{"id":"00c0f2e9-68df-4490-9a34-91ebf76b30ef","arxiv_id":"2607.20042","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A deep-unfolded, slack-free version of the Ohzeki method finds optimal knapsack solutions in about half the outer iterations of fixed-step baselines.","lead":"This paper offers a way to solve optimization problems with limits (like knapsack packing) without adding extra 'slack' bits, using a learned, iteratively updated penalty instead of two hand-tuned constants. It matters because fewer qubits and less tuning could make such constraints cheaper to run on quantum annealers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"DU-UPOM's per-iteration MCMC budget is unreported; comparing 'iterations' across arms with different sampler costs does not establish that the learned schedule accelerates convergence.","rationale":"I read the paper as claiming that DU-UPOM learns a step-size schedule that accelerates UPOM. The numerical evidence is Figs. 3–4, and the comparison is only meaningful if outer iterations have identical compute cost or if total MCS is explicitly reported. Section III.B fixes baseline UPOM at 1000 MCS per iteration; Section IV.C mentions only that training sampler calls used 100 MCMC samples. The evaluation budget for DU-UPOM is absent. Without it, 'iteration' is not a valid common cost axis. If DU-UPOM uses 100 samples per iteration, then it differs from baseline UPOM not only in the learned schedule but also in MCMC noise and per-iteration compute; the speed-up might stem from either factor. The learned η_t shown in Fig. 5 is real, but it does not establish that the speed-up comes from the schedule alone. The reader's suspicion is close, but the direction is off: fewer samples per iteration would lower DU-UPOM's total work, not raise it; the true issue is that the cost is unreported, making the comparison unauditable. I would not reject: the method is plausible, the variance-based backprop is standard, and the qualitative advantage may hold. The manuscript needs to state the evaluation MCS budget and ideally include a same-budget fixed-step baseline before the speed-up claim is accepted. Secondary concerns—the overlap with LOM and the absence of an explicit train/test split—are also worth addressing, but the sampler-budget omission is the most load-bearing for the headline speed-up claim.","tokens_in":11591,"tokens_out":8622,"duration_ms":88340,"concrete_test":"Report the number of MCMC samples used per DU-UPOM outer iteration in Figs. 3–4. Then rerun Figure 3 with DU-UPOM forced to use the same 1000 MCS per iteration as the baselines, keeping the same learned η_t sequence. If DU-UPOM still reaches zero MSE for all 500 instances within ~15 iterations, the speed-up is real. If the curve shifts right to ~30 iterations, the reported advantage is an artifact of the per-iteration budget difference. If DU-UPOM was already evaluated at 1000 MCS/iteration, simply state this; the concern resolves.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The speed-up claim rests on the iteration axis of Figs. 3–4. For fixed-step UPOM and the other untrained baselines, Sec. III.B fixes the sampler budget at 1000 MCS per iteration. For DU-UPOM, Sec. IV.C states only that each training sampler call used 100 MCMC samples; the budget used to produce the evaluation curves is never given. If the plotted DU-UPOM iterations also use 100 samples, then the comparison changes two variables at once: the learned step-size schedule and the per-iteration MCMC noise/computation. The observed reduction from ~30 to ~15 iterations then does not isolate the learned schedule; it may reflect a cheaper, noisier sampling loop rather than a genuinely better update rule. Conversely, if the budget is 1000, the comparison is clean and the claim would be supported. Since the abstract/conclusion claim that DU accelerates UPOM via the learned step-size schedule, an unreported difference in per-iteration sampler cost makes the central quantitative claim uninterpretable. Note: the reader's phrasing—'fewer samples per iteration would weaken the total-work claim'—is not quite right; fewer samples would reduce DU-UPOM's total work. The real defect is that the total sampler cost is simply not reported for one arm of the comparison.