{"id":"99efb401-b0c3-403a-9726-43a261857408","arxiv_id":"2411.15769","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"New trust-region and Levenberg-Marquardt algorithms for nonconvex-strongly concave minimax problems achieve the best known outer complexity and improve Hessian-vector product complexity to O(epsilon^-1.75).","lead":"This paper introduces two second-order optimization algorithms for nonconvex-strongly concave minimax problems, a gradient-norm-regularized trust-region method and a Levenberg-Marquardt method with negative curvature correction. It proves iteration bounds matching the best known second-order results and shows the inexact versions need fewer Hessian-vector products than previous methods.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The inexact HVP complexity claims depend on running CG/Lanczos with the Chebyshev-approximated Hessian \\hat H_t, but the paper never proves the subproblem termination conditions (2.34)-(2.36) and (3.20)-(3.22) survive this replacement.","rationale":"I read the paper in good faith and checked the algebraic skeleton of the exact-method proofs. The descent lemmas for GRTR and LMNegCur (Lemmas 2.5-2.8 and 3.1-3.3) are internally consistent in outline: the set partitions F/G, the function-decrease budget, and the gradient contraction argument all fit together, and the quoted outer iteration complexity O~(ell^1.5 rho^0.5 mu^-1.5 epsilon^-1.5) follows if the parameter choices are instantiated. I found no fatal contradiction in those parts. The advertised improvement over IMCN/ICLM, however, is not the outer iteration count but the Hessian-vector product exponent. That claim lives in Corollaries 2.1 and 3.1, and there the proof is a one-paragraph composition of external results: the Chebyshev approximation of (nabla_yy^2 f)^{-1} and the CG/Lanczos complexity bounds from [8,47,48]. The section where these are joined is precisely where the paper is least explicit. In particular, the subproblem conditions (2.34)-(2.36) and (3.20)-(3.22) are written with the exact H_t, while the implementable algorithm necessarily uses \\hat H_t; the error \\|H_t-\\hat H_t\\| is controlled by Lemma 2.4 only up to \\epsilon_2, and the descent proofs already consume an \\epsilon_2/2\\|s_t\\|^2 term. Whether the additional cross-term is absorbed by the constants in (2.37) and (3.23) is not shown, and this is a genuine correctness risk rather than a stylistic complaint. This concern is more specific than the reader's stated weakest assumption about unknown Lipschitz constants: even if every constant is known exactly, the inexact HVP guarantee still depends on the unproved compatibility of the approximate Hessian with the subproblem solvers. The reader's verdict of CONDITIONAL remains appropriate, so I do not propose to change it; the paper should supply the missing derivation or state and prove a lemma that quantifies how the subproblem termination conditions are met with \\hat H_t.","tokens_in":30635,"tokens_out":29554,"duration_ms":240718,"concrete_test":"Independently re-derive Lemma 2.9 under Algorithm IGRTR using \\hat H_t: run the CG termination residual in (2.35) on \\hat H_t, propagate the extra \\|H_t-\\hat H_t\\|\\|s_t\\| term through (2.40), and verify that with the parameter choices (2.37) the descent and contraction bounds (2.38)-(2.39) still hold. Repeat the same check for condition (3.22) in Lemma 3.4 with (3.23). If the extra term is not absorbed by the existing slack, Corollaries 2.1 and 3.1 are unproved and the claimed Hessian-vector product improvement over IMCN/ICLM is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central novelty is the improved Hessian-vector product exponent \\epsilon^{-1.75}. The proof in Corollary 2.1 (and Corollary 3.1) stacks two external results: the Chebyshev inverse-Hessian approximation used to build \\hat H_t (Lemma 8 of [32], costing \\tilde O(\\sqrt{\\kappa}) HVPs per application) and the CG/Lanczos subproblem solver of [8,47,48], with per-iteration cost min{n, \\tilde O(L_1^{0.5}L_2^{-0.25}\\epsilon^{-0.25})}. What is never shown is that the subproblem termination conditions (2.34)-(2.36) and (3.20)-(3.22), which are stated for exact H_t, are still satisfied when the solver works with \\hat H_t. If CG is run on \\hat H_t, the residual condition should be stated for \\hat H_t; converting back to H_t adds an error \\|H_t-\\hat H_t\\|\\|s_t\\|, which can be as large as 2\\epsilon_2\\|s_t\\|. The descent proofs in Lemma 2.9 and Lemma 3.4 only reserve \\epsilon_2/2\\|s_t\\|^2 for the approximation error coming from Lemma 2.4; it is not checked whether the slack in the constants (2.37) and (3.23) absorbs this second source. If it is not absorbed, the per-iteration HVP count or the outer iteration descent may fail, and the advertised \\epsilon^{-1.75} exponent would not follow. Alternatively, if exact H_t is used inside CG, the composition of the \\tilde O(\\sqrt{\\kappa}) Chebyshev work per HVP with the quoted CG iteration count needs a joint proof that the appendix does not give.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two second-order algorithms for nonconvex-strongly concave minimax problems: GRTR, a trust-region method with a gradient-norm-regularized Hessian and radius proportional to the square root of the gradient norm, and LMNegCur, a Levenberg-Marquardt method with negative-curvature correction. It claims that both methods find an O(ε,√ε)-second-order stationary point in Õ(ℓ^{1.5}ρ^{0.5}μ^{-1.5}ε^{-1.5}) outer iterations, and that the inexact variants IGRTR and ILMNegCur require only Õ(ℓ^{2.25}ρ^{0.25}μ^{-1.75}ε^{-1.75}) Hessian-vector products and Õ(ℓ^2ρ^{0.5}μ^{-2}ε^{-1.5}) gradient ascent steps with high probability. The analysis follows the standard monotone-descent plus gradient-contraction template, with inner-loop accelerated gradient ascent to approximate the maximizer y*(x_t). Numerical experiments on synthetic problems, a sinusoidal perturbation problem, and an adversarial deep learning problem are reported.","tokens_in":30987,"tokens_out":28547,"duration_ms":213464,"significance":"If fully established, the outer-iteration bounds match the best known complexity for this problem class and the Hessian-vector product exponent improves on IMCN and ICLM, which would be a meaningful contribution. The algorithms are clearly specified and the experiments give qualitative evidence of practical efficiency. However, two technical gaps currently block acceptance: a scaling error in a key descent lemma and an unproven step in the inexact Hessian-vector product analysis. The paper also omits the proof of a main theorem. These issues should be addressed before the claims can be relied upon.","major_comments":[{"comment":"The stationarity and descent claims are not correct as stated. Substituting λ_t ≤ √L2 ε, σ∥g_t∥^{1/2} ≤ (√L2/2)√ε, and ϵ2 = (√L2/12)√ε into the inequality preceding (2.25) gives a lower bound of order -√L2√ε I, not -√L2 ε I; the displayed constant 19/12 cannot be obtained for small ε. Moreover, the descent P(x_{t+1}) ≤ P(x_t) - (1/(128√L2))ε^{3/2} for λ_t ≥ √L2 ε does not follow by 'replacing ∥g_t∥ with ε': with ∥s_t∥ = r ε^{1/2}, the negative term in (2.16) is -1/2(σ∥g_t∥^{1/2}+λ_t)∥s_t∥^2, which is only O(√L2∥g_t∥^{1/2}ε + √L2 ε^2) and can be much smaller than ε^{3/2} when ∥g_t∥ is small. Since the partition into F and G in (2.27)-(2.28) requires this ε^{3/2} descent, Theorem 2.1 is not justified.","section":"Section 2.1, Lemma 2.6 and Eq. (2.25)"},{"comment":"The Hessian-vector product complexity is not supported. The algorithm descriptions in Section 2.2 state the subproblem termination conditions (2.34)-(2.36) in terms of H_t, but Corollary 2.1 replaces the inverse (∇^2_yy f)^{-1} inside H_t by a Chebyshev approximation, yielding \\hat H_t. The paper does not specify whether the CG/Lanczos procedures operate on H_t or \\hat H_t, and does not prove that the termination conditions survive the replacement. If \\hat H_t is used, the descent bounds (2.40) and (3.25) gain an extra term (1/2)∥H_t-\\hat H_t∥∥s_t∥^2, which is not absorbed by the constants in (2.37) and (3.23). If H_t is used exactly, the cost of computing H_t is not counted. In either case the advertised Õ(ε^{-1.75}) exponent is not established.","section":"Section 2.3, Corollary 2.1 and Section 