{"id":"9111efc1-b4b8-43f0-b5da-532958a60904","arxiv_id":"2505.23673","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A new multi-round algorithm, MR-LPF, achieves regret of order sqrt(Gamma(T)T) up to log factors for preference-based Bayesian optimization with binary human feedback, removing the extra kernel-complexity and link-curvature factors present in prior bounds.","lead":"The paper presents a new algorithm for Bayesian optimization where the only feedback is a human preference between two candidate choices, and proves it matches the performance guarantees of standard optimization that gets exact numeric scores. If correct, this means preference-only feedback can be as sample-efficient as direct score feedback for a large class of common kernels.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.7's proof relies on the Loewner bound G_t(theta*,theta_t) >= kappa^-1 V_t, which fails when the logistic fit leaves the utility range; without it, the confidence intervals, the kappa_r=6 argument, and Theorem 4.1 are unsupported.","rationale":"The reader's CONDITIONAL verdict identifies the Loewner bound in Theorem 4.7 as the weakest assumption. My stress-test agrees and sharpens it: the bound is not merely unproved; it is false for the unconstrained logistic minimizer. I verified algebraically on a one-point dataset that the integrated sigmoid derivative alpha can drop below 1/kappa, making (41) fail. This matters because the confidence interval is the engine of the elimination argument: Lemma B.2, the kappa_r=6 reduction, and the final regret sum all use it. The claim that MR-LPF matches scalar-feedback BO is therefore not established by the paper as written. The issue is potentially fixable by adding a bound on the parameter set or using a self-normalized concentration argument that does not require (41), but as written the central theorem lacks a valid proof. I therefore keep the reader's CONDITIONAL verdict; the requested numerical check would settle whether the gap is fatal or merely a missing lemma.","tokens_in":22850,"tokens_out":16731,"duration_ms":159581,"concrete_test":"Construct a one-dimensional toy instance of the confidence-bound problem: take a single observed pair z_1 with phi(z_1)=1, f=0 (so theta*=0, kappa=4), lambda=0.05, and y_1=0. Solve theta_t = argmin -log(1-mu(theta)) + (lambda/2)theta^2; the first-order condition mu(theta_t) + lambda theta_t = 0 gives theta_t approx -2.13. Compute alpha = Integral_0^1 mu'(nu theta_t) dnu = (mu(theta_t)-1/2)/theta_t approx 0.185. Since 1/kappa = 0.25, and G_t = alpha + lambda I, V_t = 1 + kappa lambda, the claimed Loewner inequality G_t >= kappa^-1 V_t fails numerically (0.235 < 0.30). If this failure reproduces, the proof of Theorem 4.7 must be revised, and the main theorem cannot be accepted without an additional assumption on theta_t.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.7 (Appendix C) uses inequality (41), G_t(theta*,theta_t) >= kappa^-1 V_t, attributed to Lemma 12 of Pasztor et al. (2024). For the dueling kernel and sigmoid link, this inequality is equivalent to requiring alpha_i = Integral_0^1 mu'(nu theta_t^T phi_i + (1-nu) theta*^T phi_i) dnu >= 1/kappa for every observed point. Here kappa is defined in (2) from the true utility range h(X x X), but theta_t is the regularized logistic minimizer (7), and nothing confines theta_t^T phi_i to that range. For a single observation with y_1=0 and lambda=0.05, the first-order condition mu(theta_t) + lambda theta_t = 0 gives theta_t approx -2.13, so alpha = (mu(theta_t)-1/2)/theta_t approx 0.185 < 1/4 = 1/kappa. The Loewner inequality (41) therefore fails for admissible datasets generated by MR-LPF's within-round selection rule, which is independent of y. Since both the stochastic and bias terms in Theorem 4.7 replace G_t by kappa^-1 V_t via (41), the confidence bound (17) is not proved. The elimination rule (12) and the round-reduction kappa_r=6 argument in (27)-(28) rely on these intervals, so the regret bound of Theorem 4.1 has no valid foundation as written. A repair requires either a proof of (41) under a bound on ||theta_t||, or a different confidence-interval argument that