{"id":"591b525c-81eb-41f6-b32b-581241cd6891","arxiv_id":"2501.12666","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Eigen-SAM improves Sharpness-Aware Minimization by explicitly aligning the perturbation with the top Hessian eigenvector, supported by a third-order SDE analysis and consistent small accuracy gains on CIFAR, SVHN, and Fashion-MNIST.","lead":"Using a new third-order stochastic differential equation, this paper argues that Sharpness-Aware Minimization reduces the top Hessian eigenvalue only when its perturbation direction aligns with the leading eigenvector, an alignment that is weak in practice. It then introduces Eigen-SAM, which estimates the leading eigenvector and adds its gradient-orthogonal part to the perturbation, yielding consistent but modest accuracy gains over SAM on image benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.1.1's proof bounds scalar Rayleigh-quotient error, not the vector drift error; the eigenvalue-descent interpretation is asserted through a fixable but unproven bound.","rationale":"The paper's central empirical thread is credible: Eigen-SAM shows consistent modest gains over SAM across several architectures and datasets, the code is released, and the intermittent power-method overhead is small. The third-order SDE is a meaningful extension of Compagnoni et al., and the moment-matching derivation itself is valid. The reader's weakest-assumption emphasis on 'frozen perturbation' is partially incorrect because the SAM update genuinely does not backpropagate through the perturbation direction, so the expansion in Lemma B.3 needs no gradient of u_γ. The actual soft spot is the identification of the resulting drift with -∇\\tilde f_SAM and, in Corollary 4.1.1, with -∇Eλ₁: the proof bounds the wrong quantity. This is fixable and, under the stated alignment condition, the missing error terms are O(ρ³)=O(η), so the conclusion survives in the same weak-approximation sense as the rest of the theorem. The verdict therefore remains conditional: the theory needs a corrected proof or an explicit statement that Eq. 5 holds only up to O(η) drift error, and the unverified SDE-simulation details in §6.1 should be reported before the empirical validation of the SDE is accepted.","tokens_in":21428,"tokens_out":11830,"duration_ms":119752,"concrete_test":"Independently derive the vector bound used by Corollary 4.1.1: for unit u,v1 with ‖u-v1‖ ≤ δρ, prove that ‖E[∇³f_γ(u,u) - ∇³f_γ(v1,v1)]‖ ≤ 2C δρ and ‖E[uᵀ∇²f_γ ∂_i u]‖ ≤ C' δρ under the eigenvalue-gap assumption. If these bounds hold, the corollary is valid at O(η) weak-approximation level and the paper's gap is expository; if they fail, the implicit-sharpness-reduction claim loses its basis. For numerical confirmation, compute both norms on the MNIST MLP used in §6.1 at checkpoints with measured alignment and verify that the drift error scales as ρ³ for the reported ρ=0.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The SDE moment expansion in Lemma B.3 is sound: because SAM's update is defined without differentiating through the perturbation direction u_γ = ∇f_γ/‖∇f_γ‖, the one-step conditional expectation of the update is exactly -η E[∇f_γ(x+ρu_γ(x))], and a Taylor expansion in ρ with u_γ(x) held fixed gives the stated drift to order ρ². The reader's 'frozen perturbation' concern is therefore partly misplaced for the SDE itself. The load-bearing problem is the next step: Theorem B.4 writes the drift as -∇\\tilde f_SAM with \\tilde f_SAM = f + ρE‖∇f_γ‖ + (ρ²/2)E[u_γᵀ∇²f_γ u_γ], but the true gradient of E[u_γᵀ∇²f_γ u_γ] contains the extra term 2E[u_γᵀ∇²f_γ ∂_i u_γ], so the drift is exactly -∇\\tilde f_SAM only when that term vanishes. It vanishes if u_γ is precisely an eigenvector, and under alignment 1-O(ρ) it is only O(ρ), making the drift error O(ρ³)=O(η) when ρ=O(η^{1/3}). Thus the eigenvalue-descent reading in Corollary 4.1.1 is valid only as an order-1 weak approximation, not as an exact identity. The proof in Appendix B does not show this: Eqs. 40-43 bound scalar quantities such as uᵀHu - λ₁, whereas the corollary needs a vector bound on E[∇³f_γ(u,u) - ∇³f_γ(v₁,v₁)]. The missing bound is easy to supply by symmetry and Lipschitz continuity of the third-order tensor, but as written the central corollary is asserted rather than derived.