3.3, Corollary 3.1"},{"comment":"The proof is omitted with the comment that it is 'almost identical' to Theorem 2.1. This is not self-evident, since Lemma 3.1 has a different descent inequality (3.7) and the G set in (3.19) uses contraction factor 1/2 instead of 1/3. The bound on consecutive G-iterations, which in Lemma 2.8 depends on the contraction factor and the uniform bound G, must be verified for the new constants \\bar C_1, \\bar C_2, \\bar G. The theorem should be proved or the missing steps supplied.","section":"Section 3.1, Theorem 3.1"}],"minor_comments":[{"comment":"There are numerous typos and duplicated phrases, e.g., 'satifies' in Assumption 2.1, 'respectivelly' in Section 1.2, and 'for a given xt' twice in the description of Algorithm 2.","section":"Throughout"},{"comment":"The baseline algorithm is called 'IMINIMAX-TR' in this section, while earlier it is 'MINIMAX-TR'; please make the notation consistent.","section":"Section 4.3"},{"comment":"The parameter choices in (2.11), (2.37), (3.5), and (3.23) require knowledge of L1, L2, and P(x0)-P*; the paper should explicitly state that the methods are not parameter-free and discuss the practical implications.","section":"Sections 2.1, 2.3, 3.1, 3.3"},{"comment":"The second term should be 2ϵ2 ε^{1/2}∥s_t∥ before using ε ≤ ∥g_t∥; the displayed bound relies on ε < 1 and should be clarified.","section":"Eq. (2.41)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains promising ideas and the broad proof strategy is standard, but the technical issues are substantial. The scaling error in Lemma 2.6 may be a systematic typo (ε vs √ε) across the thresholds λ_t, but the authors must correct it consistently and reprove the affected lemmas. The Hessian-vector product gap requires either a full error analysis for \\hat H_t or a restatement of the algorithm to avoid the ambiguity. I do not see grounds for rejection if these points can be fixed, but they are not cosmetic. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth a serious look: it adapts the universal trust-region idea to nonconvex-strongly-concave minimax and gets the best known outer-iteration complexity while improving the Hessian-vector-product exponent for the inexact variants from ε^-2 to ε^-1.75. The GRTR and LMNegCur designs are sensible, and the negative-curvature correction in LMNegCur is a reasonable alternative to trust-region subproblem solving. The complexity table is honest and the comparisons to [32,7,62] are correct.\n\nThe soft spots are where the proof gets thin. The main one is the gap the stress-test note flags. The inexact HVP bounds in Corollaries 2.1 and 3.1 rely on running CG/Lanczos on the Chebyshev-approximated Hessian \\hat H_t, but the subproblem exit conditions in (2.34)–(2.36) and (3.20)–(3.22) are stated for the exact H_t. The proof that the iteration complexity survives replacement shows \\|\\hat H_t - ∇²P(x_t)\\| ≤ ε2, but it never checks that the descent lemmas (2.9, 3.4) go through when the computed step satisfies the conditions only with \\hat H_t. The extra error is on the order of ε2‖s_t‖², and the constant slack in (2.37) and (3.23) is too tight to absorb it without adjustment. I checked the arithmetic for the ‖gt‖≥ε case: doubling the ε2 term in (2.40) turns the negative descent bound positive with the stated constants. The fix is likely a smaller ε2 by a constant factor, which does not change the exponent, so the result may still be right. But as written, the advertised ε^-1.75 claim is not proven.\n\nOther less serious issues: Theorem 3.1 is stated without proof, and several lemmas are described as \"almost identical\" to earlier ones; that is acceptable if true, but one of those omitted details is exactly where the gap hides. The exact LMNegCur analysis counts the Lanczos cost as at most n iterations and then ignores it; that is fine for an iteration-complexity claim but should be stated explicitly. The numerical section is single runs, no error bars, no code, and some hyperparameters are described as \"showing the most stable and optimal performance\" — not a reproducible claim. These are fixable.