{"id":"5ec5fbdb-57d8-4940-9743-7b0ec35b2534","arxiv_id":"2607.23388","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Directional Influence Function estimates training-point impact on constrained learners by linearizing the variational inequality of optimality and solving a small QP.","lead":"Classical influence functions break on constrained ML models because removing data can move both the loss and the feasible set. This paper derives a Directional Influence Function from variational-inequality sensitivity analysis that stays feasible and matches leave-one-out retraining on constrained regression and fairness CNNs.","discovery_kind":"new_method","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"Prop. 9's error bound is vacuous as stated, and the \"DIF recovers LOO\" claim is only certified locally: nothing controls the finite 1/N removal when the active constraint set changes along the path — a regime the experiments never stress.","rationale":"The reader identified SOSC/LICQ as the weakest assumption and noted Prop. 9 is \"slightly underspecified.\" I partially agree: SOSC is a genuine limitation for the deep-learning claim, but it is stated openly as a hypothesis of Thm. 8, so the conditional claim is internally sound on that front. The softer spot I find more load-bearing for the strongest claim is the gap between the local (infinitesimal) accuracy actually proved in B.2 and the finite 1/N removal used in validation, compounded by Prop. 9 being vacuous as stated (an O(∥∆ε∥) bound on a difference of two O(∆ε) quantities). This is a correctness-of-scope issue, not a contradiction: the math (VI linearization, critical cone, QP equivalence) is standard Dontchev–Rockafellar machinery and appears correctly specialized, and the toy example genuinely demonstrates the failure of IF and the fix. My concern does not overturn the paper; it delimits the regime in which the headline empirical claim is supported and points at a cheap, decisive diagnostic (active-set stratification) the authors can run with their existing CVXPY pipeline. Hence the verdict stays CONDITIONAL — the condition just shifts from \"SOSC holds\" toward \"active set stable over the removal, or error quantified when it is not.\"","tokens_in":24521,"tokens_out":3850,"duration_ms":102146,"concrete_test":"Re-run the §5.1 constrained-regression LOO trials, recording the active/binding partition at θ̂ and at each retrained LOO solution, and stratify Fig. 3(b) residuals by whether the partition changed. Then add targeted removals designed to force a flip (e.g., remove the point with the largest multiplier-weighted constraint gradient contribution to a binding constraint, or shrink N so 1/N is large enough that removals deactivate constraints). If DIF error concentrates in the active-set-change subset and grows from ~0 to O(∥∆θ_LOO∥), the \"recovers LOO\" claim holds only for activity-preserving removals and Prop. 9 needs an explicit smallness condition; if errors stay small under flips, the concern does not land.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central empirical claim is that DIF recovers leave-one-out solution changes, i.e., accuracy at the finite perturbation ∆ε̄ = (−1/N₀, …, −1/Nₘ), not at infinitesimal t. The theory certifies only local first-order accuracy: Appendix B.2 shows S(ε̄+∆ε̄) − s̄(−∇εf·∆ε̄) = o(∥∆ε̄∥), with no quantification of how small ∆ε̄ must be. Prop. 9, the paper's stated accuracy result, says ∥∆θ − ∆θ̂∥ ≤ M∥∆ε̄∥ — but this follows trivially from the triangle inequality, since both ∆θ (Lipschitz solution map) and ∆θ̂ (positively homogeneous in ∆ε̄, Cor. 3) are individually O(∥∆ε̄∥). So Prop. 9 certifies nothing about approximation quality; the real guarantee is the o(·) term whose neighborhood is uncontrolled.\n\nWhy this matters specifically here: the DIF is computed from the critical cone at ε̄ (frozen active/binding/non-binding partition). The linearized VI (15) correctly handles first-order activation of currently non-binding active constraints, but if along the ray from ε̄ to ε̂ an inactive constraint becomes active, or a binding constraint goes slack beyond first order, the solution path has a kink at some t* < 1 and the first-order extrapolation error at t = 1 is O(∥∆θ∥), not o(1/N). Influence estimation is most interesting precisely for high-influence points, which are the ones most likely to flip constraint activity. The validation does not probe this: the regression task (n=1000, d=5, 1/N tiny) likely keeps the active set stable across all 100 trials, which would explain the near-perfect y=x alignment in Fig. 3(b) without exercising the failure mode; no active-set diagnostics are reported. The CNN experiment sidesteps solution comparison entirely (loss proxy), so it cannot detect this either. The reader's SOSC concern is about existence in non-convex settings; this concern is about accuracy at finite step even in the convex setting where SOSC holds.