{"id":"78f2af3c-7c13-4976-9404-e2f4eb10edb7","arxiv_id":"2607.04828","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Mixed-precision Newton for optimization has limiting accuracy set mainly by gradient error and working precision, not Hessian precision, with explicit rate and accuracy bounds covering inexact and Gauss–Newton methods.","lead":"This paper gives precise error bounds for Newton's method when gradient, Hessian, and updates use different floating-point precisions. The analysis shows you can often cheapen the Hessian solve without losing final accuracy, and it covers inexact and Gauss–Newton variants too.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly identifies both the strongest claim (Theorem 2.3 and the first-order separation of Hessian error from limiting accuracy) and the genuine soft spot (purely local analysis requiring ν_i < 1 and a sufficiently close start). That soft spot is already acknowledged by the authors and does not invalidate the local result that is proved. The derivation is standard, the special cases recover known theory, the experiments measure the modeled quantities rather than fitting, and code is available. No stronger load-bearing concern surfaces on a second pass; the verdict ACCEPT with high confidence therefore stands.","tokens_in":23827,"tokens_out":486,"duration_ms":6070,"concrete_test":"Independently re-derive the first-order forms (2.12)–(2.13) from the exact expressions (2.7)–(2.9) under the regime ν_i \to 0 and α_i||x̂_i-x*|| \to 0; confirm that ε^H_i drops out of γ_i while remaining only in β_i. If the algebra holds, the headline independence claim is secure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 2.3 (error bound with attainable accuracy governed by γ_i independent of ε^H_i to first order) is correctly derived under the stated local assumptions. The proof in Appendix A.1 follows the standard Newton residual expansion (Lemmas A.1–A.2), carefully separates the three error sources of Error Model 2.1, and recovers classical exact-arithmetic, inexact-Newton, and Gauss–Newton rates as special cases. The reader already flags the strictly local regime (ν_i < 1, good start, no globalization) as the weakest assumption; the paper itself documents when the theory is conservative (SINREG n=8, some CUTEst cases). That limitation is real but does not undermine the local claim that is actually proved and experimentally checked. No hidden inconsistency, circularity, or unsupported leap appears in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper develops a local error analysis of Newton’s method for unconstrained optimization under a general Error Model 2.1 that separately bounds absolute gradient error, relative Hessian (formation/solve) error, and relative update error. Theorems 2.3 and 2.4 give explicit bounds on the iterate error and gradient norm, showing that attainable accuracy is governed to first order by γ_i ≈ ε^g_i ∥H(x̂_i)^{-1}∥ + ε_i ∥x*∥ (independent of the Hessian error), while Hessian inexactness mainly enters the linear rate factor β_i ≈ ε^H_i κ(H). The framework recovers classical exact-arithmetic Newton, inexact Newton (via Rigal–Gaches), and Gauss–Newton rates as special cases and is used to propose mixed-precision variants. Extensive numerical experiments on synthetic least-squares problems and a CUTEst subset validate the bounds when the local assumptions hold and illustrate precision-assignment trade-offs.","tokens_in":24075,"tokens_out":1060,"duration_ms":16305,"significance":"Mixed-precision algorithms are well developed in numerical linear algebra but largely lack a unifying local theory for second-order nonlinear optimization. This paper supplies that theory with a clean separation of error sources, recovers known rates as special cases, and yields concrete guidelines (evaluate the gradient at least as accurately as the working precision; tolerate Hessian error inversely proportional to κ(H)). The analysis is deterministic and local rather than worst-case global complexity, which yields sharper, more informative bounds. Strengths include independent measurement of ε^g_i and ε^H_i (extended-precision reference and Rigal–Gaches), publicly available code, and experiments that falsifiably check when the theory is tight versus conservative. The contribution is solid and of clear interest to the optimization and high-performance computing communities.","major_comments":[{"comment":"The central claim of Theorem 2.3 is correctly derived under the stated local hypotheses (H(x*) nonsingular, H Lipschitz near x*, ν_i = ε^H_i κ(H(x̂_i)) < 1, and a sufficiently small starting ball). The Appendix A.1 proof follows the standard residual expansion (Lemmas A.1–A.2) and cleanly separates the three error sources; classical rates are recovered when