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes Sharpness-Aware Minimization (SAM) through a third-order stochastic differential equation (SDE). The main theoretical claim is that, when the SAM perturbation vector aligns with the top Hessian eigenvector, the SDE drift explicitly contains the gradient of the top eigenvalue, implying that SAM implicitly regularizes sharpness. The paper further proposes Eigen-SAM, which adds the gradient-orthogonal component of an estimated top eigenvector to the perturbation, and reports experiments on CIFAR, Fashion-MNIST, SVHN, and fine-tuning tasks where Eigen-SAM consistently outperforms SAM. The paper also proves a PAC-Bayes generalization bound involving the top eigenvalue and proves a convergence rate for Eigen-SAM.","tokens_in":21841,"tokens_out":15735,"duration_ms":144858,"significance":"If the theoretical claims are correct, the paper provides a more refined continuous-time model of SAM than existing second-order SDEs and identifies perturbation-eigenvector alignment as the key mechanism, leading to a novel and simple algorithm. The empirical contribution is solid: experiments are repeated with confidence intervals, sensitivity analysis is provided, Hessian spectra are shown, and code is available, which makes the observed gains of Eigen-SAM over SAM credible. The theoretical part, however, has load-bearing gaps: the proof of Corollary 4.1.1 does not establish the required vector drift bound, and the SDE drift is presented as the gradient of a scalar while the derivation uses a frozen-perturbation gradient that differs from the full gradient. These issues are fixable but must be addressed for the paper's central interpretation to be rigorous.","major_comments":[{"comment":"The SDE drift is defined as -∇f̃_SAM, where f̃_SAM is a scalar function, but Lemma B.3 computes the discrete one-step mean using the gradient of the perturbed loss with the perturbation direction held fixed, yielding the frozen-gradient vector ∂e_i f̃_SAM. These two vector fields are not equal in general; the full gradient of f̃_SAM contains additional terms from differentiating ∇fγ/‖∇fγ‖ with respect to x. The proof of Theorem B.4 compares the SDE one-step mean to the discrete one-step mean without accounting for this difference. Under the alignment assumption the difference is O(ρ³), which is O(η) when ρ=O(η^{1/3}) and therefore still compatible with an order-1 weak approximation, but the statement of the theorem as an exact SDE with a scalar potential is not justified. Please either define the SDE drift explicitly as the frozen-gradient vector field (which may not be a gradient) or state and prove the O(ρ³) error in the theorem statement.","section":"Theorem B.4 (Eq. 29) and Lemma B.3 (Eq. 16)"},{"comment":"The proof of the corollary bounds scalar differences of Rayleigh quotients, such as ‖E[uᵀ∇²fγu] - E[v1ᵀ∇²fγv1]‖ = O(ρ³) in Eq. (40). However, the corollary claims a replacement in the drift vector field: the term ρ²/2 E[∇³fγ(u,u)] must be replaced by ρ²/2 ∇Eλ1(∇²fγ). This requires a vector bound on ‖E[∇³fγ(u,u)] - E∇λ1(∇²fγ)‖, which is not shown. While Eqs. (42)-(43) provide a vector bound for the first-order term, no such vector bound is given for the third-order term. The missing bound can be supplied using the eigengap (Assumption B.2) and Lipschitz continuity of the third-order tensor, but as written the eigenvalue-descent interpretation in Eqs. (5)-(6) is asserted rather than derived. The corollary should also explicitly state that the replacement holds up to O(ρ³) (or O(ρ⁴) for the stronger alignment).","section":"Corollary 4.1.1, Appendix B (Eqs. 40-43)"},{"comment":"The equality ρ²/2 E[∇³fγ(∇fγ,∇fγ)/‖∇fγ‖²] = ρ²/2 ∇E[∇fγᵀ∇²fγ∇fγ/‖∇fγ‖²] is stated to hold because ∇fγ is treated