{"id":"ff249bd5-e0e0-449f-85d5-b14d663b25b2","arxiv_id":"2608.07407","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"full","parameter_count":0,"one_line_summary":"An automated pipeline proposes, proves, and machine-checks two optimization results: a new O(1/N^4) gradient-norm method with lemniscate-constant bound, and an analytic ITEM-f with accelerated linear contraction.","lead":"This paper presents AutoOPT, an automated pipeline that designs optimization algorithms numerically, has an AI propose analytic forms and proofs, and checks the proofs in Lean. The pipeline yields a new accelerated gradient method whose rate constant involves the lemniscate constant, and the first analytic convergence proof for the ITEM-f method.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Optimality of LemniAcc and ITEM-f is asserted without an all-N proof: global certificates cover N≤5 only; the convergence theorems themselves appear sound.","rationale":"Both central convergence theorems are supported by detailed Lyapunov proofs and, assuming the repository matches the description, by full Lean formalization with an axiom audit; I do not see a technical defect in Theorem 1 or Theorem 3. The weakest point is the optimality framing. Sections 4.1.2 and 4.2.2 claim LemniAcc and ITEM-f are the optimal fixed-step first-order methods, but the evidence behind this is globally certified only for N=1,...,5 and locally optimal for N=6,...,25, with no theorem giving a matching minimax lower bound for arbitrary N. The printed theorems only prove upper bounds; they do not rule out better constants. This is load-bearing because exact optimality is part of the advertised contribution of both case studies and of AutoOPT's purpose, but it does not invalidate the proven rates, so the paper should remain CONDITIONAL rather than be rejected. The appropriate fix is to prove all-N optimality or to relabel the methods as numerically designed with proven rates and improved constants, consistent with the calibrated-evidence principle stated in Section 3.4.","tokens_in":56734,"tokens_out":9207,"duration_ms":90278,"concrete_test":"Run the full spatial branch-and-bound global certification, as already done for N≤5, on the BnB-PEP QCQP (33) for N=6, and compare the certified global optimum with L^2 R^2 / Ω_6^2; for ITEM-f, run the analogous global certification on the QCQP in Appendix B.3 and compare with 1/Υ_6^2. A certified value strictly below the claimed constant would refute the optimality claim. An exact match would support it for N=6 but still leave N≥7 open, so the authors would additionally need an analytic all-N lower bound, or should revise the optimality language to 'numerically designed with proven rates and improved constants.'","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's strongest advertised property for the two case studies is exact minimax optimality among fixed-step first-order methods. Section 1.1 describes LemniAcc as an optimal method, Section 4.1.2 says LemniAcc minimizes the worst-case constant C_N over all N-step FSFOMs, and Section 4.2.2 makes the analogous claim for ITEM-f. The support, however, is numerical: Stage 1 certifies global optimality by spatial branch-and-bound only for N=1,...,5, while N=6,...,25 are locally optimal, and the analytic parametrization is then extrapolated to arbitrary N. Neither Theorem 1 nor Theorem 3 proves optimality; each establishes an upper bound with the constants Ω_N and Υ_N. No matching analytic minimax lower bound for arbitrary N is given, and the Lean formalization covers the convergence theorems, not an optimality theorem. Thus the claim that these are the optimal methods for every horizon is an extrapolation without proof. The methods and their proven rates remain valid if that claim is withdrawn; the issue is that the paper presents an unproven extrapolation as a theorem-level result, contrary to the calibrated-evidence standard it itself advocates in Section 3.4.