the error terms vanish. No load-bearing algebraic error was found.","section":null},{"comment":"Section 4.2.1 and Figure 4.2 (SINREG, n=8, κ(H)≈4×10^5) and the CUTEst discussion (MEXHAT, PENALTY1, BROWNAL) show that the assumptions of Theorems 2.3–2.4 can fail while the method still converges, and conversely that the theory correctly flags some failures. Because the paper advertises “precise guidelines for assigning a different precision,” a short, explicit discussion of how a practitioner should use (or not use) the ν_i < 1 and θ_i < θ_max tests when they are known to be conservative would strengthen the practical takeaway. This is a clarification, not a flaw in the local claim.","section":null}],"minor_comments":[{"comment":"Throughout the manuscript many cross-references appear as “??” (e.g., “?? presents the error analysis”, “See ??.”, Algorithm and theorem pointers). These must be resolved before publication.","section":null},{"comment":"Conclusion: “sacrifing” → “sacrificing”.","section":null},{"comment":"Section 1.3 / Definition 2.2: the Lipschitz definition is standard; a brief remark that the same constant LH is used for both the residual lemma and the rate coefficients would help readers track constants.","section":null},{"comment":"Figures 4.1–4.3: the transparent dashed bound curves are helpful; ensuring they are distinguishable in grayscale (or adding markers) would improve accessibility.","section":null},{"comment":"Section 3.2: the relation η_i ≈ ζ_i c_solver u_H is a useful rule of thumb; a one-sentence pointer to the corresponding experiment (Figure 4.5 / 4.6) already present later would tighten the narrative.","section":null},{"comment":"Related-work paragraph on first-order mixed-precision methods ([28,33]) is brief; a sentence contrasting local second-order rates with the global complexity bounds typical of those works would better position the contribution.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically sound and fills a genuine gap. The many unresolved “??” cross-references look like a conversion artifact rather than missing content; once fixed, the paper is close to ready. Scope fits a numerical-optimization or computational-mathematics journal well. I see no novelty or citation concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful takeaway is simple: under their three-source error model, attainable accuracy of Newton is governed to first order by the gradient error and the working precision, not by Hessian inexactness; the latter mainly degrades the linear rate factor. That is the practical rule implementers can use.\n\nWhat is new is the specialization of Tisseur-style floating-point Newton analysis to unconstrained optimization, with gradient and Hessian approximation sources folded in (Rigal–Gaches for inexact Newton, residual term S for Gauss–Newton) and mixed-precision algorithms stated for both. The appendix proofs are standard Lipschitz + Neumann arguments; they recover the classical exact, inexact, and Gauss–Newton rates as special cases, which is the right sanity check. Experiments measure ε^g and ε^H independently (extended-precision reference, Rigal–Gaches) rather than fitting the bounds, and the code is linked. The CUTEst profile and the inexact/Gauss–Newton interplay plots are clear and match the first-order predictions when the assumptions hold.\n\nThe soft spot is real but already flagged by the authors: the analysis is strictly local (nonsingular H(x*), Lipschitz neighborhood, good start, ν_i < 1 every step). No globalization. On some ill-conditioned SINREG and CUTEst instances the theory is conservative and fails to guarantee convergence that is still observed. That does not break the local claim that is actually proved; it just means the paper is about the asymptotic regime, not a black-box solver.\n\nCitation pattern is ordinary background (Higham, Nocedal–Wright, Tisseur, Dembo et al.). No circularity.\n\nThis is for people who implement second-order methods on modern hardware or who care about mixed-precision numerical optimization. It deserves a serious referee. I would accept it for peer review and would cite the precision-assignment rules if I were writing a mixed-precision Newton or Gauss–Newton code.","headline":"Solid local error analysis that cleanly separates gradient, Hessian, and update errors for mixed-precision Newton, with usable precision-assignment rules and honest experiments.","tokens_in":24628,"tokens_out":494,"would_cite":true,"duration_ms":6170,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C30","65K05","65G50","65F10"],"pacs":[],"model":"grok-4.5","headline":"Newton's method can run the Hessian in low precision without hurting final accuracy; the gradient precision and working precision set the floor.","keywords":["Newton's