{"id":"a612cf96-8528-4815-9c50-4d04034be44e","arxiv_id":"2507.11895","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In high-dimensional regression, classical influence functions underestimate true leave-one-out influence by a per-point factor, and the proposed Newfluence estimator corrects this bias.","lead":"This paper shows that classical influence functions, a standard tool for explaining machine learning model decisions, systematically underestimate how much each training point matters when models have many parameters. The authors propose a corrected estimator, Newfluence, that keeps the same computational cost and is far more accurate in high-dimensional settings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1's consistency claim rests entirely on Lemma 3.3 of the authors' own preprint (Zou et al., 2025); if that lemma's hypotheses are stricter than A1–A4/B1–B2, the Newfluence error bound is unsupported.","rationale":"I read the strongest claim as the asymptotic consistency of Newfluence and the factor-(1−Hii) bias formula for classical influence functions. Both parts hinge on the same single-Newton-step error bound, introduced in Appendix B.1 via Lemma 3.3 of Zou et al. (2025). This is a genuine load-bearing dependency: the paper's proof is not self-contained at its critical point. I checked whether any other internal flaw would independently invalidate the theorem. The Woodbury derivation of Newfluence is algebraically sound, Hii satisfies 0 ≤ Hii < 1 by the Sherman–Morrison form with a positive semidefinite remainder, and the concentration bounds used in the proof are standard. The numerical experiments support the theory but do not verify the imported lemma. Therefore the manuscript is neither clearly wrong nor fully established; its correctness is conditional on the external lemma. The reader's weakest assumption identified exactly this dependency, so I agree with the reader's assessment. A conditional verdict is appropriate: the paper should either include the lemma and its proof or point to a publicly inspectable version whose hypotheses match A1–A4/B1–B2, and the LASSO example in the introduction should be reconciled with the smoothness assumptions. I would not change the verdict.","tokens_in":11238,"tokens_out":5508,"duration_ms":58576,"concrete_test":"Inspect Lemma 3.3 of arXiv:2505.07640 and attempt to re-derive the bound ||bβ/i − β~Newton/i|| = oP(PolyLog(n)/sqrt(n)) using only Assumptions A1–A4 and B1–B2 of this paper, with m=t=1 in the lemma. If the derivation requires additional hypotheses, such as λ√n→∞, a uniform lower bound on the Hessian, or an initialization within O(1/sqrt(n)) of β*, then Theorem 2.1 must be restated with those conditions. As a supporting check, run the logistic-ridge setup at n=p=2000 and record max_i ||bβ/i − β~Newton/i||; the empirical rate should decay like n^{-1/2} up to log factors if the lemma holds. The decisive check is the proof itself, not the simulation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that a single Newton step from the full-model optimum approximates the leave-one-out optimum to oP(PolyLog(n)/sqrt(n)). This is exactly what Lemma 3.3 of Zou et al. (2025) is cited for in Appendix B.1, yet the lemma is neither stated nor proved in this manuscript, and it is not independently verified. The proof of Part 1 of Theorem 2.1 is the sole anchor for Parts 1 and 3: if the lemma's assumptions include extra conditions not listed in A1–A4 and B1–B2, the claimed oP(PolyLog(n)/n) error for Newfluence and the (1−Hii) bias formula for classical influence functions do not follow. The most likely additional requirements would be a lower bound on the regularization parameter, a stronger Hessian-Lipschitz condition, or a basin-of-attraction assumption on the initialization. I found no internal inconsistency in the rest of the proof: the Woodbury expansion is correct, Hii is well defined and lies in [0,1), and the concentration arguments in Lemma C.4 are standard. The experiments with logistic ridge are consistent with the theory but do not settle the lemma's validity. Thus the manuscript's correctness is conditional on an unverified external result at its load-bearing point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies regularized empirical risk minimization in the proportional asymptotics n,p→∞ with n/p→γ. It claims that in this regime the classical influence function I_IF of Koh and Liang systematically underestimates the true leave-one-out influence I_True by a per-point factor 1−H_ii, with error o_P(PolyLog(n)/n), while the proposed Newfluence estimator, a one-step Newton correction using the Woodbury formula, satisfies |I_New−I_True|=o_P(PolyLog(n)/n) at the same computational cost. Theory is developed for GLMs with smooth, strongly convex separable regularizers and Gaussian features; simulations on logistic ridge regression show near-perfect Kendall τ for Newfluence and degraded τ for I_IF at λ=0.01, with both performing well at λ=10.","tokens_in":11574,"tokens_out":8476,"duration_ms":94827,"significance":"If Theorem 2.1 holds, this is a valuable and surprising result: the classical influence function carries a datapoint-dependent multiplicative bias that does not vanish when p is comparable to n, and the bias can be removed by a parameter-free correction requiring no retraining. The Woodbury-based derivation is clean, the estimator has no tuned parameters, and the experiments directly test the ranking claim. The principal weakness is that the proof of Part 1 rests on Lemma 3.3 of a same-author preprint that is not stated or proved here, so the significance is conditional on that external result being valid and its hypotheses matching A1-A4 and B1-B2.","major_comments":[{"comment":"The proof of Part 1 is entirely based on Lemma 3.3 of Zou et al. (2025), a preprint by the same authors that is neither stated nor proved in this manuscript. That lemma supplies the key bound ||bβ/i − tildeβ^Newton/i|| = o_P(PolyLog(n)/√n), from which both the Newfluence consistency (Part 1) and the bias formula (Part 3) follow. If the lemma has extra hypotheses beyond A1-A4 and B1-B2—for example a lower bound on λ, a stronger Hessian-Lipschitz condition, or an initialization/basin assumption—the central claims are unsupported as stated. Please include the full lemma statement and proof, or an independent proof under A1-A4/B1-B2, before the theorem can be accepted.","section":"Appendix B.1 / Theorem 2.1"},{"comment":"The claim 'we claim without proof that, for most commonly used regularizers we have ||∇^2 r(β)|| = O_P(1)' is used to show H_ii is bounded below and hence (1−H_ii)^{-1}=O_P(1) in the proof of Part 2. Remark 2.2 also asserts H_ii=Θ_P(1/(1+λ)) and max_j r¨_j(β_j)=O_P(1) without proof. These bounds are load-bearing for the nonvanishing-bias conclusion. Please prove them under the paper's assumptions or restrict the regularizer class (e.g., to ridge/elastic net with uniformly bounded Hessian); otherwise the factor 1−H_ii could be degenerate or the inverse unstable.","section":"Appendix A.1 / Remark 2.2"}],"minor_comments":[{"comment":"The sentence 'by choosing γ < 1, we it will even cover the situation where the number of features are less than the number of observations' is garbled and the direction is wrong: γ<1 corresponds to p>n. Please fix.","section":"Section 2.1"},{"comment":"In the proof of Lemma C.4, the definition of z should be z=√p x_i (or x_i/√p depending on convention), not z=n^{-1/2} x_i; as written the scaling does not produce the quoted χ² tail probability.","section":"Lemma C.4"},{"comment":"In the chain proving Part 1, the displayed inequality should use |x_0^T(·)| rather than x_0^T(·) before the Cauchy-Schwarz step, since the polynomial-growth factor multiplies an absolute value.","section":"Appendix B.1"},{"comment":"The symbol PolyLog(n) is used for different polynomials in different places. Please state whether the theorem's o_P(PolyLog(n)/n) bound holds for some fixed polynomial of log n or for every polynomial, and use consistent notation.","section":"Notation / throughout"},{"comment":"The name is inconsistently typeset as NEWFLUENCE and Newfluence; please unify.","section":"Section 4 / Abstract"}],"recommendation":"major_revision","confidential_remarks":"The main theorem's proof depends on a same-author preprint (Zou et al., 2025) that is not included in the submission. For a journal review, please ensure this preprint is available to reviewers or require the authors to include the lemma and