as a perturbation vector independent of Xt. This equality is correct only if the gradient on the right-hand side does not differentiate through the normalized perturbation. In standard calculus, the right-hand side would contain additional terms involving ∂(∇fγ/‖∇fγ‖)/∂x. The paper should make this frozen-gradient convention explicit in the main text, in the definition of f̃_SAM in Theorem B.4, and in the proof of Lemma B.3, since the subsequent interpretation of the drift as a gradient of a scalar potential relies on this convention.","section":"Section 4.1, Eq. (4)"}],"minor_comments":[{"comment":"In the statement of Theorem 3.1, the term L/(2√n)√(d log(1+‖x‖²/(dσ²))) + O(1) + 2 log(1/δ) + 4 log(n+d) is typeset so that the square root appears to cover only d log(...), whereas the proof in Appendix C places the O(1)+... terms inside the square root. Please make the formula consistent.","section":"Theorem 3.1 (main text)"},{"comment":"In the proof of Theorem D.2, the mini-batch variance bound σ² is used to bound E‖∇f(xt+ρϵ)-∇f(xt)‖², but this is not the variance of the mini-batch gradient. The correct bound follows from β-smoothness: E‖∇f(xt+ρϵ)-∇f(xt)‖² ≤ β²ρ²E‖ϵ‖² = β²ρ²(1+α²). The final bound should therefore contain β²ρ²(1+α²) rather than β²(ρ²+α²).","section":"Theorem D.2 (Appendix D)"},{"comment":"Table 1 states that training was conducted for 200 epochs, but Appendix E says that SGD was allowed to train for twice the number of epochs. Please clarify the exact training protocol for each method.","section":"Table 1 and Appendix E"},{"comment":"The numerical simulation of the third-order SDE is not fully described. Please specify the discretization scheme (e.g., Euler-Maruyama), the number of trajectories used, and how the diffusion term was simulated so that the results are reproducible.","section":"Section 6.1"},{"comment":"The claim of 'state-of-the-art' performance is too strong given that comparisons are made only against SGD and SAM, not against other SAM variants such as GSAM or ASAM. I suggest rephrasing to 'outperforms SAM and SGD on the tested benchmarks.'","section":"Tables 1 and 2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a reasonable contribution to the understanding of SAM, with a novel algorithm and credible empirical evidence. The main theoretical result, however, is not rigorously established as written because of the missing vector bound in Corollary 4.1.1 and the conflation of full and frozen gradients in the SDE statement. These are fixable without changing the paper's scope. The empirical gains of Eigen-SAM are modest but consistent, and code is provided. I recommend major revision rather than rejection, because the theoretical gaps are local and repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper is worth engaging. It does two things, one theoretical and one practical. The theoretical piece is a third-order SDE for SAM that explicitly contains a sharpness-reduction term once the perturbation aligns with the top Hessian eigenvector. The practical piece is Eigen-SAM, a cheap modification that re-estimates the top eigenvector every p steps and adds its gradient-orthogonal component to the SAM perturbation. Empirically, the gains over plain SAM on CIFAR and fine-tuning tasks are consistent and shown with confidence intervals. That is a real result, and the paper earns credit for shipping code and for including the counterexample experiments in Appendix A (Reverse-SAM, EGR).\n\nThe main soft spot is exactly where the stress-test lands. The SDE moment expansion in Lemma B.3 is fine: because SAM's implementation does not differentiate through the perturbation direction, the one-step conditional expectation is -η E[∇f_γ(x+ρu_γ)], and the Taylor expansion holds with u_γ fixed. The problem is the next step, where the drift is rewritten as -∇\\tilde f_SAM with \\tilde f_SAM = f + ρE||∇f_γ|| + (ρ²/2)E[u_γᵀH_γu_γ]. The true gradient of the third term contains 2E[u_γᵀH_γ ∂_i u_γ]; that term is O(ρ) under alignment 1-O(ρ), so the drift error is O(ρ³)=O(η) when ρ=O(η^{1/3}). The corollary's eigenvalue-descent reading therefore holds as an order-1 weak approximation, not as an exact identity. Appendix B's proof only bounds scalar quantities like uᵀHu - λ₁, when the corollary needs a vector bound on the difference of third-order tensor contractions. The missing bound is easy to supply by symmetry and Lipschitz continuity, so this is fixable, but as written the central corollary is asserted.