method","mixed precision","error analysis","floating-point arithmetic","inexact Newton","quasi-Newton","Gauss–Newton","local convergence"],"falsifier":"On a problem whose Hessian condition number is moderate, deliberately inflate only the Hessian arithmetic error (or the CG residual tolerance) while keeping gradient and working precision fixed; if the final attainable accuracy worsens by more than a constant factor while the linear rate slows as predicted, the claimed separation of γ from ε^H is false.","tokens_in":24746,"feed_emoji":"√","tokens_out":723,"duration_ms":6384,"temperature":0.7,"pith_summary":"Newton's method is expensive because every step builds a Hessian and solves a linear system. This paper shows that those costly steps can be done in lower floating-point precision (or with deliberate approximation) without spoiling the final answer quality. A single error model covers rounding errors, inexact linear solves, and quasi-Newton Hessian approximations at once. The analysis proves that the sequence still contracts until it reaches a limiting accuracy that depends, to first order, only on the gradient error and the working precision used to store the iterates—not on the Hessian error. Hessian inexactness mainly slows the linear rate of approach. The same bounds recover classical exact-arithmetic results for inexact and Gauss–Newton methods and immediately yield mixed-precision variants of those algorithms. Experiments on synthetic problems and a CUTEst suite confirm that the predicted rates and accuracy floors appear in practice, and that many problems tolerate half-precision Hessians with only modest iteration inflation.","feed_headline":"Hessian can run in low precision; gradient sets the accuracy floor","feed_subtitle":"A single error model shows attainable accuracy depends on gradient and working precision, not Hessian error","key_machinery":"Error Model 2.1 together with Theorems 2.3 and 2.4: the model injects three controlled perturbations into the classical Newton step; the theorems convert those perturbations into explicit α_i, β_i, γ_i (and analogous gradient-norm factors) that bound both the iterate error and the gradient norm until a precision-dependent floor is reached.","core_discovery":"Under a three-source error model (gradient absolute error, Hessian relative error, and update rounding), mixed-precision Newton satisfies a local bound whose quadratic, linear, and constant terms cleanly separate the effects of each error. The attainable accuracy is governed by the constant term γ_i ≈ ε^g_i ‖H^{-1}‖ + ε_i ‖x*‖ and is therefore independent of the Hessian perturbation to first order; Hessian error appears only in the linear rate factor β_i ≈ ε^H_i κ(H).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Gradient error sets Newton accuracy floor; Hessian error does not","Low-precision Hessian safe in Newton; gradient sets the accuracy limit","Newton attainability ignores Hessian error; gradient precision rules","Mixed-precision Newton: Hessian stays cheap while gradient sets floor","Hessian runs low-precision; only gradient and eps control final accuracy"],"cache_read_input_tokens":15744,"weakest_assumption_plain":"Everything is local: the Hessian must be nonsingular and Lipschitz near the minimizer, the starting point must already lie inside a small enough ball, and at every step the product of the Hessian relative error and the condition number must stay strictly below one.","fun_headline_variants_meta":{"raw":{"variants":["Gradient error sets Newton accuracy floor; Hessian error does not","Low-precision Hessian safe in Newton; gradient sets the accuracy limit","Newton attainability ignores Hessian error; gradient precision rules","Mixed-precision Newton: Hessian stays cheap while gradient sets floor","Hessian runs low-precision; only gradient and eps control final accuracy"]},"model":"grok-4.5","effort":"low","cost_usd":0.004732,"raw_usage":{"total_tokens":1348,"prompt_tokens":743,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":47320000,"prompt_tokens_details":{"text_tokens":743,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":537,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":743,"tokens_out":68,"duration_ms":4183,"temperature":1.0,"reasoning_tokens":537,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T12:51:45.550781+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On a problem whose Hessian condition number is moderate, deliberately inflate only the Hessian arithmetic error (or the CG residual tolerance) while keeping gradient and working precision fixed; if the final attainable accuracy worsens by more than a constant factor while the linear rate slows as predicted, the claimed separation of γ from ε^H is false.","supporting_citations":[],"review_version":1}