{"id":"9b44bd77-1007-4366-99ec-90c19076540d","arxiv_id":"2411.10658","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Distributed optimization algorithms derived from optimal control theory that claim superlinear convergence and avoid Hessian inversion.","lead":"The authors recast multi-agent optimization as an optimal control problem and use maximum-principle techniques to design distributed update rules for each agent. They claim these rules converge superlinearly, meaning the error shrinks faster than linearly, without inverting Hessian matrices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's superlinear proof is invalid: the factorization in (39) is algebraically wrong and the Taylor remainder is never controlled.","rationale":"The paper's advertised central advance is the superlinear convergence of DOAOC, stated in Theorem 2. The reader's verdict rejected the paper partly because the proof of Theorem 2 contains a specific algebraic error in the definition of G. My independent reading confirms that the derivation in (39) is algebraically incorrect: the matrix factored out as (I - \\eta h*)^k G has an extra \\eta on the B^T P B term relative to the paper's definition of G. Even if that typo were corrected, the proof still relies on a first-order Taylor approximation of \\bar d_k around x* while ignoring the k-dependent higher-order terms; no bound is given on the remainder, so the claimed inequality (36) is not established. This is a load-bearing concern because the central claim depends on it, and it is independent of the separate issue of whether Lemma 3's transformation from optimal control to iteration (19) is valid. I do not think the paper's algorithmic idea is necessarily unsalvageable, but as written the proof of the main advertised theorem is invalid. Since the reader already assigned REJECT for reasons that include this defect, my stress-test does not change the verdict: the rejection stands, though the underlying algorithm might be repairable with corrected algebra, a rigorous remainder bound, and numerical verification. My agreement with the reader is partial rather than full because the reader's identified weakest assumption was the deferred proof of Lemma 3, while I find the most direct load-bearing failure inside Theorem 2's own proof.","tokens_in":11634,"tokens_out":4095,"duration_ms":43068,"concrete_test":"Re-derive (39) symbolically from (37)-(38): factor (I - \\eta h*)^{k+1} - \\eta(I - \\eta h*)^k B^T P B exactly and compare it with (I - \\eta h*)^k G for G as defined in the paper. In addition, take a two-agent scalar quadratic problem, compute the exact local error operator and the norm of the Taylor remainder as a function of k, and check whether the paper's r2 c^k bound is satisfied; if the exact factor differs from \\|G\\| or the remainder is not o(c^k\\|x(k)-x*\\|), Theorem 2's proof fails as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Even accepting Lemma 3, the proof of Theorem 2 does not establish (36). In the derivation from (37)-(38), the paper writes x(k+1)-x* = (I + \\bar d'_k(x*))(x_k - x*) and then, with G := I - (\\eta h* + B^T P B), claims the factor equals (I - \\eta h*)^k G. Direct algebra gives instead (I - \\eta h*)^{k+1} - \\eta(I - \\eta h*)^k B^T P B = (I - \\eta h*)^k [I - \\eta(h* + B^T P B)], so the displayed equality in (39) is false unless B^T P B = 0. The same line also replaces \\bar d_k(x_k) by its first-order Taylor expansion about x* without bounding the remainder uniformly in the inner-loop count k; this is not a removable '\\approx' because the algorithm's direction is generated by up to k recursive inner updates. Consequently the derived contraction factor r2 c^k is not justified, and the central superlinear claim for DOAOC remains unproved. A secondary gap: c = \\|I - \\eta h*\\| < 1 is asserted for 0 < \\eta < 1, whereas the standard sufficient condition is \\eta < 2/m2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two distributed optimization algorithms, DOCMC and DOAOC, by reformulating the minimization of an average of strongly convex smooth local objective functions as a discrete-time optimal control problem. The DOCMC algorithm uses a parameter server and requires inversion of a matrix that combines the Hessian and a Riccati-term; the DOAOC algorithm replaces that inverse by a scalar step η and relies only on neighbor communication plus average-consensus evaluations of the gradient and Hessian. The paper claims convergence for DOCMC and superlinear convergence for both DOCMC (Theorem 1) and DOAOC (Theorem 2).","tokens_in":11875,"tokens_out":6974,"duration_ms":70603,"significance":"If the superlinear claims were rigorously established, the DOAOC algorithm would be a notable contribution: it is a distributed second-order-type method that avoids Hessian inversion, uses only a scalar