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes UPOM, a reformulation of unbalanced penalization (UP) for inequality-constrained combinatorial optimization, in which the two static penalty coefficients of original UP are replaced by an auxiliary-variable update in the spirit of the Ohzeki method, and the squared residual term is removed from the sampling Hamiltonian. It further proposes DU-UPOM, a deep-unfolded version that learns the step-size schedule of the auxiliary-variable update from training instances using variance-based backpropagation through the nondifferentiable MCMC sampler. The methods are evaluated on random binary knapsack instances of size N=200, with 500 instances and exact dynamic-programming reference optima, and compared against original UP, fixed-step UPOM, a Lagrangian Ohzeki method (LOM), and an Ohzeki method with slack variables. The reported results indicate that UPOM outperforms tuned original UP in cumulative MCS, and that DU-UPOM reaches optimal solutions in fewer iterations than the fixed-step baselines.","tokens_in":11871,"tokens_out":6603,"duration_ms":70738,"significance":"If the claims are fully supported, the paper offers a useful step toward inequality-constrained COPs on quantum annealers without slack variables and without the squared residual term in the sampler Hamiltonian, thereby reducing both logical-variable and embedding overhead. The variance-based gradient formula for training through an MCMC sampler is a valuable mechanism, and the use of 500 random instances with exact DP references is a solid experimental setup. The work also extends the authors' earlier DUOM framework from equality to inequality constraints. However, the central quantitative claim about DU-UPOM's faster convergence is currently not fully interpretable because the per-iteration MCMC budget for DU-UPOM's evaluation is not reported, and the train/test separation is not documented. These are fixable experimental-protocol issues, but they are load-bearing for the paper's main conclusion.","major_comments":[{"comment":"The per-iteration sampling budget for DU-UPOM during evaluation is never stated. Section III.B says 'One UPOM iteration uses 1000 MCS,' and Section IV.C says 'Each training sampler call used 100 MCMC samples,' but the evaluation curves in Figs. 3–4 use 'iteration' as the horizontal axis without specifying whether DU-UPOM's plotted iterations use 1000 MCS, 100 MCS, or some other budget. If the evaluation budget differs from the baselines, the comparison changes two variables at once: the learned step-size schedule and the per-iteration sampling cost/noise. The statement in Section IV.C that 'reducing the number of iterations also reduces the number of sampler calls' assumes equal per-iteration cost. Please report the per-iteration MCS budget for DU-UPOM in the evaluation, and ideally also present results as a function of cumulative MCS or wall-clock time so that the speed-up claim is unam","section":"§IV.C, Figs. 3–4"},{"comment":"The separation between training and test instances is not documented. The text says 'All methods are evaluated on 500 instances' and that training used 'datasets consisting of minibatches of random knapsack instances,' but it does not state that the 500 evaluation instances are disjoint from the training instances. If the same instances were used for both training and evaluation, the reported speed-up of DU-UPOM could reflect overfitting to the training set rather than generalization to new instances from the same distribution. Please specify how the training and evaluation sets were generated, confirm that they are disjoint, and report the number of training instances.","section":"§IV.C"},{"comment":"The training loss L_DU-UPOM includes a nonsmooth positive-part term, but the paper does not give the explicit gradient of this loss with respect to the final auxiliary variable u^(T). Equation (21) provides ∂⟨F⟩/∂u via the variance, which is needed for the chain rule, but the backpropagation also requires ∂L_DU-UPOM/∂u^(T), e.g., -β Cov_Q(f0 + ρ[F-W]_+^2, F) for this loss. Without this expression, the claim that the 'same backpropagation mechanism' applies is incomplete and the training procedure is not fully reproducible. Please provide the explicit gradient formula, including how the positive part is handled.","section":"§IV.A, Eq. (IV.A)"}],"minor_comments":[{"comment":"The annotation 'initial value = 1.0 × 10 1' appears to be missing a superscript minus sign; the text states the initial step size is 1.0×10^{-1}. Please correct the figure label.","section":"Fig. 5"},{"comment":"The sentence 'One UPOM iteration uses 1000 MCS' is stated in the UPOM paragraph. Please also state explicitly whether the LOM and slack-variable baselines in Section IV.C use the same per-iteration MCS budget, since Figs. 3–4 compare them on the iteration axis.","section":"§III.B"},{"comment":"The training data description is vague: 'datasets consisting of minibatches of random knapsack instances.' Please report the total number of training instances, how minibatches are sampled, and whether the same random seed was used for training and evaluation.","section":"§IV.C"},{"comment":"The definition of \\hat{x}(t) as 'the best candidate recorded up to the current computational budget' should be made more precise: is it the best feasible solution among all sampled candidates up to iteration t? How are infeasible samples treated in the MSE and optimal-instance-rate metrics?","section":"Eq. (25)"},{"comment":"The phrase 'other baseline' should be 'other baselines' for grammatical consistency.