\n\nOther soft spots are minor. The SDE simulation in Section 6.1 lacks enough implementation detail to reproduce (how is the SDE integrated, what is the diffusion discretization?). The 'state-of-the-art' claim is overreach given the baseline set: no comparison with other SAM variants or methods with similar computational overhead. The sensitivity analysis is thin but adequate. The limitation section is honest about the cost.\n\nBottom line: the empirical contribution stands on its own, and the theory is close but needs a repaired proof. I would send it to peer review; I would ask for a proof of the missing vector bound and a modest tightening of claims.","headline":"A useful third-order SDE for SAM and a simple, empirically supported Eigen-SAM tweak, but the headline eigenvalue-descent corollary is asserted rather than proved; fixable, not fatal.","tokens_in":22380,"tokens_out":2410,"would_cite":true,"duration_ms":21398,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"SAM's implicit bias includes a sharpness-reduction term once perturbation aligns with the top Hessian eigenvector, and Eigen-SAM enforces that alignment.","keywords":["Sharpness-Aware Minimization","third-order SDE","top Hessian eigenvalue","implicit regularization","perturbation-eigenvector alignment","Eigen-SAM","generalization bound","flat minima"],"falsifier":"Train a network with SAM while logging $\\mathrm{Align}(\\epsilon^{\\mathrm{SAM}}_\\gamma, v_1)$ and the projection of the SDE's cubic drift term onto $\\nabla\\lambda_1$. The central claim predicts that whenever alignment is $1-O(\\rho)$, that projection equals $\\frac{\\rho^2}{2}\\|\\nabla\\lambda_1\\|^2$ up to $O(\\rho^3)$; observing a mismatch growing like $\\rho^2$ at such alignments would falsify Corollary 4.1.1. A complementary check: compare Eigen-SAM to SAM with the top-eigenvalue component removed from the perturbation; if accuracy and final $\\lambda_1$ do not separate, the explicit regularization is not carrying the gain.","tokens_in":21224,"feed_emoji":"🎯","tokens_out":6817,"duration_ms":62157,"temperature":0.7,"pith_summary":"This paper tries to explain what Sharpness-Aware Minimization (SAM) actually does to the loss landscape, beyond the standard story that it penalizes gradient norm. It derives a third-order stochastic differential equation for SAM's discrete updates and shows that, when the perturbation direction lines up with the Hessian's leading eigenvector, SAM's drift contains an explicit gradient of the top eigenvalue, so the method implicitly minimizes sharpness. The paper then reports that this alignment is poor in practice and proposes Eigen-SAM, which periodically estimates the leading eigenvector and adds its gradient-orthogonal component to the perturbation to force the alignment. If the derivation holds, Eigen-SAM is a direct way to turn SAM's implicit sharpness bias into an explicit one, and the experiments support that with consistent accuracy gains over SAM on several image benchmarks.","feed_headline":"New analysis reveals the term that lets SAM shrink the top Hessian eigenvalue","feed_subtitle":"A third-order SDE exposes the eigenvalue-descent drift, and Eigen-SAM adds the eigenvector to make it work.","key_machinery":"The load-bearing object is the third-order stochastic differential equation (Theorem 4.1): an order-1 weak approximation of discrete SAM with drift $-\\nabla\\tilde f^{\\mathrm{SAM}}$ and curvature-dependent diffusion $\\Sigma^{\\mathrm{SAM}}$. The measure of sharpness is the largest Hessian eigenvalue $\\lambda_1(\\nabla^2 f(x))$, and the