step η, and has an error bound whose contraction factor decays as c^k. The algorithmic construction is clearly motivated, and the authors are explicit about connections to their prior work. However, the proof of the central superlinear result contains a concrete algebraic error and an uncontrolled Taylor approximation, the key transformation in Lemma 3 is deferred to unpublished prior work, and the consensus argument in Lemma 4 is not rigorous. These issues leave the main claims unproved in the present form.","major_comments":[{"comment":"The factorization in Eq. (39) is algebraically incorrect. The paper defines G = I - (ηh* + B^T P B), but the displayed equality requires G = I - η(h* + B^T P B). Direct algebra gives (I - ηh*)^{k+1} - η(I - ηh*)^k B^T P B = (I - ηh*)^k [I - η(h* + B^T P B)], not (I - ηh*)^k [I - (ηh* + B^T P B)]. The claimed factorization holds only in the trivial case B^T P B = 0. Since the superlinear contraction bound r2 c^k depends on this factorization, Theorem 2 is not proved.","section":"IV, proof of Theorem 2, Eq. (39)"},{"comment":"The proof replaces the direction d̄_k(x(k)) by its first-order Taylor expansion about x* and writes '≈' in Eq. (39), but no bound on the Taylor remainder is provided. The map d̄_k depends on k through up to k inner recursions, so the remainder can in principle depend on k and may destroy the claimed c^k contraction. This is not a removable notational shortcut: the polynomial (I - ηh*)^k is exactly the source of the superlinear factor, and an uncontrolled remainder could contain terms that are not dominated by c^k.","section":"IV, proof of Theorem 2, Eqs. (37)-(39)"},{"comment":"The condition 'selecting 0 < η < 1 and c = ||I - ηh*|| < 1' is not sufficient as stated. Under Assumption 2, the Hessian h* satisfies m1 I ≤ h* ≤ m2 I, and the standard sufficient condition for ||I - ηh*|| < 1 is 0 < η < 2/m2, which is not implied by η < 1 when m2 > 2. The theorem statement should state the correct step-size condition and the proof should verify it; the present condition is inadequate.","section":"IV, statement of Theorem 2"},{"comment":"The central transformation from the non-causal optimal controller (12) to the implementable iteration (19) is not proved in this paper; the proof of Lemma 3 says only 'referring to [14]'. Likewise, Lemma 4 imports the convergence of the centralized subproblem from Lemma 2 of [23], and Remark 5 states that (23) is exactly Algorithm II in [23]. Since [14], [22], and [23] are preprint references and the distributed extension is the claimed contribution, these deferred arguments are load-bearing. The paper needs either self-contained proofs of these steps or a precise statement of which properties of the prior work are used and a verification that they continue to hold in the presence of the consensus terms.","section":"III-B, Lemma 3 and Remark 5"},{"comment":"The consensus claim (26) is not established. Equation (25) mixes a matrix expression involving ((Γ_P + h(k))^{-1}Γ_P)^{k+1}Γ_P^{-1} L̄ with the vector x(k), and the passage to W = I - εL̄ 'satisfying the properties of P_ε in [20]' is not justified. The matrix L̄ = B^T P B R^{-1} is asserted to be a Laplacian, and the matrix ((Γ_P + h(k))^{-1}Γ_P)^{k+1} is asserted to be symmetric, but neither assertion is proved, and the cited properties in [20] require conditions that are not verified here. Thus the contraction in (26) does not follow from the displayed derivation.","section":"IV, Lemma 4"}],"minor_comments":[{"comment":"The heading 'Dissussion of the DOCMC and DOAOC Algorithm' contains a typo and should read 'Discussion'.","section":"III-C heading"},{"comment":"The loop body in Step 8 has a duplicated line number '8:' inside the for loop, which makes the pseudocode confusing.","section":"Algorithm 1, Step 8"},{"comment":"The phrase 'convergence and superlinear convergence rate are also analyzed stringently' is grammatically awkward; the manuscript would benefit from a language edit throughout.","section":"Abstract"},{"comment":"The display of Eq. (29) breaks the norm inequality across a line with the multiplication sign '(30)' on the following line; this should be formatted as a single chain of inequalities.","section":"IV, Theorem 1, Eqs. (29)-(30)"},{"comment":"In Eq. (27), the factor ((Γ_P + h(k))^{-1}R)^{k+1} appears, but the derivation and surrounding text suggest it should be ((Γ_P + h(k))^{-1}Γ_P)^{k+1}. Please correct the typo.","section":"IV, Eq. (27)"},{"comment":"The notation x̄(k) is used before it is defined. The later definition x̄(k) = (1/n)11^T x(k) should be introduced before the inequality in (25).","section":"IV, Lemma 4"},{"comment":"The conclusion states 'global superlinear convergence', but the theorems only provide a local contraction bound under the condition that the iterates converge. The qualifier 'local' or 