avoids this Loewner bound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies Bayesian optimization from preference feedback (BOHF) under the Bradley-Terry-Luce (BTL) model. It introduces MR-LPF, a multi-round elimination algorithm that queries the pair of currently plausible actions with maximum kernel variance and, at the end of each round, eliminates actions whose upper confidence bound against every surviving action is below 1/2. The main theorem (Theorem 4.1) claims a regret bound of order O~(sqrt(Gamma(T) T log|X|)), improving on the O~(Gamma(T) kappa^2 sqrt(T)) and O~((Gamma(T) T)^{3/4}) bounds of Pasztor et al. (2024) and Xu et al. (2024), respectively. The proof strategy exploits the fact that, within a round, query locations are chosen independently of the labels, yielding tighter logistic-regression confidence intervals (Theorem 4.7). Corollaries give sample-complexity statements, and experiments compare MR-LPF against MaxMinLCB on synthetic kernels and the Yelp dataset.","tokens_in":23238,"tokens_out":13177,"duration_ms":130520,"significance":"If Theorem 4.1 were fully proved, the paper would settle an important open question: preference feedback would impose no asymptotic penalty over scalar feedback in kernelized BO, at least for the BTL model. The multi-round structure is a clean and potentially transferable idea, and the within-round separation of query design from labels is a genuine methodological contribution. The paper also responsibly flags in Remark 4.6 that its lower-bound comparison to conventional BO is informal. However, as I detail below, the confidence-interval proof contains a substantial unproved Loewner bound, the finite-action requirement is not stated in the setup, and the regret definition is inconsistent with the proof. These issues prevent the main claim from being accepted as written, despite the value of the algorithmic idea.","major_comments":[{"comment":"Equation (1) defines regret as R(T) = sum_t [P(x* ≻ x_t) + P(x* ≻ x'_t) - 1/2]. As printed, this quantity can be negative when both selected actions are much worse than the optimum, and it is not bounded by 1/2 per step. The proof, however, repeatedly uses per-step bounds of the form P(x* ≻ x) - 1/2 (see Eqs. (18), (30), (31)), which correspond to the average sub-optimality gap (P(x* ≻ x_t) + P(x* ≻ x'_t))/2 - 1/2. The theorem is therefore stated for a different quantity than the one analyzed. This is fixable by changing the definition (and the abstract/introduction accordingly), but the current mismatch makes the main theorem ill-posed.","section":"Section 2.1, Eq. (1); Appendix B, Eq. (18)"},{"comment":"The setup explicitly allows a continuous action set X, but Theorem 4.1 and Lemma B.1 rely on a union bound over the |X| action pairs, with beta(r) in Eq. (14) containing log(2R|X|/delta). For continuous X, log|X| is undefined and the stated regret bound has no meaning. The theorem should either explicitly assume that X is finite or provide a separate discretization argument. As written, the advertised applicability to continuous domains is not supported by the analysis.","section":"Section 2.1 and Theorem 4.1"},{"comment":"The proof uses the Loewner bound G_t(theta*, theta_t) ⪰ kappa^{-1} V_t, citing Pasztor et al. (2024, Lemma 12). The coefficient alpha(z_i; theta*, theta_t) in the definition of G_t is the average of mu' along the segment between theta*^T phi(z_i) and theta_t^T phi(z_i), and (41) requires this average to be at least kappa^{-1}. The kappa in Eq. (2) is defined from the true utility range h(X × X), but theta_t is the regularized logistic minimizer of Eq. (7), and the paper gives no bound on theta_t^T phi(z_i). For a one-point dataset with y_1=0, the first-order condition is mu(theta_t) + lambda theta_t = 0 (up to the feature norm); as lambda -> 0, theta_t -> -infinity and the averaged derivative alpha tends to 0, so for any fixed kappa the inequality alpha >= kappa^{-1} fails for sufficiently small lambda. Such a dataset is admissible in MR-LPF because within a round the query locations are chosen from the variance alone, independently of the labels. Since both the stochastic and bias bounds in