization to inclusions: Bregman splitting unifies proofs","Reflexive Banach spaces: Bregman splitting convergence proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000305,"raw_usage":{"total_tokens":1729,"prompt_tokens":903,"completion_tokens":826,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":745}},"tokens_in":519,"tokens_out":826,"duration_ms":8403,"temperature":1.0,"reasoning_tokens":745,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:57:57.555201+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test: take $X=\\mathbb{R}^2$, $A$ the normal cone of a cone, $B$ a monotone but non-cocoercive operator, and $f=\\frac12\\|\\cdot\\|^2$; scan a grid over $x,y\\in C$ and $z\\in S$ and numerically evaluate the difference between the two sides of (1.1). If any triple gives a positive remainder, the theory's main assumption is violated. In the minimization setting, run Algorithm 3.8 on a convex problem satisfying (3.21) and check whether the objective error decays faster than $1/n$; a plateau at order $1/n$ would contradict the claimed $o(1/n)$ rate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Variable-metric forward-backward splitting in Hilbert spaces, recovered as a special case in Corollary 3.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier Bregman forward-backward splitting for convex minimization, generalized in Theorem 3.9 with sharper rates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Standard Hilbert-space forward-backward theory and technical lemmas, including Lemma 5.31, used throughout the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Legendre function theory used to control boundedness and weak cluster points via Bregman distances."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Euclidean Bregman descent lemma and finite-dimensional convergence rates that Theorem 3.9 improves."},{"cited_title":"Mercier , T opics in Finite Element Solution of Elliptic Problems (Lectures on Mathematics, no","cited_arxiv_id":null,"evidence_quote":"Classical forward-backward method with cocoercive $B$, recovered as a special case of the new framework."}],"review_version":1}