\n\nThe paper deserves peer review. The central idea is real, the gap is likely patchable, and the complexity comparison is meaningful. I would send it out with a request to fix the Chebyshev/termination-condition accounting, supply the missing proofs, and make the numerics reproducible. A reader in the optimization subfield gets useful information from this paper; anyone else can skip.","headline":"A genuinely useful complexity improvement, but the proof of the advertised ε^-1.75 HVP bound has a real gap around the Chebyshev-approximated Hessian that needs fixing before the claim stands.","tokens_in":31597,"tokens_out":6633,"would_cite":false,"duration_ms":51663,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C47","90C26","90C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Both GRTR and LMNegCur reach an $O(\\epsilon,\\sqrt{\\epsilon})$-second-order stationary point in $\\tilde{O}(\\ell^{1.5}\\rho^{0.5}\\mu^{-1.5}\\epsilon^{-1.5})$ outer iterations, while the inexact variants use only…","keywords":["nonconvex-strongly concave minimax","gradient norm regularization","trust-region method","Levenberg-Marquardt method","negative curvature","second-order stationary point","iteration complexity","Hessian-vector product"],"falsifier":"Run GRTR on a nonconvex-strongly concave instance with known constants, for example the synthetic problem (4.1) with $n=10$, $L=1$, $\\gamma=1$, and count outer iterations until $\\|\\nabla P(x)\\le \\xi\\epsilon$ and $\\nabla^2P(x)\\succeq -\\theta\\sqrt{\\epsilon}I$; if for some $\\epsilon$ the count exceeds a fixed multiple of $\\sqrt{L_2}(P(x_0)-P^*)\\epsilon^{-3/2}\\ln(1/\\epsilon)$ with $L_2=\\rho(1+\\kappa)^3$, Theorem 2.1's bound is violated, and the same counting test on LMNegCur would settle Theorem 3.1.","tokens_in":30352,"feed_emoji":"⚖️","tokens_out":11647,"duration_ms":91513,"temperature":0.7,"pith_summary":"The paper establishes that second-order methods for nonconvex-strongly concave minimax problems can reach a second-order stationary point at the best known outer-iteration count while spending less work per iteration. The GRTR algorithm adds a Hessian regularization and a trust-region radius that both scale with the square root of the gradient norm; the LMNegCur algorithm replaces the trust-region subproblem by a Levenberg-Marquardt linear system, using a negative-curvature step when the approximate Hessian is too negative. Both are proved to find an $O(\\epsilon,\\sqrt{\\epsilon})$-second-order stationary point within $\\tilde{O}(\\ell^{1.5}\\rho^{0.5}\\mu^{-1.5}\\epsilon^{-1.5})$ outer iterations. The inexact variants IGRTR and ILMNegCur preserve this bound and need only $\\tilde{O}(\\ell^{2.25}\\rho^{0.25}\\mu^{-1.75}\\epsilon^{-1.75})$ Hessian-vector products, improving the $\\epsilon$ dependence of earlier inexact cubic-regularization methods. The significance is a concrete algorithm pair that combines the robustness of trust-region and Levenberg-Marquardt steps with a matching complexity guarantee and cheaper per-iteration linear algebra.","feed_headline":"Second-order minimax methods match record rate, cut Hessian calls","feed_subtitle":"New trust-region and Levenberg-Marquardt variants reach second-order accuracy with cheaper Hessian-vector products.","key_machinery":"The load-bearing object is the gradient-norm regularized trust-region subproblem (2.3): it uses $\\tilde H_t = H_t + \\sigma\\|g_t\\|^{1/2}I$ in the quadratic model and a ball of radius $r\\max\\{\\|g_t\\|^{1/2},\\epsilon^{1/2}\\}$. The dual multiplier $\\lambda_t$ in its optimality conditions classifies each iteration into a descent case or a contraction case, which is what converts the function-value gap into an iteration count. For LMNegCur the corresponding mechanism is the negative-curvature test on $\\lambda_{\\min}(H_t)$: if it is at most $-\\frac12\\sqrt{L_2}\\max\\{\\|g_t\\|,\\epsilon\\}$, the algorithm steps along a unit curvature direction of length $\\sqrt{\\max\\{\\|g_t\\|,\\epsilon\\}/L_2}$; otherwise it solves the LM system with regularization $\\sqrt{L_2\\|g_t\\|}I$. The inner loop of accelerated gradient ascent (length $N_t$ chosen in (2.8)) is what makes the approximate gradient and Hessian accurate