Eqs. (42) and (43) rely on (41), Theorem 4.7 is not proved as stated, and the regret bound of Theorem 4.1 does not follow from the given argument. A repair is needed, for example a norm bound on theta_t, a lower bound on lambda, or a different confidence-interval argument.","section":"Appendix C, Eq. (41)"},{"comment":"The abstract and Section 1 claim that the order-optimal sample complexities of conventional BO are recovered, but Remark 4.6 acknowledges that the comparison with Scarlett et al. (2017) is not strictly valid: the lower bound there assumes Gaussian noise while BTL preference feedback corresponds to Gumbel/logistic noise, and the reduction changes constants and even the value of the target function. Since the paper provides only an informal justification, the word 'recovered' in the abstract overstates what is proved. I recommend either providing a formal lower bound or tempering the optimality claim in the abstract and introduction.","section":"Remark 4.6 and Abstract"}],"minor_comments":[{"comment":"There is a typo in 'Bradeley-Terry-Luce'; it should be 'Bradley-Terry-Luce'.","section":"Section 2.1"},{"comment":"The text contains 'Supvariations of UCB'; this should likely be 'Subvariations' or a related intended term.","section":"Section 1.1.1"},{"comment":"The proof of Lemma B.1 is only sketched ('follows from Theorem 4.7, a union bound ...'). Please spell out the union bound over rounds and pairs and the precise role of the kappa_r = 6 bound derived in Eqs. (27)-(28).","section":"Appendix B, Lemma B.1"},{"comment":"The notation Gamma_{(4 lambda)}(T) in the theorem is introduced without explanation; since the definition in Eq. (13) is Gamma_lambda(T), the reader should be told that the scaling 4 lambda arises from kappa_r >= 4 for r >= 2.","section":"Theorem 4.1 and Eq. (13)"},{"comment":"The experiments compare MR-LPF only against MaxMinLCB, not against POP-BO of Xu et al. (2024), even though the related-work table discusses both. Adding the second baseline would make the experimental comparison align with the paper's claimed improvements.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The Loewner inequality gap in Appendix C is the main obstacle. The counterexample I describe is not a contrived lower-bound construction: a sequence of labels all equal to 0 has positive probability under the algorithm's query rule, and the regularized logistic minimizer can then leave the range where the derivative of the sigmoid is bounded below by kappa^{-1}. I still believe the algorithmic idea may be salvageable with an additional argument (e.g., controlling ||theta_t|| or choosing lambda appropriately), so I recommend major revision rather than rejection. The authors should also fix the regret definition and state a finite-action assumption before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is this: MR-LPF is a genuine algorithmic contribution, and the paper's main regret bound is clearly new relative to Pasztor et al. and Xu et al. The round structure that makes query points within a round independent of labels is a nice trick, and it does remove the sqrt(Gamma(T)) factor and the kappa^2 dependence in the confidence width. The paper is readable, the experiments are honest, and the authors flag Remark 4.6 as informal rather than pretending it is a formal lower bound. Credit where it is due: this is a good idea, and the proof strategy is coherent up to one load-bearing step.\n\nThat step is the problem. Theorem 4.7 uses G_t(theta*, theta_t) >= kappa^{-1} V_t (Eq. 41, cited to Lemma 12 of Pasztor et al.) to bound both the stochastic and bias terms in the confidence interval. The stress-test note gives a concrete admissible dataset where this inequality fails: a single observation with y=0 and lambda=0.05 yields theta_t approx -2.13, so the integrated sigmoid derivative alpha is approx 0.185, less than 1/kappa = 0.25. The selection rule within a round is independent of y, so this is a dataset the algorithm can actually produce. If (41) fails, the confidence bound, the elimination rule, the kappa_r=6 argument, and Theorem 4.1 all lose their foundation. This is not a small gap; it is the bridge between the logistic regression estimator and the kernel confidence intervals. A repair would need either a proof of (41) under a norm bound on theta_t or a different argument that avoids the Loewner bound entirely.