bridge from the cubic term to eigenvalue descent is the eigenvalue-differentiation identity $\\nabla^2 f_\\gamma(x)(v_1,v_1) = \\nabla \\lambda_1(\\nabla^2 f_\\gamma(x))$ (Magnus, 1985), valid under an eigenvalue-gap assumption. The alignment measure $\\mathrm{Align}(\\epsilon, v_1) := 1 - \\min_{s\\in\\{\\pm1\\}}\\|\\epsilon/\\|\\epsilon\\| - s v_1\\|$ quantifies when the perturbation points along the top eigenvector; Corollary 4.1.1 turns that alignment into the explicit eigenvalue-gradient drift.","core_discovery":"The central discovery is that SAM's training dynamics, modeled to third order, are driven by the effective objective $\\tilde f^{\\mathrm{SAM}}(x) = f(x) + \\rho\\mathbb{E}\\|\\nabla f_\\gamma(x)\\| + \\frac{\\rho^2}{2}\\mathbb{E}\\frac{\\nabla f_\\gamma^\\top \\nabla^2 f_\\gamma(x)\\nabla f_\\gamma}{\\|\\nabla f_\\gamma\\|^2}$, with the cubic term acting as a gradient of the top Hessian eigenvalue whenever the perturbation $\\nabla f_\\gamma/\\|\\nabla f_\\gamma\\|$ is aligned with the leading eigenvector $v_1(\\nabla^2 f_\\gamma(x))$. Corollary 4.1.1 states that if alignment is at least $1-O(\\rho)$, the SDE drift becomes $\\nabla f + \\rho\\nabla\\mathbb{E}\\|\\nabla f_\\gamma\\| + \\frac{\\rho^2}{2}\\nabla\\mathbb{E}\\lambda_1(\\nabla^2 f_\\gamma)$, so SAM implicitly regularizes both gradient norm and sharpness. On the practical side, the paper shows the alignment is typically insufficient in real networks and introduces Eigen-SAM, which adds $\\alpha\\,\\mathrm{sign}(\\langle\\nabla f_\\gamma,\\hat v\\rangle)\\hat v_\\perp$ to the SAM perturbation, where $\\hat v$ is a power-method estimate of the top eigenvector.","pith_inferences":["Beyond the paper, if the third-order SDE is the right model, SAM's benefit over SGD should be reproducible by any optimizer that penalizes $\\lambda_1$ directly; Eigen-SAM is one such optimizer, and the paper's Hessian spectrum plots predict that flatter spectra, not just lower loss, carry the generalization gain.","Beyond the paper, the frozen-perturbation step that makes the cubic term a gradient suggests a boundary: for large $\\rho$, the true $x$-dependence of $\\nabla f_\\gamma/\\|\\nabla f_\\gamma\\|$ should show up as a drift that is not a gradient, so sharpness descent should degrade; measuring the projection of the residual onto $\\nabla\\lambda_1$ as $\\rho$ grows would locate that boundary.","Beyond the paper, the eigenvalue-gap assumption limits the corollary to regions where $v_1$ is well-defined; near eigenvalue crossings the alignment measure loses meaning, and Eigen-SAM's eigenvector estimate may flip, so performance gains may depend on the training trajectory staying away from degeneracy."],"forward_implications":["SAM can be understood as jointly descending loss, gradient norm, and top Hessian eigenvalue, with the three contributions at scales $1$, $\\rho$, and $\\rho^2/2$, respectively, provided alignment holds.","Measured alignment in practice is poor, so standard SAM does not efficiently minimize $\\lambda_1$; this explains why second-order SDE theories that only see gradient-norm regularization miss part of SAM's behavior.","Eigen-SAM's perturbation, formed from the top-eigenvector estimate's gradient-orthogonal component, provably improves alignment for a wide range of $\\alpha$ (Proposition D.1) and reaches the same order of convergence as SAM on smooth non-convex objectives (Theorem D.2).","With eigenvector re-estimation every $p=100$ steps and $q=5$ power iterations, the added cost is roughly $q/p$ times a gradient, and increasing $p$ to 1000 keeps most of the test-accuracy gain."],"supporting_citations":[{"why":"introduces SAM and the perturbed-loss objective that the paper analyzes and extends.","marker":"Foret et al. (2021)"},{"why":"provides the second-order SDE for SAM that the paper expands to third order and compares against.","marker":"Compagnoni et al. (2023)"},{"why":"supplies the continuous-time SDE approximation framework and the moment-matching lemmas used in the proofs.","marker":"Li