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The manuscript introduces the Directional Influence Function (DIF), an extension of classical influence functions (Koh & Liang, 2017) to constrained empirical risk minimization, where data perturbations affect both the objective and the feasible region. The authors formulate the KKT optimality conditions of the perturbed constrained problem as a variational inequality over R^d × R^m_+, linearize it around the reference solution, and show (Prop. 6, Thm. 8) that under LICQ and a second-order sufficient condition the solution map is directionally differentiable, with the directional derivative given by the θ-component of the unique solution of an auxiliary VI. They further show (Thm. 10) that this auxiliary VI is exactly the KKT system of an explicitly constructed QP (18), making DIF computable with standard solvers, and (Prop. 11) that DIF reduces to classical IF when no constraints are active. A toy ℓ1-constrained regression example demonstrates non-existence of the two-sided derivative and infeasibility of the IF step. Validation consists of 100 leave-one-out trials on a small constrained linear regression (d=5, n=1000), where DIF tracks LOO retraining almost exactly while IF and penalty-IF are biased, and a fairness-constrained CNN on MNIST where DIF-predicted test-loss changes on one misclassified test point correlate with retraining at r=0.90.","tokens_in":25012,"tokens_out":3664,"duration_ms":97711,"significance":"If the results hold, this is a useful and, to my knowledge, the first rigorous treatment of data attribution for constrained learning. The strengths are concrete: (i) the derivation is a genuine parameter-free application of the Dontchev–Rockafellar sensitivity theory for generalized equations (critical-cone reduction, single-valued Lipschitz localization of (A + N_K)^{-1}), not a heuristic; the appendix proofs are careful and appear correct under the stated assumptions; (ii) the estimator is computable via an explicit QP whose KKT system is proved equivalent to the auxiliary VI — no tuning parameters in the estimator itself; (iii) the toy example cleanly exhibits the two failure modes of IF (non-existence of the derivative, infeasible estimate), making the motivation falsifiable and checkable; (iv) the LOO comparisons are external benchmarks (independent retraining), not circular fits. The constrained-learning formulation (Chamon & Ribeiro) is increasingly used for fairness/safety, so a correct attribution tool there has practical value. The main limitations to significance are the narrowness of the empirical validation (one synthetic regression, one CNN/dataset/test point) and th","major_comments":[{"comment":"As stated, Prop. 9 (∥∆θ − ∆θ̂∥ ≤ M∥∆ε̄∥) is vacuous as an accuracy result: both ∆θ (Lipschitz solution map, established in Lemma 21) and ∆θ̂ (positively homogeneous in ∆ε̄, Cor. 3) are individually O(∥∆ε̄∥), so the bound follows from the triangle inequality with no content about approximation quality. The actual guarantee proved in Appendix B.2 is the first-order statement S(ε̄+∆ε̄) − s̄(−∇εf·∆ε̄) = o(∥∆ε̄∥) (Eq. 66–67). Prop. 9 should be restated as this o(·) statement (which is the honest analog of the first-order justification of classical IF), or the authors should provide a quantitative remainder (e.g., O(∥∆ε̄∥²) under additional second-order regularity of the active-set structure). As written, a reader could reasonably believe the paper certifies linear-rate accuracy at the finite perturbation ∆ε̄ = (−1/N₀,…,−1/N_m) used in all experiments, which it does not.","section":"§4.2, Proposition 9"},{"comment":"DIF is computed from the critical cone at ε̄, i.e., the active/binding/non-binding partition is frozen at the reference solution. The linearized VI (15) correctly captures first-order activation of non-binding active constraints, but if along the ray from ε̄ to ε̂ an inactive constraint becomes active or a binding constraint becomes slack beyond first order, the solution path has a kink at some t* < 1 and the extrapolation error at t = 1 is O(∥∆θ∥), not o(1/N). High-influence points — the ones attribution is most interested in — are precisely those most likely to flip constraint activity. The validation does not probe this regime: with n=1000, d=5 and 1/N perturbations, the active set plausibly remains stable across