\n\nOther issues are minor by comparison. The printed regret definition in Eq. (1) does not match the proof's usage of average gap over the two actions; the algorithm implicitly assumes a finite action set for the union bound even though the problem statement allows continuous X; and the abstract's phrase \"order-optimal sample complexities of conventional BO are recovered\" overstates what Remark 4.6 actually claims, since the lower-bound comparison is informal and uses a different noise model. These are fixable. The Loewner gap is not obviously fixable without a new idea.\n\nWho is this paper for? Researchers working on preference-based BO and dueling bandits. It deserves a serious referee: the problem is important, the algorithm is clever, and the gap is specific enough that a good referee can tell whether a repair is plausible. I would not cite the main theorem until the confidence bound is fixed, but I would bring the paper to a reading group to discuss the Loewner issue and the round-design idea. Send it to review, and ask the referee to focus on Appendix C.","headline":"The paper has a real algorithmic idea and a real gap: the confidence bound in Theorem 4.7 leans on an unproved Loewner inequality that fails for small lambda, which collapses the regret bound as written.","tokens_in":23774,"tokens_out":1330,"would_cite":false,"duration_ms":14793,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Bayesian optimization with only pairwise preference feedback can achieve the same regret rate as conventional scalar-feedback Bayesian optimization, and proves it for a multi-round elimination algorithm.","keywords":["Bayesian optimization","preference feedback","Bradley-Terry-Luce model","regret bounds","kernel methods","multi-round algorithms","maximum information gain","sample complexity"],"falsifier":"Run MR-LPF on a small finite grid with a Mat\\'ern kernel and small regularization $\\lambda$, and at each round compute the smallest eigenvalue of $G_t(\\theta_\\star,\\theta_t) - \\kappa^{-1}V_t$; a single negative eigenvalue in a trajectory where the true parameter is known refutes Eq. (41) and with it the confidence bound of Theorem 4.7, independently of any regret experiment.","tokens_in":22670,"feed_emoji":"⚖️","tokens_out":12305,"duration_ms":116164,"temperature":0.7,"pith_summary":"The paper tries to establish that in Bayesian optimization with binary preference queries, a carefully structured algorithm can match the regret rate of conventional scalar-feedback Bayesian optimization: $\\tilde{O}(\\sqrt{\\Gamma(T)T})$. If true, this means pairwise \"which is better?\" queries are as informative, in a worst-case sense, as direct numeric evaluations of the hidden utility function. The proposed algorithm, MR-LPF, works in rounds, queries the most uncertain pair inside a shrinking set of plausible optima, and eliminates actions whose preference confidence bounds fall below $\\tfrac12$. The paper further derives sample complexities for linear, squared-exponential, and Mat\\'ern kernels that match, up to logarithms, the known lower bounds for scalar-feedback Bayesian optimization.","feed_headline":"Match scalar-feedback regret with preference queries alone","feed_subtitle":"New multi-round algorithm proves pairwise comparisons match direct-function-value rates in Bayesian optimization.","key_machinery":"The carrying mechanism is MR-LPF itself: rounds of increasing size ($N_1 = \\lceil\\sqrt{T}\\rceil$, $N_r = \\lceil\\sqrt{N_{r-1}T}\\rceil$), pair queries chosen within the active set $\\mathcal{M}_r$ by maximizing kernel uncertainty $\\sigma_{n-1,r}(x,x')$, and end-of-round elimination that keeps an action $x$ only if $\\mu(h(x,x')) + \\beta^{(r)}\\sigma(x,x') \\ge 1/2$ for every rival $x'$. Three ingredients do the analytical work: the dueling kernel $k((x,x'),(u,u')) = k(x,u)+k(x',u')-k(x,u')-k(x',u)$, which preserves the RKHS norm and makes ridge-type regression applicable to utility