enough for Lemma 2.4 and hence for both algorithms.","core_discovery":"In the paper's own terms, gradient-norm regularization is sufficient to carry a second-order minimax method to an $O(\\epsilon,\\sqrt{\\epsilon})$-second-order stationary point at rate $\\tilde{O}(\\ell^{1.5}\\rho^{0.5}\\mu^{-1.5}\\epsilon^{-1.5})$. At iteration $t$, accelerated gradient ascent on $y$ supplies an approximate gradient $g_t$ and Hessian $H_t$ that are within $\\epsilon_1$ and $\\epsilon_2$ of the true ones for $P(x)=\\max_y f(x,y)$. GRTR then solves a trust-region subproblem with regularized Hessian $H_t+\\sigma\\|g_t\\|^{1/2}I$ and radius $r\\max\\{\\|g_t\\|^{1/2},\\epsilon^{1/2}\\}$; the optimality conditions show that the iteration either cuts the function value by order $\\|g_t\\|^{3/2}$ or contracts $\\|g_t\\|$ by a factor of $1/3$, and the two cases are balanced by the function-value gap $P(x_0)-P^*$. LMNegCur uses the same approximate information but detects negative curvature and takes a step of length $\\sqrt{\\max\\{\\|g_t\\|,\\epsilon\\}/L_2}$ along it, otherwise solving the linear system $(H_t+\\sqrt{L_2\\|g_t\\|}I)s=-g_t$. The inexact variants apply trust-region Newton-CG and randomized Lanczos/CG solvers, giving the Hessian-vector product bound stated in Corollaries 2.1 and 3.1.","pith_inferences":["The same descent-versus-contraction dichotomy suggests that gradient-norm regularization could be ported to stochastic or Hessian-free settings, but strong concavity in $y$ is used decisively in Lemma 2.4, so such extensions would need a new inner-loop accuracy argument.","The paper does not give an adaptive way to set $\\epsilon_1,\\epsilon_2,\\sigma,r$ when the problem constants are unknown; a natural next step is a parameter-free or self-tuning version of GRTR, and its absence is the main practical gap.","The $\\epsilon^{-1/4}$ improvement in Hessian-vector products should make IGRTR and ILMNegCur noticeably cheaper than IMCN and ICLM on large-scale problems where matrix-vector products dominate; this is directly testable by benchmarking Hessian-vector product counts rather than wall-clock time on the adversarial deep learning problem in Section 4.3.","Because LMNegCur detects negative curvature early via the Lanczos procedure, its saddle-escaping behavior may be less sensitive to trust-region radius and hard-case geometry than GRTR; the numerical experiments point in that direction, though the paper does not isolate this as a separate claim."],"forward_implications":["Both GRTR and LMNegCur match the best known outer-iteration complexity $\\tilde{O}(\\ell^{1.5}\\rho^{0.5}\\mu^{-1.5}\\epsilon^{-1.5})$ for finding an $O(\\epsilon,\\sqrt{\\epsilon})$-second-order stationary point.","The inexact variants bring the Hessian-vector product count down to $\\tilde{O}(\\ell^{2.25}\\rho^{0.25}\\mu^{-1.75}\\epsilon^{-1.75})$, improving the $\\epsilon$ exponent from $-2$ to $-1.75$ relative to the IMCN and ICLM algorithms.","The total number of gradient ascent steps is $\\tilde{O}(\\ell^2\\rho^{0.5}\\mu^{-2}\\epsilon^{-1.5})$ for all four variants, matching earlier algorithms.","GRTR needs no function values of $P(x)=\\max_y f(x,y)$ and no bounded-gradient assumption, unlike MINIMAX-TRACE; its trust-region radius adapts to the gradient norm instead of staying fixed.","LMNegCur avoids solving a trust-region subproblem altogether: each iteration requires at most one minimum-eigenvalue estimate and one positive-definite linear solve, so the hard case of trust-region subproblems does not arise."],"supporting_citations":[{"why":"supplies the universal trust-region framework whose gradient-regularized subproblem GRTR generalizes to the minimax setting.","marker":"[20]"},{"why":"defines the MCN and IMCN baselines and the Chebyshev approximation of the inverse Hessian used in the inexact complexity corollaries.","marker":"[32]"},{"why":"establishes the smoothness and Lipschitz properties of $P$ and $H$ used as Assumption 2.1 and Lemmas 2.1-2.2.","marker":"[7]"},{"why":"provides the MINIMAX-TR and MINIMAX-TRACE trust-region baselines whose radius strategy and complexity GRTR improves.","marker":"[62]"},{"why":"supplies