all 100 trials, which would mechanically produce the near-perfect y=x alignment in Fig. 3(b). The paper should include at least one experiment that stresses activity changes","section":"§5.1, Fig. 3 (and §4.2, Def. of D via critical cone)"},{"comment":"The CNN experiment does not specify how the QP (18) is actually computed in the deep setting. The QP requires ∇²_θθ L(ε̄, θ̄, λ̄) for a 7-layer convnet, which is large and generally indefinite; LICQ and SOSC (Assumptions 1–2) are invoked for existence but are neither verified nor discussed for this non-convex problem, and Thm. 10's QP-VI equivalence as proved assumes the KKT characterization is valid, which for an indefinite Hessian QP needs care (stationary points need not be optimal; the proof appeals to KKT sufficiency). Please state: what Hessian approximation (Gauss–Newton? damping?) is used, how the QP is solved at this scale, what τ and multiplier values were obtained, and whether the active/binding partition was extracted from the primal–dual run. Without these details the r=0.90 result is not reproducible and the claim that DIF is 'efficient' (abstract) is unsupported — there is","section":"§5.2, constrained CNN"},{"comment":"The CNN validation evaluates influence only through the loss change on a single misclassified test point, justified by the possibility that retraining moves to a different basin. This weakens the central claim ('recovers leave-one-out retraining results') to a one-dimensional projection of it. At minimum, the experiment should be extended to a modest set of test points (or aggregate test-risk change per group, which is the quantity the fairness constraints actually control), and the identity of the test point and the selection procedure for the '100 most influential training samples' (influential by what measure?) should be described. As it stands, one point and one dataset leave open the possibility that the agreement is idiosyncratic.","section":"§5.2, Fig. 4"}],"minor_comments":[{"comment":"The constraint row for j ∈ I_Inactive is labeled 'free', which is confusing: for inactive constraints the corresponding row should simply be absent (equivalently ζ_j = 0). As written, 'free' suggests an unconstrained equality/inequality row is present in the QP. Please clarify the formulation.","section":"§4.3, QP (18)"},{"comment":"Notation inconsistency: the QP variable is ω, but the theorem statement mixes w⋆ and ω⋆ ('(w⋆, ζ⋆) also satisfies...', '(w*, ζ*) = (∆θ̂, ∆λ̂)'). Unify to ω.","section":"§4.3, Theorem 10"},{"comment":"Two glitches: (i) 'Since ¯η = G(0)' should read ¯η = G^{-1}(0); (ii) the text says 'Substituting (61) to (63) yields' but the inequality being substituted into is (62) — the displayed equation numbered (63) appears to be the result, and the numbering between (62) and (63) is off. Also 'satisfies ˚a that' (typo) in (53).","section":"Appendix B.1, Lemma 21"},{"comment":"Assumption 1 refers to 'I Activate' while the rest of the paper uses I_Active. Also, the phrase 'if the limit exists' in Definition 2 combined with the generalized limit (direction may vary in a neighborhood) deserves one sentence clarifying that Thm. 8 establishes existence of exactly this stronger (Hadamard-type) directional derivative, not merely the classical one.","section":"§4.2, Theorem 8"},{"comment":"Prop. 11 (DIF reduces to IF when I_Active = ∅) is stated without proof. The argument is short (with no active constraints, D = {0} × ... collapses and the auxiliary VI reduces to the stationarity equation A∆η̂ + µ∆ε̄ = 0, whose θ-block is the IF formula); include it in the appendix for completeness.","section":"§4.2, Proposition 11"},{"comment":"The claim that equality constraints 'can be transformed into inequalities' is fine for the primal problem, but the sensitivity analysis treats multipliers of two-sided inequalities as free only in the binding case; a brief remark that equalities end up in I_Binding with free multipliers (as in the regression experiment, A_eq ∆θ = 0 in (118)) would prevent confusion.","section":"§2, footnote 1"},{"comment":"Fig. 1's axis labels appear garbled in the rendered version (e.g., '2 = 0', '1 + 2^2 = 0.5'); please regenerate. In Fig. 3, report a quantitative summary (slope/intercept of a regression of estimate on LOO, or mean relative error) in addition to the scatter, so the comparison with IF/penalty-IF bias is quantified.","section":"Figures 1 and 3"},{"comment":"No code or reproducibility statement is given. Given that all experiments use standard tools (CVXPY, MNIST, a small convnet), releasing the QP construction and experimental scripts would substantially strengthen the