differences; the maximum information gain $\\Gamma_\\lambda(T)$, whose sublinear growth for a kernel family guarantees sublinear regret; and the Loewner inequality $G_t(\\theta_\\star,\\theta_t) \\succeq \\kappa^{-1}V_t$ taken from the prior MaxMinLCB analysis, which converts a mean-value-theorem derivative integral into the ridge covariance $V_t = S_t + \\kappa\\lambda I$ and yields the confidence bound $|\\mu(h_t(z))-\\mu(h(z))| \\le \\beta\\sigma_t(z)$ with $\\beta$ free of $\\sqrt{\\Gamma(T)}$.","core_discovery":"The central claim is that the Multi-Round Learning from Preference-based Feedback (MR-LPF) algorithm achieves cumulative regret $R(T) \\le 2CR\\,\\beta^{(R)}(\\delta)\\sqrt{\\Gamma_{4\\lambda}(T)}\\,(T^{1/2}+1)$ with probability at least $1-\\delta$, which simplifies to $\\tilde{O}(\\sqrt{\\Gamma_\\lambda(T)T\\log(|\\mathcal{X}|/\\delta)})$. This is the same rate as order-optimal scalar-feedback BO and improves over the two existing bounds $\\tilde{O}(\\Gamma(T)\\kappa^2\\sqrt{T})$ and $\\tilde{O}((\\Gamma(T)T)^{3/4})$, removing both the $\\kappa^2$ dependence and the extra $\\sqrt{\\Gamma(T)}$ factor. From the regret bound, the paper obtains simple-regret sample complexities that match, up to logarithmic factors, the known lower bounds for scalar-feedback BO with Mat\\'ern kernels, with the caveat that the comparison is informal because the preference-noise model is Gumbel-like rather than Gaussian. The improvement comes from the round structure: observation points are selected by pure variance, independent of past feedback values, which yields tighter confidence intervals; and elimination keeps only nearly tied actions, so the sigmoid curvature bound $\\kappa_r$ collapses to a fixed value of $6$ after the first round.","pith_inferences":["A direct but untested consequence of the paper's own logic is that the query count needed to reach fixed $\\epsilon$ should trace the $\\epsilon^{-(2+d/\\nu)}$ curve kernel by kernel; measuring this across SE, Mat\\'ern with several $\\nu$, and neural tangent kernels would provide a scaling check the paper does not report.","The independence assumption on which the tight confidence bound rests is specific to a learner that ignores past outcomes within a round; an adversary who conditions action choices on observed preferences would break that independence, so the near-optimal rate likely does not extend to adversarial or non-stationary preference feedback.","If Gumbel-noise lower bounds were derived for scalar-feedback BO, the paper's informal tightness argument would become a formal equivalence, converting 'same sample complexity as scalar feedback' from a suggestive match into a proven one; this is the missing step the authors explicitly acknowledge.","The $\\kappa_r = 6$ argument predicts that after one round of pure uncertainty exploration the surviving candidates are nearly tied in preference probability; this structural prediction could be checked directly by inspecting the empirical distribution of $\\mu(h(x,x'))$ over surviving pairs in the paper's experiments."],"forward_implications":["For linear kernels, MR-LPF finds an $\\epsilon$-good action with $T = \\tilde{O}(d\\log(1/\\delta)/\\epsilon^2)$ preference queries; for squared-exponential kernels, $T = \\tilde{O}(\\log(1/\\delta)/\\epsilon^2)$; for Mat\\'ern kernels of smoothness $\\nu$, $T = \\tilde{O}(\\log(1/\\delta)/\\epsilon^{2+d/\\nu})$, each matching the corresponding scalar-feedback rate up to logs.","Because $\\kappa$ disappears from the dominant regret term, a preference-learning algorithm no longer needs to know or control the maximum sigmoid curvature of the utility range to get near-optimal guarantees.","The regret bound is sublinear for every kernel whose maximum information gain grows sublinearly; for Mat\\'ern and neural tangent kernels, where $\\Gamma(T)$ can grow faster than $\\sqrt{T}$, the prior bounds could become vacuous while this one remains sublinear.","Outputting any action that survives the final round yields simple regret at most $2\\beta^{(R)}C\\sqrt{R\\Gamma_{4\\lambda}(T)/T}$, so the final recommendation, not only the cumulative query loss, reaches $\\epsilon$ accuracy at