et al. (2017)"},{"why":"prior theory claiming SAM regularizes the Hessian spectrum near the minimizer manifold, contrast case for the paper's richer drift.","marker":"Wen et al. (2022)"},{"why":"derives related discrete SAM dynamics that oscillate along the top eigenvector, used to position Corollary 4.1.1.","marker":"Bartlett et al. (2023)"},{"why":"is the eigenvalue-differentiation identity that turns the cubic term into the gradient of the top eigenvalue.","marker":"Magnus (1985)"},{"why":"the general PAC-Bayes theorem on which the new top-eigenvalue generalization bound is built.","marker":"Alquier et al. (2016)"},{"why":"provides the weak-approximation theorem (Lemma B.2) that establishes the order-1 error of the SDE.","marker":"Mil'shtein (1986)"},{"why":"documents gradient alignment with the top Hessian eigenvector, the condition under which the cubic term becomes eigenvalue descent.","marker":"Arora et al. (2022)"}],"fun_headline_variants":["Eigen-SAM: the eigenvector fix that unlocks SAM's sharpness control","Third-order SDE reveals SAM's true drift, and how to repair it","Explicit top-eigenvalue targeting beats implicit SAM regularization","Alignment is key: Eigen-SAM makes SAM's sharpness descent explicit","SAM's missing piece: top-eigenvector alignment, now added"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation treats the SAM perturbation direction as frozen, independent of $x$, when forming the SDE drift; if the true dependence of that direction on $x$ contributes at the same order as the cubic term, the drift is not a true gradient and the eigenvalue-descent reading collapses.","fun_headline_variants_meta":{"raw":{"variants":["Eigen-SAM: the eigenvector fix that unlocks SAM's sharpness control","Third-order SDE reveals SAM's true drift, and how to repair it","Explicit top-eigenvalue targeting beats implicit SAM regularization","Alignment is key: Eigen-SAM makes SAM's sharpness descent explicit","SAM's missing piece: top-eigenvector alignment, now added"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000776,"raw_usage":{"total_tokens":3466,"prompt_tokens":1011,"completion_tokens":2455,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":2360}},"tokens_in":627,"tokens_out":2455,"duration_ms":19951,"temperature":1.0,"reasoning_tokens":2360,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:56:34.806638+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train a network with SAM while logging $\\mathrm{Align}(\\epsilon^{\\mathrm{SAM}}_\\gamma, v_1)$ and the projection of the SDE's cubic drift term onto $\\nabla\\lambda_1$. The central claim predicts that whenever alignment is $1-O(\\rho)$, that projection equals $\\frac{\\rho^2}{2}\\|\\nabla\\lambda_1\\|^2$ up to $O(\\rho^3)$; observing a mismatch growing like $\\rho^2$ at such alignments would falsify Corollary 4.1.1. A complementary check: compare Eigen-SAM to SAM with the top-eigenvalue component removed from the perturbation; if accuracy and final $\\lambda_1$ do not separate, the explicit regularization is not carrying the gain.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"is the eigenvalue-differentiation identity that turns the cubic term into the gradient of the top eigenvalue."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces SAM and the perturbed-loss objective that the paper analyzes and extends."},{"cited_title":"M., Biggio, L., Orvieto, A., Proske, F","cited_arxiv_id":null,"evidence_quote":"provides the second-order SDE for SAM that the paper expands to third order and compares against."},{"cited_title":"L., Long, P","cited_arxiv_id":null,"evidence_quote":"derives related discrete SAM dynamics that oscillate along the top eigenvector, used to position Corollary 4.1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the general PAC-Bayes theorem on which the new top-eigenvalue generalization bound is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the weak-approximation theorem (Lemma B.2) that establishes the order-1 error of the SDE."}],"review_version":1}