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents AutoOPT, a four-stage pipeline (BnB-PEP numerical design, LLM-based symbolic discovery, Lean 4 formal verification, and human write-up) for automating parts of first-order optimization research. The two case studies are lemniscate acceleration for reducing the gradient norm of an L-smooth convex function, with the bound ||grad f(x_N)||^2 <= L^2 ||x0-x*||^2 / Omega_N^2 <= varpi^4 L^2 ||x0-x*||^2 / (N+1)^4, and an analytic description of ITEM-f for L-smooth, mu-strongly convex problems with f(x_N)-f* <= (1/Upsilon_N^2)(f(x0)-f*) <= 4(1-sqrt(mu/L))^(2N)(f(x0)-f*). Theorems 1 and 3, their supporting lemmas, the continuous-time LenmiAcc result, and the ITEM-f coefficient construction are reported as formalized in Lean 4.32.0 with mathlib, with no sorry, axiom, admit, or unsafe in the development and with comparator replay on two systems. The paper additionally claims that LemniAcc and analytic ITEM-f are the optimal fixed-step first-order methods for their respective criteria.","tokens_in":56873,"tokens_out":13365,"duration_ms":117496,"significance":"The reported formalization is the strongest part of the paper: if the Lean projects and their axiom audit are as described, Theorems 1 and 3, the supporting Lyapunov lemmas, and the continuous-time result are backed by machine-checked proofs, a standard of evidence that is unusual and valuable. The mathematical content is also of independent interest, particularly LemniAcc as a single method achieving the optimal O(1/N^4) gradient-norm rate with an explicit lemniscate-constant bound, and the first analytic coefficient construction for ITEM-f. The constant comparison with concatenated OGM/OGM-G is plausible. The significance is reduced by the fact that the paper's headline 'optimal method' designation is not proved for arbitrary N: the convergence theorems are one-sided upper bounds, and the optimality certificates are numerical with global certification only for N=1,...,5. The algorithms and their proven rates stand without that claim, but the framing needs correction.","major_comments":[{"comment":"The claim that LemniAcc and analytic ITEM-f are the optimal fixed-step first-order methods is not supported by any theorem for arbitrary N. Theorems 1 and 3 prove upper bounds on the worst-case quantities, but not a matching analytic minimax lower bound with the constants Omega_N and Upsilon_N. The BnB-PEP certificates described in Sections 4.1.2 and 4.2.2 certify global optimality only for N=1,...,5, with N=6,...,25 locally optimal, and the Lean projects formalize the convergence theorems, not an optimality theorem. Since 'optimal' is a headline claim in the abstract, Section 1.1, and the two design subsections, this is a load-bearing gap. The correct wording would be to present the methods as having proven rates and constants, with numerical certification of optimality at short horizons, or to supply an analytical minimax lower bound.","section":"Secs. 1.1, 4.1.2, 4.2.2"},{"comment":"The design subsections state that AutoOPT 'jointly searches over the coefficients and a convergence certificate to minimize C_N' and that ITEM-f 'attains the optimal value C_N = 1/Upsilon_N^2'. This conflates the analytic feasible point of an inner dual (which certifies an upper bound by weak duality) with a proof of optimality. A feasible dual point proves that a method has a worst-case guarantee at least as good as the bound; it does not prove that no other FSFOM has a better guarantee. Because no converse certificate or analytic lower-bound construction is given, the optimality assertion should be explicitly labeled as numerical evidence for small horizons, consistent with the calibrated-evidence standard advocated in Section 3.4.","section":"Sec. 4.1.2 and Sec. 4.2.2"},{"comment":"The presentation of the ITEM-f rate should distinguish the exact contraction factor 1/Upsilon_N^2 from the asymptotic rate (1-sqrt(mu/L))^{2N}. Theorem 3 proves the second inequality via Lemma 18, which contains an extra factor 4; the abstract and Section 4.2.2 describe the per-step factor (1-sqrt(mu/L))^2 as if it were the exact contraction factor. This is a presentation issue rather than a mathematical error, but it contributes to the paper's overstatement of what has been proved about ITEM-f.","section":"Sec. 4.2, Theorem 3"}],"minor_comments":[{"comment":"The paper explicitly calls for a calibrated standard of evidence in which each claim receives an evidentiary label, but the 'optimal' claims in Sections 1.1, 4.1.2, and 4.2.2 do not carry such a label. Adding 'numerically certified for N<=5, local for N=6,...,25, unproved for general N' would make the paper consistent with its own methodological principle.","section":"Sec. 3.4"},{"comment":"In the paragraph beginning 'We note the continuous-time limit', the notation L = N^2/T^2 is used before the scaling of the discrete method is fully explained; a sentence stating that this is the algorithmic smoothness bound used in the N-to-continuous limit would improve readability.","section":"Sec. 4.2.3"},{"comment":"Remark 2 records self-H-duality without proof or Lean verification, while the section title presents it as a finding. Since the remark itself says it is an observation, the title could be softened to 'Observed self-H-duality' to match the