proof in an appendix. This is not a question of motivation but of verifiability of the central result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a clean, useful result: classical influence functions in high-dimensional GLMs systematically underestimate true leave-one-out influence by a per-point factor (1 - H_ii), and a one-Newton-step correction called Newfluence removes that bias. The bias formula and the consistency guarantee are genuinely new, and the connection to the approximate leave-one-out cross-validation literature is apt. The simulations with logistic ridge match the theory and show a dramatic improvement in rank correlation.\n\nWhere it gets shaky: the proof of Part 1 of Theorem 2.1, which anchors everything, relies entirely on Lemma 3.3 of the authors' own preprint (Zou et al., 2025) for the error of a single Newton step. That lemma is not stated, proved, or independently verified in this manuscript, and its hypotheses may be stricter than A1-A4/B1-B2. If the lemma fails, both the Newfluence consistency claim and the (1-H_ii) bias formula are unsupported. This is a real soft spot, not a manufactured one. There are also two smaller issues: the appendix claims ||nabla^2 r(beta)|| = O_P(1) for 'most commonly used regularizers' without proof, and the paper mentions LASSO as an example despite Assumption A2 requiring a twice-differentiable regularizer (the experiments only cover logistic ridge, so the mismatch is never tested). The concluding remarks acknowledge the smoothness limitation, which is honest, but the LASSO mention early on is misleading.\n\nAll that said, the central idea is likely right. The Woodbury expansion is correct, H_ii is well defined in [0,1), and the concentration arguments are standard. The paper is worth serious referee time, but the authors need to either prove the Newton-step lemma or point to a publicly inspectable version. Right now the proof is conditional on an external result at its load-bearing point.\n\nFor a reader: anyone working on influence functions, data valuation, or high-dimensional interpretability gets value here. I'd cite the bias formula if the lemma checks out, but I wouldn't build on it until then. Send it to review, but insist the key lemma be included or verified.","headline":"Clean bias decomposition and a plausible fix for influence functions in high dimensions, but the main proof leans on an unverified lemma from the authors' own preprint.","tokens_in":599,"tokens_out":655,"would_cite":true,"duration_ms":22339,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F35","62J07","62R07"],"pacs":[],"model":"deepseek-v4-flash","headline":"In high dimensions, the classical influence function undercounts true leave-one-out influence per data point, and Newfluence removes the bias at the same cost.","keywords":["influence functions","leave-one-out","high-dimensional statistics","generalized linear models","interpretability","Newton step","Woodbury identity","regularized empirical risk minimization"],"falsifier":"For square-loss linear regression with Gaussian features and fixed $n/p$, compute $I_{\\mathrm{IF}}$, $H_{ii}$, and exact leave-one-out influence $I_{\\mathrm{True}}$ for every training point. The paper's Part 1 and Part 3 imply $(1-H_{ii})^{-1}I_{\\mathrm{IF}}-I_{\\mathrm{True}}=o_P(\\mathrm{PolyLog}(n)/n)$; if the residual stays of order $1/\\sqrt{n}$ instead, the central claim is false. A simpler empirical falsifier is to rerun the paper's logistic-ridge simulation and check whether Newfluence's Kendall $\\tau$ remains near 1 while classical IF falls to roughly 0.88.","tokens_in":11034,"feed_emoji":"🎯","tokens_out":9638,"duration_ms":103351,"temperature":0.7,"pith_summary":"Machine-learning interpretability tools routinely use influence functions to estimate how much removing one training point changes the prediction on a test point. The standard formula, which avoids retraining by making a first-order Taylor approximation around the fitted model, was designed for settings where the number of parameters is far smaller than the number of observations. This paper argues that in modern high-dimensional regimes, where the parameter count is comparable to the sample size, that formula develops a systematic per-data-point bias: it