'under convergence' should be added.","section":"V, Conclusions"},{"comment":"The introduction announces numerical examples in Section V, but no numerical section appears in the manuscript text; either include the experiments or remove the announcement.","section":"I, Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends critically on the authors' own unpublished preprints [14], [22], and [23] at exactly the points where the proofs are needed. This is both a self-containment concern and a novelty-disclosure concern: if those preprints are not available to the reader, the present paper cannot be evaluated on its own. I would ask the editor to require the authors to supply the missing proofs or to explicitly state and verify the assumptions imported from the prior work. The algebraic error in Theorem 2 is locally fixable, but the uncontrolled Taylor remainder and the unproved Lemma 3 are substantial; I therefore recommend major revision rather than rejection, provided the authors can supply rigorous arguments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for passing along 2411.10658. My short take: the paper's central superlinear claims don't hold as written, but the framework isn't empty.\n\nWhat's actually new: the paper recasts distributed optimization as a discrete-time optimal control problem and uses the discrete maximum principle to derive second-order iterations that avoid Hessian inversion. Lemma 1's derivation of the Riccati-based controller is standard but clean, and Lemma 2's observation that the non-causal controller averages gradients is a useful connection to consensus protocols. The authors also deserve credit for being transparent about lineage: Remarks 4 and 5 say explicitly that when agent states coincide, recursions (21) and (23) reduce to Algorithms I and II from the authors' own [14] and [23]. So the genuinely new algorithmic piece is the consensus-error feedback term B^T P e(k) in the distributed setting. That's a modest but real contribution.\n\nThe problems are load-bearing. Theorem 2's proof for DOAOC has a concrete algebra error. The paper defines G = I - (ηh* + B^T P B), but the factor that actually appears in the line before (39) is I - η(h* + B^T P B); the displayed factorization only works if B^T P B = 0. The same proof replaces d_k(x(k)) by its first-order Taylor expansion about x* without bounding the remainder uniformly in k, and d_k is built from up to k inner recursions, so the remainder isn't clearly negligible. Also, c = ||I - ηh*|| < 1 does not follow from 0 < η < 1 alone; the correct sufficient condition is η < 2/m2. Theorem 1 has the same uncontrolled Taylor issue. Lemma 3, the key bridge from the non-causal controller to the implementable iteration, is deferred to [14] without a proof here, and Lemma 4's consensus proof is compressed and leans on \"[23]\". The text says numerical examples appear in Section V, but the version I read ends at the conclusions—no experiments.\n\nSo the superlinear convergence advertised in the title is not established. The idea might be salvageable: fix the G definition, add a genuine uniform bound on the Taylor remainder, and provide experiments. As is, I couldn't rely on the results.\n\nMy verdict: reject, but with encouragement to resubmit after fixing proofs and adding experiments. It's a borderline desk reject; I'd only send it out if you expect the authors to engage seriously with referees. I wouldn't cite it in current form. Good reading-group material for spotting proof gaps.","headline":"Reasonable optimal-control framing for distributed optimization, but the central superlinear proof has an algebraic error and the algorithms largely reduce to the authors' prior work.","tokens_in":12446,"tokens_out":4190,"would_cite":false,"duration_ms":40093,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","93A16","49K15","68W15","90C53"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that distributed optimization can be recast as an optimal control problem, and that the resulting algorithms converge superlinearly to the global minimizer of a sum of strongly convex functions.","keywords":["distributed optimization","optimal control","maximum principle","superlinear convergence","multi-agent consensus","average gradient","second-order optimization","convex optimization"],"falsifier":"Take a two-agent balanced strongly connected graph, choose strongly convex quadratics with known minimizer $x^*$, pick $\\eta$ satisfying the theorem's condition, and record the ratios $\\|x(k+1)-x^*\\|/\\|x(k)-x^*\\|$ along DOAOC iterations. The theorem predicts these ratios are bounded by $r_2 c^k$, so the logarithm of the ratio should fall off linearly with $k$; observing ratios that stay bounded away from zero, or divergence for an $\\eta$ inside the stated range, would disprove the central