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The core idea is plausible and within the journal's scope, and the missing experimental details are likely straightforward to provide. The main risk is that the DU-UPOM speed-up may be an artifact of an unreported difference in per-iteration MCMC budget or of train/test overlap. I recommend a major revision requiring the authors to report the missing protocol details and, if necessary, rerun comparisons at equal cumulative sampler cost. The reliance on the authors' previous DUOM papers is natural, but independent validation would strengthen the submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe paper combines unbalanced penalization with the Ohzeki method to handle inequality constraints without slack variables, and then applies deep unfolding to learn the step-size schedule. The core idea is sound and the math checks out: the variance-based gradient derivation is correct, and the knapsack experiments are clearly described (500 instances, exact DP optima). The method does remove the squared residual term from the sampler Hamiltonian and reduces the two-parameter tuning of UP to a learned schedule. That is a real, if contained, contribution.\n\nThe main soft spot is the comparison between DU-UPOM and the untrained baselines. The paper fixes 1000 MCS per iteration for UPOM, LOM, and the slack-variable method, but never states the per-iteration sampling budget for DU-UPOM during evaluation. Training uses 100 MCMC samples per call, and if the evaluation curves in Figs. 3–4 also use 100 samples per iteration, then DU-UPOM is being compared at a different computational cost per iteration than the baselines. The 'fewer iterations' claim then does not establish that the learned schedule converges faster; it may just reflect a cheaper, noisier sampler loop. This is the paper's most important missing number.\n\nSecond, UPOM as defined in Eqs. (18)–(20) is, up to sign, the unprojected version of the Lagrangian Ohzeki method (LOM) cited as [36]. The paper compares against LOM but does not disclose this overlap. That doesn't invalidate the deep-unfolding contribution, but it should be stated clearly.\n\nThird, the paper never explicitly separates training and test instances, and no code or data are provided. That makes it hard to assess overfitting and reproducibility.\n\nThe experiments otherwise support the qualitative behavior, and the learning of a nonuniform step-size schedule (Fig. 5) is a plausible mechanism for acceleration. The paper would benefit from reporting the evaluation sampling budget, stating the relation to LOM, and providing code/data.\n\nThis one deserves a serious referee, but would need those clarifications before the speed-up claim can be taken at face value. I'd send it out, with a request to address the budget and the LOM overlap.","headline":"Useful extension of Ohzeki/UP for inequality constraints, but the speed-up claim is undercut by an unreported per-iteration sampling budget.","tokens_in":12431,"tokens_out":2225,"would_cite":true,"duration_ms":20787,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C27","81P68","68T07"],"pacs":["03.67.Lx"],"model":"deepseek-v4-flash","headline":"This paper claims that inequality-constrained combinatorial optimization can be solved without slack variables by folding the squared penalty away into an auxiliary-variable update, and that the resulting step-size schedule can be learned f","keywords":["quantum annealing","combinatorial optimization","inequality constraints","unbalanced penalization","Ohzeki method","deep unfolding","knapsack problem","step-size learning"],"falsifier":"Re-run the Fig. 3-4 comparison with total Monte Carlo steps on the horizontal axis instead of iterations. If DU-UPOM's total MCS to reach zero MSE is not smaller than fixed-step UPOM's, the headline speed-up claim is an artifact of the axis choice.","tokens_in":11416,"feed_emoji":"⚛️","tokens_out":6168,"duration_ms":52983,"temperature":0.7,"pith_summary":"The paper is trying to establish that inequality constraints — the kind that force extra slack bits and squared penalty couplings onto a quantum annealer — can be handled by a slack-free reformulation called UPOM. UPOM merges the asymmetric linear penalty of unbalanced penalization with the Ohzeki method's auxiliary-variable update, so only a single step-size schedule remains to be chosen. The paper then claims that deep unfolding can learn that schedule from training instances, and that the learned DU-UPOM reaches optimal solutions for random knapsack problems in roughly half the iterations of fixed-step baselines. A sympathetic reader would care because this removes two tuning burdens at once: the two static penalty coefficients of original UP and the quadratic residual term that inflates qubit connections.","feed_headline":"Inequality constraints solved slack-free by one learned schedule","feed_subtitle":"Learned