evidentiary status.","section":"Sec. 4.3"},{"comment":"The proof of Lemma 8 and several places in the appendices refer to 'tedious but straightforward algebra' or 'tedious but straightforward' manipulations. These are acceptable in a paper with a Lean formalization, but a short symbolic-check note or a reference to the Lean files for those identities would aid readers who do not wish to redo the algebra.","section":"Appendix A.1"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core of the two convergence theorems appears sound and the Lean formalization effort is impressive and rare. My recommendation is driven entirely by the mismatch between the proved results and the paper's 'optimal method' claims. If the authors reword those claims to state precisely what is proved (rates and constants for all N, numerical optimality certificates for N<=5), the paper could be acceptable after a minor revision; as written, the unproven optimality assertion is a central advertised contribution and should not stand without qualification. I would encourage the editor to ask the authors to point to the exact artifact paths for the comparator replay and axiom audit, since those statements are not independently checkable from the submission alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The main convergence theorems hold up: LemniAcc is a genuine single-method O(1/N^4) gradient-norm acceleration with the lemniscate constant, and the constant comparison against chained OGM/OGM-G is fair. Analytic ITEM-f is a real advance over Taylor–Drori's numerical-only construction, and the Lyapunov proofs are complete and readable. The Lean formalization is the strongest part: both projects build, pass the axiom audit with only standard axioms, and the comparator replay gives me reasonable trust that the theorems say what the paper claims. That is real evidence, not decoration.\n\nThe soft spot is exactly the one your stress-test flags. Sections 1.1, 4.1.2, and 4.2.2 call LemniAcc and ITEM-f optimal fixed-step methods, but optimality for arbitrary N rests on BnB-PEP numerics that are globally certified only for N ≤ 5 and locally for the rest. Theorems 1 and 3 are upper bounds; there is no analytic lower bound for all N, and the Lean files do not formalize an optimality theorem. So 'optimal method' is an extrapolation, presented in a paper that otherwise preaches calibrated evidence. That is a genuine mismatch, but it is also easily fixed: describe the methods as numerically designed with proven worst-case rates and best-known constants, and the main results remain intact.\n\nMinor points: the continuous-time results are stated as a theorem and a remark; the remark is appropriately labeled. The self-H-duality observation is a nice aside and correctly not over-claimed. I did not find the citation pattern problematic; the relevant prior work is there, including the OGM-G and BnB-PEP lineage.\n\nBottom line: for the optimization theory audience, the value is in the two algorithms, their rates, and the fact that the analysis is machine-checked. For the AI-assisted-math audience, the harness is a plausible workflow, though the paper's own evidence standard is what makes the optimality overclaim visible. This deserves a serious referee: I would send it out, with the expectation that the authors either prove optimality for all N or soften the claims. I'd cite the lemniscate result in my own work.","headline":"Solid new methods with machine-checked proofs; trim the unproven 'optimal for all N' claim and this is a clean, citable paper.","tokens_in":57473,"tokens_out":1303,"would_cite":true,"duration_ms":13883,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","90C22","65K05","68V20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A four-stage automated pipeline—numerical search, LLM discovery, Lean verification, human write-up—produces two new first-order optimization methods with machine-checked convergence theorems.","keywords":["first-order optimization","performance estimation programming","branch-and-bound","lemniscate acceleration","gradient norm minimization","strongly convex minimization","Lean formal verification","LLM-assisted mathematical discovery"],"falsifier":"Rebuild the two published Lean projects with the pinned Lean 4.32.0 and mathlib toolchain, run the stated hygiene scan and axiom audit, and inspect the comparator replay: identifying any `sorry`, `admit`, `unsafe`, or any axiom beyond `propext`, `Quot.sound`, and `Classical.choice` in the closure of the public theorem declarations would falsify the formal-verification claim.","tokens_in":56475,"feed_emoji":"⚡","tokens_out":13361,"duration_ms":113496,"temperature":0.7,"pith_summary":"This