claim.","tokens_in":11384,"feed_emoji":"📈","tokens_out":9362,"duration_ms":80354,"temperature":0.7,"pith_summary":"The paper tries to establish a new route to distributed optimization: instead of tuning step sizes and inverting Hessians, each agent treats its update as a control input and the overall task as an optimal control problem. Applying the maximum principle to the resulting multi-agent system yields an iterative algorithm, DOAOC, whose error contracts as $\\|x(k+1)-x^*\\| \\le r_2 c^k \\|x(k)-x^*\\|$ with $c=\\|I-\\eta h^*\\|<1$. Because $c^k \\to 0$ geometrically, the method is superlinear: the contraction ratio shrinks on every step. A central-server variant, DOCMC, is given the same treatment with a similar bound. If these claims hold, distributed optimization over balanced, strongly connected networks can reach the optimal value at a rate usually reserved for Newton-type methods, without ever forming the inverse Hessian.","feed_headline":"Superlinear distributed optimization from optimal control","feed_subtitle":"Agents treat updates as control inputs; the maximum-principle solution shrinks error at a rate that tightens every step.","key_machinery":"The machinery is a discrete-time linear system $x(k+1)=x(k)+u(k)$ combined with a per-agent cost that penalizes consensus errors $e_{ij}(k)=x_i(k)-x_j(k)$, control effort, and the local objective $f_i$. The maximum principle turns this into forward-backward equations whose solution is the controller (12), with a Riccati equation (11) carrying the consensus information. A matrix $M(l)$ appearing in the controller is shown to converge to $\\frac{1}{n}\\mathbf{1}\\mathbf{1}^T$, so the optimal control naturally uses the average gradient. The non-causal future-dependent controller is converted, via Lemma 3, into the iterative updates (19) and then (22), where the averaged Hessian $h(k)=\\frac{1}{n}\\sum_i \\nabla^2 f_i(x_i(k))$ plays the role of second-order information and a scalar $\\eta$ replaces the inverse matrix. This replacement is what yields the contraction factor $(I-\\eta h^*)^k$ in Theorem 2.","core_discovery":"On the paper's own terms, the central discovery is that the maximum principle applied to the consensus-plus-optimization cost functional (8) produces controllers whose limiting form uses the average gradient and average Hessian, and that the non-causal controller (12) can be converted into a causal, implementable iteration. The result is an algorithm that updates each agent by $x_i(k+1)=x_i(k)+d^k_i(k)$, where each inner loop step is $d^{l+1}_i(k)=d^l_i(k)-\\eta g_i(k)-\\eta h_i(k)d^l_i(k)$ and the initial direction uses the consensus error. Theorem 2 states that under strong convexity and smoothness assumptions and with $\\eta$ chosen so $\\|I-\\eta h^*\\|<1$, any convergent run satisfies $\\|x(k+1)-x^*\\| \\le r_2 c^k \\|x(k)-x^*\\|$, i.e. superlinear convergence, because the prefactor $c^k$ forces the contraction ratio to zero as $k$ grows.","pith_inferences":["The theorem's condition $\\|I-\\eta h^*\\|<1$ is an asymptotic condition at the optimum; the authors do not give a practical rule for selecting $\\eta$ before knowing $h^*$, so a line search or adaptive estimate would be needed in applications.","Because DOAOC uses only the average Hessian through consensus, the method could plausibly be combined with finite-time average consensus protocols; the convergence proof would then need to account for the transient consensus error, a testable extension the paper does not address.","The proof structure suggests the superlinear rate is inherited from the centralized Algorithm II of the cited prior work, with the consensus term acting as a stabilizer; if that is so, improving the centralized inner loop would immediately improve the distributed algorithm."],"forward_implications":["If the superlinear bound is correct, agents reach the global minimizer of (1) with a contraction ratio that itself decays geometrically, so the number of accurate digits grows at least linearly with iterations.","Because the algorithm avoids Hessian inversion, per-iteration cost stays close to gradient-type methods while the convergence rate approaches Newton-type behavior.","The consensus error term in the control law means the algorithm can tolerate agents' states being unequal, whether from initialization or communication disturbances, without losing the optimization objective.","The DOCMC central-server variant provides the same superlinear guarantee for parameter-server architectures, making the result portable to machine-learning deployments that already use a central node."],"supporting_citations":[{"why":"Supplies the methodology that converts the non-causal optimal controller (12) into the implementable iteration (19); Lemma 3 is explicitly deferred to this reference.","marker":"[14]"},{"why":"Provides Algorithm II of the previous