approximates $(1-H_{ii})$ times the true influence, so influential points can appear unimportant. The paper introduces Newfluence, a single-Newton-step estimator with the same computational cost that is shown to estimate the true leave-one-out influence consistently. If the argument is correct, a widely used attribution technique has a high-dimensional blind spot, and a simple correction closes it.","feed_headline":"Influence functions undercount; Newfluence restores their accuracy","feed_subtitle":"In high dimensions the standard formula misses a per-point term; one Newton step and a Woodbury correction remove the bias.","key_machinery":"The load-bearing object is a rank-one-corrected Newton step. Starting from the full-data fit $\\hat\\beta$, one Newton step toward the leave-one-out minimizer is $\\tilde\\beta^{\\mathrm{Newton}}_{/i}=\\hat\\beta+\\dot{\\ell}_i(\\hat\\beta)G^{-1}x_i/(1-H_{ii})$, obtained by applying the Woodbury identity to the leave-one-out Hessian. Newfluence defines $I_{\\mathrm{New}}(z_i,z_0)=\\ell_0(\\tilde\\beta^{\\mathrm{Newton}}_{/i})-\\ell_0(\\hat\\beta)$. This carries the argument because it approximates the retrained prediction without retraining, and because the Woodbury derivation exposes exactly where the classical approximation goes wrong: the standard formula omits the $1/(1-H_{ii})$ factor and therefore reports $(1-H_{ii})I_{\\mathrm{True}}$ instead of the true influence. The consistency proof additionally relies on a bound showing that the single Newton step is close to the true leave-one-out retrained estimate.","core_discovery":"The paper's central claim is an asymptotic identity for generalized linear models with Gaussian features and a smooth strongly convex regularizer. As $n,p\\to\\infty$ with $n/p\\to\\gamma$, the classical influence estimate $I_{\\mathrm{IF}}(z_i,z_0)$ equals $(1-H_{ii})\\,I_{\\mathrm{True}}(z_i,z_0)+o_P(\\mathrm{PolyLog}(n)/n)$, where $H_{ii}=x_i^\\top G^{-1}x_i\\,\\ddot{\\ell}_i(\\hat\\beta)$ is the leverage-like diagonal entry of $XG^{-1}X^\\top\\mathrm{diag}(\\ddot{\\ell}_i)$ and $G$ is the Hessian of the regularized loss. Since $H_{ii}$ is typically of constant order, the classical estimator understates true influence, and the distortion varies from point to point. The paper's proposed Newfluence estimator evaluates the test loss at a single Newton step from the full-data solution, $\\hat\\beta + \\dot{\\ell}_i(\\hat\\beta)G^{-1}x_i/(1-H_{ii})$, and is proven to satisfy $|I_{\\mathrm{New}}(z_i,z_0)-I_{\\mathrm{True}}(z_i,z_0)|=o_P(\\mathrm{PolyLog}(n)/n)$. In logistic ridge experiments at $n/p=1/2$ with weak regularization, Newfluence achieves near-perfect Kendall rank correlation with exact leave-one-out influence while the classical approximation drops to about 0.88.","pith_inferences":["The same $1/(1-H_{ii})$ correction should apply to other leave-one-out diagnostics built on influence-style approximations, such as fast estimates of prediction error or cross-validated risk, so the bias documented here may be more widespread than influence attribution alone.","Because $H_{ii}$ acts as a per-point leverage, the correction grows as the model approaches interpolation; the paper's strong-convexity assumption keeps $1-H_{ii}$ bounded away from zero, but behavior near the interpolation boundary is a natural place to stress-test the method.","The single-Newton-step construction is model-agnostic, so the correction could be adapted to non-convex models by applying the step in the last linear layer or a local quadratic surrogate, and then checked against brute-force leave-one-out on small networks."],"forward_implications":["Influence-function rankings computed from the classical formula are not merely noisy in high dimensions; they are systematically biased, so data debugging, data valuation, and model-alignment decisions built on them should be rechecked.","Newfluence corrects the bias at essentially the same computational cost as the classical formula, since the same Hessian inverse is reused and each point needs only a rank-one update.","In low-dimensional or strongly regularized models, where $\\mathrm{df}/p$ is small, classical influence functions remain accurate, and Newfluence matches them; the correction matters exactly when model