the stated sample counts."],"supporting_citations":[{"why":"Defines the BOHF formalization and the MaxMinLCB baseline whose regret bound this paper improves, and supplies the dueling kernel and the Loewner inequality used at Eq. (41).","marker":"Pasztor et al. (2024)"},{"why":"POP-BO is the second baseline, whose $(\\Gamma(T)T)^{3/4}$ bound and regret definition are the comparison points for Theorem 4.1.","marker":"Xu et al. (2024)"},{"why":"Supplies the multi-round elimination template (BPE) and the round-count bound $R \\le \\lceil\\log_2\\log_2 T\\rceil+1$ used in the regret summation.","marker":"Li & Scarlett (2022)"},{"why":"Introduces the maximum information gain $\\Gamma(T)$ and its role in GP-UCB regret analysis, the complexity measure in which Theorem 4.1 is stated.","marker":"Srinivas et al. (2010)"},{"why":"Provides the eigenvalue-decay bounds for linear, squared-exponential, and Mat\\'ern kernels that convert $\\Gamma_\\lambda(T)$ into the sample-complexity expressions of Corollary 4.5.","marker":"Vakili et al. (2021b)"},{"why":"Establishes lower bounds for scalar-feedback GP bandits that the paper's sample complexities are informally compared against as an optimality check.","marker":"Scarlett et al. (2017)"},{"why":"Logistic bandit analysis showing the regret dependence on $\\kappa$ can be removed; the confidence-interval proof adapts its techniques.","marker":"Faury et al. (2020)"},{"why":"Provides kernel bandit confidence bounds used as a component in the proof of Theorem 4.7.","marker":"Whitehouse et al. (2024)"}],"fun_headline_variants":["Preference queries achieve scalar-feedback regret","Near-optimal regret from human preference feedback alone","Pairwise comparisons match direct-value BO rates","Tighter BO regret bounds from preference queries"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole bound hinges on the inequality $G_t(\\theta_\\star,\\theta_t) \\succeq \\kappa^{-1}V_t$, which requires the integrated sigmoid slope between the true and fitted preference functions to stay above $1/\\kappa$; a regularized logistic fit can leave the bounded utility interval where the sigmoid is flat, so this slope bound is not automatic and the proof collapses if it fails.","fun_headline_variants_meta":{"raw":{"variants":["Preference queries achieve scalar-feedback regret","Near-optimal regret from human preference feedback alone","Pairwise comparisons match direct-value BO rates","Tighter BO regret bounds from preference queries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1448,"prompt_tokens":1048,"completion_tokens":400,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":344}},"tokens_in":664,"tokens_out":400,"duration_ms":4998,"temperature":1.0,"reasoning_tokens":344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:42:18.791078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run MR-LPF on a small finite grid with a Mat\\'ern kernel and small regularization $\\lambda$, and at each round compute the smallest eigenvalue of $G_t(\\theta_\\star,\\theta_t) - \\kappa^{-1}V_t$; a single negative eigenvalue in a trajectory where the true parameter is known refutes Eq. (41) and with it the confidence bound of Theorem 4.7, independently of any regret experiment.","supporting_citations":[{"cited_title":"M., and Seeger, M","cited_arxiv_id":null,"evidence_quote":"Introduces the maximum information gain $\\Gamma(T)$ and its role in GP-UCB regret analysis, the complexity measure in which Theorem 4.1 is stated."},{"cited_title":"Lower bounds on regret for noisy gaussian process bandit optimization","cited_arxiv_id":null,"evidence_quote":"Establishes lower bounds for scalar-feedback GP bandits that the paper's sample complexities are informally compared against as an optimality check."},{"cited_title":"Improved optimistic algorithms for logistic bandits","cited_arxiv_id":null,"evidence_quote":"Logistic bandit analysis showing the regret dependence on $\\kappa$ can be removed; the confidence-interval proof adapts its techniques."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides kernel bandit confidence bounds used as a component in the proof of Theorem 4.7."}],"review_version":1}