paper. Relatedly, cite or contrast with concurrent data-attribution surveys (e.g., Lin et al. 2024 is cited for diffusion; a sentence situating DIF among TracIn/Datamodels-style methods for the constrained case would help readers).","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The variational-analysis component is strong and clearly benefits from the co-authorship of a leading figure in that field; the proofs in Appendix B look correct to me under the stated LICQ/SOSC hypotheses. My concerns are concentrated in the gap between the local theory and the finite-perturbation empirical claims (Prop. 9's presentation, untested active-set-flip regime) and in the thinness and under-specification of the CNN experiment. None of these seem unfixable within the manuscript's scope — the active-set stress test in particular requires only the existing CVXPY pipeline — but they are load-bearing for the 'reliable tool for data attribution' claim, hence major rather than minor revision. Fit for the journal is good assuming the empirical section is strengthened."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"Punchline: this is a real methods contribution. Classical IF breaks under hard constraints (feasibility and one-sided derivatives), and they fix it by treating optimality as a VI, linearizing, and reading the directional change off a QP on the critical cone. Rockafellar on the author list shows in the appendix; the LICQ/SOSC existence argument and the QP equivalence are carefully done, and Prop. 11 (DIF collapses to IF when nothing is active) is the right sanity check.\n\nWhat works: the ℓ1 toy example is the right teaching device—two-sided derivative fails, IF steps outside the ball, DIF stays on the face and matches the projected path. On convex constrained least squares (n=1000, d=5) DIF tracks LOO almost on the diagonal while IF and penalty-IF do not. That is the strongest empirical claim and it is credible for the local, active-set-stable regime.\n\nSoft spots, in proportion. Prop. 9 is essentially vacuous: both true Δθ and the positively homogeneous DIF estimate are O(‖Δε‖), so the inequality does not certify approximation quality at finite 1/N removal. The real guarantee is an uncontrolled o(·) neighborhood. The linearized cone is frozen at the original active/binding partition; if a high-influence point flips activity along the ray, first-order extrapolation can be O(1) wrong. The regression experiments never report active-set diagnostics and 1/N is tiny, so they probably never hit that regime. The CNN section switches to a test-loss proxy precisely because non-convex retraining can leave the basin; SOSC is assumed and not relaxed. No code. Those are real limits on the “recovers LOO / reliable for constrained deep nets” rhetoric, not on the core construction.\n\nWho it is for: people doing data attribution or sensitivity under fairness, safety, portfolio, or physics constraints who already speak KKT/VI. Worth a serious referee. I would bring it to reading group and cite the DIF/QP construction if I need constrained influence. Send to peer review.","headline":"Clean VI-based directional influence for constrained learning; math is solid, finite-step LOO claims and deep-net evidence are thinner than the abstract suggests.","tokens_in":23991,"tokens_out":528,"would_cite":true,"duration_ms":19486,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C31","68T05","49J40"],"pacs":[],"model":"grok-4.5","headline":"Classical influence functions break under constraints; a directional version that linearizes the optimality variational inequality recovers leave-one-out solution changes.","keywords":["influence function","constrained learning","variational inequality","data attribution","directional derivative","fairness constraints","sensitivity analysis"],"falsifier":"On a constrained linear-regression problem with known active set, compute DIF via the stated quadratic program after removing one point; if the predicted parameter change systematically deviates from the exact leave-one-out retrain while the classical influence does not, the central claim is false.","tokens_in":23676,"feed_emoji":"📐","tokens_out":844,"duration_ms":18554,"temperature":0.7,"pith_summary":"When models are trained with hard constraints for fairness, safety, or physics, removing a training point can move both the loss and the feasible set, so ordinary influence functions produce infeasible or biased parameter shifts. This paper defines the Directional Influence Function (DIF) as the directional derivative of the solution map with respect to a data-weight perturbation. It obtains that derivative by writing the constrained