step sizes find all optimal solutions in 15 iterations, half the fixed-step baseline's 30.","key_machinery":"The load-bearing object is the auxiliary-variable recursion u(t+1)=u(t)+η_t(B−⟨F⟩), paired with the linear sampler Hamiltonian H_U=f0−u F. The identity that makes training work is ∂/∂u⟨F⟩=β Var_{Q_U}[F], which allows the gradient to be computed from sampling moments rather than by differentiating through the Markov chain. With m inequalities the same derivative becomes the covariance matrix of the constraint functions.","core_discovery":"The central claim is that the two static penalty coefficients λ1, λ2 of original unbalanced penalization can be replaced by one auxiliary variable u(t) updated as u(t+1)=u(t)+η_t(B−⟨F(x)⟩_Q), while the sampler Hamiltonian contains only a linear coupling f0(x)−u(t)F(x). This removes the squared residual term from the distribution being sampled and replaces the two-parameter penalty search with a single step-size schedule. The paper further claims that the derivative of the constraint expectation with respect to the auxiliary variable equals β times the variance of the constraint under the sampling distribution, which lets the step sizes be trained by backpropagation even though the sampler it","pith_inferences":["Because the paper compares in iterations and reports 1000 MCS per iteration only for fixed-step UPOM, the practical speed-up in total sampling work would be confirmed by comparing cumulative MCS; if DU-UPOM's per-iteration sampler budget is smaller, part of the advantage is already accounted for in the iteration axis.","The variance identity suggests a natural self-tuning schedule: step size should scale roughly as the inverse of the constraint variance, so the learned schedule may be interpretable as an adaptive second-order update; that interpretation is the authors' future work, not their claim.","The same slack-free treatment should transfer from knapsack problems to other inequality-constrained COPs such as scheduling, bin packing, or portfolio selection, and to quantum-annealer execution with transfer learning, which the authors list as future work."],"forward_implications":["UPOM eliminates the two-parameter penalty tuning of original UP; a single step size (fixed or learned) governs the update, and a simple grid search sufficed in the knapsack experiments.","The sampler Hamiltonian no longer contains the squared residual term, so the additional quadratic couplings that would have to be minor-embedded on a quantum annealer disappear; combined with no slack variables, this reduces logical and physical qubit overhead.","DU-UPOM learns the step-size schedule without needing optimal solutions for training instances; the training loss is the expected penalized objective under the sampling distribution.","The learned nonuniform schedule reaches zero MSE for all 500 test instances within 15 iterations, about half the ~30 iterations needed by fixed-step UPOM and the two reference baselines.","The differentiability mechanism extends to multiple inequalities, where the needed derivatives are covariances of the constraint functions."],"fun_headline_variants":["Slack-free solver learns step sizes, cuts iterations in half","Learned steps replace penalty tuning in constraint optimization","No slack variables: one learned schedule solves knapsack faster","Cut penalty tuning out: learned step sizes beat fixed-step baseline","One learned schedule replaces two penalties, cuts iterations"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The comparison plots use 'iteration' as the cost axis, but the per-iteration sampler budget for DU-UPOM is never stated for the evaluation; if the learned method consumes fewer MCMC samples per iteration than the 1000 MCS used for the baselines, 'fewer iterations' does not by itself demonstrate less total sampling work.","fun_headline_variants_meta":{"raw":{"variants":["Slack-free solver learns step sizes, cuts iterations in half","Learned steps replace penalty tuning in constraint optimization","No slack variables: one learned schedule solves knapsack faster","Cut penalty tuning out: learned step sizes beat fixed-step baseline","One learned schedule replaces two penalties, cuts iterations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000814,"raw_usage":{"total_tokens":3424,"prompt_tokens":785,"completion_tokens":2639,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":2559}},"tokens_in":529,"tokens_out":2639,"duration_ms":19744,"temperature":1.0,"reasoning_tokens":2559,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:57:48.624370+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the Fig. 3-4 comparison with total Monte Carlo steps on the horizontal axis instead of iterations. If DU-UPOM's total MCS to reach zero MSE is not smaller than fixed-step UPOM's, the headline speed-up claim is an artifact of the axis choice.","supporting_citations":[],"review_version":1}