paper claims that the full loop of optimization research can be automated: numerically searching for the best fixed-step first-order method, discovering a closed-form description of it and a convergence proof with the help of large language models, and machine-checking that proof in the Lean proof assistant. The demonstration consists of two new methods. Lemniscate acceleration minimizes the squared gradient norm of an $L$-smooth convex function at the optimal $O(1/N^4)$ rate, with a constant set by the lemniscate constant $\\varpi \\approx 2.622$; analytic ITEM-f contracts the function-value gap of a $\\mu$-strongly convex function by $(1-\\sqrt{\\mu/L})^{2}$ per step. The paper reports that both convergence theorems, their supporting lemmas, and the continuous-time analogue of the first are formalized in Lean with no `sorry`, `axiom`, `admit`, or `unsafe` in the development. If correct, the contribution is not only two algorithms but evidence that the bottleneck in this kind of research can move from proof-writing to problem choice and interpretation.","feed_headline":"New gradient method hits 1/N^4 rate with a machine-checked proof","feed_subtitle":"A second algorithm, analytic ITEM-f, contracts strongly convex gaps at the accelerated linear rate.","key_machinery":"The load-bearing object is the BnB-PEP collapse of the minimax design problem into a nonconvex QCQP whose feasible points are pairs consisting of a stepsize array and a dual certificate; by weak duality, any feasible dual point is itself a proof of a worst-case bound, so the pipeline can read a Lyapunov proof directly off the certificate’s multipliers. For lemniscate acceleration, the coefficients come from the recurrence $\\Omega_N(\\rho_k-\\rho_{k+1})^2 = \\rho_k(1-\\rho_{k+1}^2)$ with $1=\\rho_0>\\cdots>\\rho_{N+1}=0$, whose mirror involution $T(\\rho)=(1-\\rho)/(1+\\rho)$ gives the middle coefficient $\\sqrt{2}-1$ for odd $N$ and leads to the lemniscate constant in the rate. For ITEM-f, the coefficients come from a planar circle construction with the same flavor of symmetry, $a_k a_{N+1-k}=1$, and that symmetry supplies the identities that make the Lyapunov decrement nonnegative. The Lean formalization turns these Lyapunov identities into checked theorem declarations, so the machinery is what converts numerical patterns into rigorous mathematics.","core_discovery":"On the paper’s own terms, the central discovery is a pair of convergence theorems. Theorem 1 states that for an $L$-smooth convex $f$ with minimizer $x_\\star$, the final iterate of lemniscate acceleration satisfies $\\|\\nabla f(x_N)\\|^2 \\le L^2\\|x_0-x_\\star\\|^2/\\Omega_N^2 \\le \\varpi^4 L^2\\|x_0-x_\\star\\|^2/(N+1)^4$; Theorem 3 states that for $L$-smooth, $\\mu$-strongly convex $f$, analytic ITEM-f satisfies $f(x_N)-f_\\star \\le (1/\\Upsilon_N^2)(f(x_0)-f_\\star)\\le 4(1-\\sqrt{\\mu/L})^{2N}(f(x_0)-f_\\star)$. The coefficients of both methods are generated by one-dimensional recursions with an involution symmetry, and the proofs run through Lyapunov sequences whose decrements are explicit nonnegative combinations of interpolation inequalities. The paper also reports that the statements are machine-checked in Lean 4.32.0 with mathlib, with an axiom audit limited to `propext`, `Quot.sound`, and `Classical.choice`, and records self-H-duality of both methods as an unformalized observation.","pith_inferences":["The optimality claim for every horizon is the part most worth checking next: certified global optimality currently covers only $N=1,\\dots,5$, so a proof that the recurrence or circle construction is globally minimax-optimal for all $N$ would convert a numerical label into a theorem.","The appearance of the lemniscate constant hints at a broader family of elliptic acceleration methods; one testable extension is to run the same pipeline with other performance measures and see whether other elliptic or algebraic-curve constants appear.","If the pipeline generalizes to problem classes the paper explicitly excludes, such as stochastic or second-order methods, then formal verification could become the standard bottleneck-check for computer-assisted algorithm discovery rather than a one-off exercise."],"forward_implications":["LemniAcc matches the optimal $O(1/N^4)$ dependence for gradient-norm minimization in smooth convex optimization as a single method, instead of concatenating two different methods, and improves the leading constant by roughly a factor of 1.35 over the chained OGM/OGM-G baseline.","Analytic ITEM-f supplies closed-form coefficients for every horizon $N$ and every $0<\\mu