the trust-region Newton-CG subproblem solver and stopping conditions used by IGRTR.","marker":"[8]"},{"why":"gives the accelerated gradient ascent rate in (2.2.23) that Lemma 2.4 uses to control inner-loop approximation error.","marker":"[36]"},{"why":"supplies the trust-region optimality conditions (Theorem 4.1) that Lemma 2.3 relies on.","marker":"[39]"},{"why":"gives the randomized Lanczos procedure used to estimate the minimum eigenvalue in LMNegCur and ILMNegCur.","marker":"[24]"},{"why":"provides the CG and Lanczos Hessian-vector product counts that bound the inexact variants' per-iteration cost.","marker":"[47]"},{"why":"motivates the Levenberg-Marquardt regularization with gradient norm used in the LMNegCur update.","marker":"[33]"}],"fun_headline_variants":["Minimax second-order methods match record rate, cut Hessian calls","Gradient-norm regularization speeds minimax to second-order accuracy","Inexact minimax solvers need fewer Hessian-vector products","New trust-region and LM variants reach minimax second-order points"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the user to know the strong-concavity coefficient $\\mu$, the Lipschitz constants $\\ell$ and $\\rho$, and the gap $P(x_0)-P^*$ to set the inner-loop lengths and regularization parameters; without those constants the descent and stopping arguments do not apply, and the paper gives no adaptive estimation procedure.","fun_headline_variants_meta":{"raw":{"variants":["Minimax second-order methods match record rate, cut Hessian calls","Gradient-norm regularization speeds minimax to second-order accuracy","Inexact minimax solvers need fewer Hessian-vector products","New trust-region and LM variants reach minimax second-order points"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000425,"raw_usage":{"total_tokens":2356,"prompt_tokens":1299,"completion_tokens":1057,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":915,"completion_tokens_details":{"reasoning_tokens":981}},"tokens_in":915,"tokens_out":1057,"duration_ms":8785,"temperature":1.0,"reasoning_tokens":981,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:55:32.139193+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run GRTR on a nonconvex-strongly concave instance with known constants, for example the synthetic problem (4.1) with $n=10$, $L=1$, $\\gamma=1$, and count outer iterations until $\\|\\nabla P(x)\\le \\xi\\epsilon$ and $\\nabla^2P(x)\\succeq -\\theta\\sqrt{\\epsilon}I$; if for some $\\epsilon$ the count exceeds a fixed multiple of $\\sqrt{L_2}(P(x_0)-P^*)\\epsilon^{-3/2}\\ln(1/\\epsilon)$ with $L_2=\\rho(1+\\kappa)^3$, Theorem 2.1's bound is violated, and the same counting test on LMNegCur would settle Theorem 3.1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the MCN and IMCN baselines and the Chebyshev approximation of the inverse Hessian used in the inexact complexity corollaries."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes the smoothness and Lipschitz properties of $P$ and $H$ used as Assumption 2.1 and Lemmas 2.1-2.2."},{"cited_title":"Two trust region type algorithms for solving nonconvex-strongly concave minimax problems","cited_arxiv_id":"2402.09807","evidence_quote":"provides the MINIMAX-TR and MINIMAX-TRACE trust-region baselines whose radius strategy and complexity GRTR improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the trust-region Newton-CG subproblem solver and stopping conditions used by IGRTR."},{"cited_title":"Nesterov","cited_arxiv_id":null,"evidence_quote":"gives the accelerated gradient ascent rate in (2.2.23) that Lemma 2.4 uses to control inner-loop approximation error."},{"cited_title":"Kuczy´ nski and H","cited_arxiv_id":null,"evidence_quote":"gives the randomized Lanczos procedure used to estimate the minimum eigenvalue in LMNegCur and ILMNegCur."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the CG and Lanczos Hessian-vector product counts that bound the inexact variants' per-iteration cost."},{"cited_title":"Mishchenko","cited_arxiv_id":null,"evidence_quote":"motivates the Levenberg-Marquardt regularization with gradient norm used in the LMNegCur update."}],"review_version":1}