work and its convergence proof, used here for the centralized subproblem and for the symmetry of the matrix in Lemma 4.","marker":"[23]"},{"why":"The consensus-only specialization of the controller; its form is used for the consensus error term in the distributed algorithm.","marker":"[17]"},{"why":"States the maximum principle version from which the costate and equilibrium conditions (9)-(10) are taken.","marker":"[18]"},{"why":"Supplies the Laplacian eigenvalue and consensus convergence facts used in Lemma 2 and Lemma 4.","marker":"[20]"},{"why":"Establishes the average-gradient distributed optimization idea and the consensus contraction inequality used in Lemma 4.","marker":"[19]"},{"why":"Motivates replacing the inverse matrix with a scalar eta, which defines the DOAOC iteration.","marker":"[22]"},{"why":"The distributed subgradient baseline and the consensus condition under Assumption 1; the paper contrasts its step-size-free method with this algorithm.","marker":"[7]"}],"fun_headline_variants":["Maximum principle yields superlinear distributed optimization","Agents design control inputs to shrink error superlinearly","Control-based updates achieve superlinear distributed optimization","Causal controller from maximum principle speeds consensus","Agents as controllers: superlinear distributed optimization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The superlinear claim rests on Lemma 3, which converts the maximum-principle controller into the implementable iteration; the paper does not prove this conversion but imports it from earlier work, and it is not shown that the conversion remains valid when consensus error terms are present.","fun_headline_variants_meta":{"raw":{"variants":["Maximum principle yields superlinear distributed optimization","Agents design control inputs to shrink error superlinearly","Control-based updates achieve superlinear distributed optimization","Causal controller from maximum principle speeds consensus","Agents as controllers: superlinear distributed optimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000723,"raw_usage":{"total_tokens":3189,"prompt_tokens":837,"completion_tokens":2352,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":453,"completion_tokens_details":{"reasoning_tokens":2283}},"tokens_in":453,"tokens_out":2352,"duration_ms":19810,"temperature":1.0,"reasoning_tokens":2283,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:27:41.355972+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a two-agent balanced strongly connected graph, choose strongly convex quadratics with known minimizer $x^*$, pick $\\eta$ satisfying the theorem's condition, and record the ratios $\\|x(k+1)-x^*\\|/\\|x(k)-x^*\\|$ along DOAOC iterations. The theorem predicts these ratios are bounded by $r_2 c^k$, so the logarithm of the ratio should fall off linearly with $k$; observing ratios that stay bounded away from zero, or divergence for an $\\eta$ inside the stated range, would disprove the central claim.","supporting_citations":[{"cited_title":"Optimization Methods Rooting in Optimal Control","cited_arxiv_id":"2312.01334","evidence_quote":"Supplies the methodology that converts the non-causal optimal controller (12) into the implementable iteration (19); Lemma 3 is explicitly deferred to this reference."},{"cited_title":"Superlinear Optimization Algorithms","cited_arxiv_id":"2403.11115","evidence_quote":"Provides Algorithm II of the previous work and its convergence proof, used here for the centralized subproblem and for the symmetry of the matrix in Lemma 4."},{"cited_title":"Distributed Optimal Control and Application to Consensus of Multi-Agent Systems","cited_arxiv_id":"2309.12577","evidence_quote":"The consensus-only specialization of the controller; its form is used for the consensus error term in the distributed algorithm."},{"cited_title":"Zhang, H","cited_arxiv_id":null,"evidence_quote":"States the maximum principle version from which the costate and equilibrium conditions (9)-(10) are taken."},{"cited_title":"Olfati-Saber and R","cited_arxiv_id":null,"evidence_quote":"Supplies the Laplacian eigenvalue and consensus convergence facts used in Lemma 2 and Lemma 4."},{"cited_title":"Qu and N","cited_arxiv_id":null,"evidence_quote":"Establishes the average-gradient distributed optimization idea and the consensus contraction inequality used in Lemma 4."},{"cited_title":"Distributed Optimization Algorithm with Superlinear Convergence Rate","cited_arxiv_id":"2409.12392","evidence_quote":"Motivates replacing the inverse matrix with a scalar eta, which defines the DOAOC iteration."},{"cited_title":"Nedi ´c, and A","cited_arxiv_id":null,"evidence_quote":"The distributed subgradient baseline and the consensus condition under Assumption 1; the paper contrasts its step-size-free method with this algorithm."}],"review_version":1}