complexity is non-negligible relative to the sample size.","The paper's high-dimensional asymptotic framework applies beyond influence functions and can be used to analyze other attribution tools such as Shapley values.","For logistic ridge models with $n/p=0.5$ and $\\lambda=0.01$, the experiments report near-perfect Kendall $\\tau\\approx 1.00$ for Newfluence versus $\\approx 0.88$ for classical influence functions."],"supporting_citations":[{"why":"Defines the classical influence-function approximation I_IF whose high-dimensional accuracy the paper challenges.","marker":"Koh & Liang (2017)"},{"why":"Establishes the low-dimensional influence-curve rationale that the paper argues breaks down when p is comparable to n.","marker":"Hampel (1974)"},{"why":"Supplies Lemma 3.3, the single-Newton-step error bound on which Theorem 2.1 Part 1 depends.","marker":"(Zou et al., 2025)"},{"why":"Provides the approximate leave-one-out and Newton-step technique that Newfluence adapts to influence estimation.","marker":"Rahnama Rad & Maleki (2020)"},{"why":"Reports empirical failures of influence functions in neural networks, motivating the theoretical account of the bias.","marker":"Bae et al. (2022)"},{"why":"Establishes the high-dimensional proportional-asymptotics framework and concentration tools used in the proofs.","marker":"El Karoui et al. (2013)"},{"why":"Supplies the high-dimensional logistic-regression asymptotics that set the experimental regime.","marker":"Sur et al. (2019)"}],"fun_headline_variants":["Influence functions undercount; Newfluence fixes the bias","Newfluence corrects influence functions in high dimensions","High-dim influence functions undercount; Newfluence removes bias","Influence functions fail in high dims; Newfluence succeeds","Newfluence gives accurate influence where classic fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything depends on an unverified bound, taken from the authors' earlier preprint, that one Newton step lands within a tiny error of the true retrained model; if that bound is wrong, the main consistency result is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Influence functions undercount; Newfluence fixes the bias","Newfluence corrects influence functions in high dimensions","High-dim influence functions undercount; Newfluence removes bias","Influence functions fail in high dims; Newfluence succeeds","Newfluence gives accurate influence where classic fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000648,"raw_usage":{"total_tokens":3028,"prompt_tokens":1048,"completion_tokens":1980,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":1903}},"tokens_in":664,"tokens_out":1980,"duration_ms":15867,"temperature":1.0,"reasoning_tokens":1903,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:59:41.269243+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For square-loss linear regression with Gaussian features and fixed $n/p$, compute $I_{\\mathrm{IF}}$, $H_{ii}$, and exact leave-one-out influence $I_{\\mathrm{True}}$ for every training point. The paper's Part 1 and Part 3 imply $(1-H_{ii})^{-1}I_{\\mathrm{IF}}-I_{\\mathrm{True}}=o_P(\\mathrm{PolyLog}(n)/n)$; if the residual stays of order $1/\\sqrt{n}$ instead, the central claim is false. A simpler empirical falsifier is to rerun the paper's logistic-ridge simulation and check whether Newfluence's Kendall $\\tau$ remains near 1 while classical IF falls to roughly 0.88.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the classical influence-function approximation I_IF whose high-dimensional accuracy the paper challenges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the low-dimensional influence-curve rationale that the paper argues breaks down when p is comparable to n."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports empirical failures of influence functions in neural networks, motivating the theoretical account of the bias."},{"cited_title":"J., Lim, C., and Yu, B","cited_arxiv_id":null,"evidence_quote":"Establishes the high-dimensional proportional-asymptotics framework and concentration tools used in the proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the high-dimensional logistic-regression asymptotics that set the experimental regime."}],"review_version":1}