optimality conditions as a variational inequality, linearizing it, and solving the resulting quadratic program. On constrained linear regression DIF matches leave-one-out retraining almost exactly while classical and penalty-based influence do not; on fairness-constrained CNNs it also predicts test-loss changes under sample removal. The result supplies a practical, constraint-aware tool for data attribution whenever feasibility must be respected.","feed_headline":"Influence functions fixed for constrained learning","feed_subtitle":"A directional derivative via variational inequalities matches leave-one-out retraining where classical IF fails","key_machinery":"The Directional Influence Function (DIF): the directional derivative of the solution map obtained by linearizing the variational inequality that encodes constrained optimality and solving the equivalent quadratic program whose KKT system is that linearized VI.","core_discovery":"Under standard constraint qualifications the directional derivative of a constrained learning solution with respect to data down-weighting exists, equals the primal component of the unique solution of a linearized auxiliary variational inequality, can be computed by a single quadratic program, reduces to the classical influence function when no constraints are active, and empirically recovers true leave-one-out parameter and loss changes on both convex and non-convex constrained models.","pith_inferences":["Because the construction works with any twice-differentiable losses, the same DIF QP can be attached to physics-informed networks or robust optimization without redesigning the influence estimator.","If SOSC is only marginally satisfied, the QP may become ill-conditioned; monitoring the smallest eigenvalue of the projected Hessian would give a practical reliability diagnostic the paper does not supply.","Extending the perturbation from single-point removal to group or continuous reweighting is immediate from positive homogeneity and could support continuous data valuation under constraints."],"forward_implications":["Data attribution pipelines for fairness-, safety-, or physics-constrained models can replace classical influence with DIF and obtain feasible first-order estimates without full retraining.","When all constraints are inactive DIF automatically collapses to ordinary influence, so existing unconstrained code remains correct as a special case.","The same QP yields simultaneous estimates of both primal parameter change and dual multiplier change, giving a joint sensitivity of the KKT system.","Leave-one-out debugging and data cleaning under hard constraints become computationally practical for moderate-dimensional convex problems and, via loss-change proxies, for non-convex nets."],"fun_headline_variants":["Directional Influence Function recovers leave-one-out under constraints","DIF matches retraining where classical influence functions break","Constrained learning influence via variational inequalities","One quadratic program yields true data influence with active constraints","DIF predicts loss changes in fairness-constrained CNNs"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The Lagrangian Hessian must be positive definite on the subspace orthogonal to the gradients of the binding constraints; if that second-order condition fails the directional derivative need not exist or be unique.","fun_headline_variants_meta":{"raw":{"variants":["Directional Influence Function recovers leave-one-out under constraints","DIF matches retraining where classical influence functions break","Constrained learning influence via variational inequalities","One quadratic program yields true data influence with active constraints","DIF predicts loss changes in fairness-constrained CNNs"]},"model":"grok-4.5","effort":"low","cost_usd":0.00369,"raw_usage":{"total_tokens":1172,"prompt_tokens":781,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":36904000,"prompt_tokens_details":{"text_tokens":781,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":334,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":781,"tokens_out":57,"duration_ms":5638,"temperature":1.0,"reasoning_tokens":334,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T23:34:22.987162+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On a constrained linear-regression problem with known active set, compute DIF via the stated quadratic program after removing one point; if the predicted parameter change systematically deviates from the exact leave-one-out retrain while the classical influence does not, the central claim is false.","supporting_citations":[],"review_version":1}