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REVIEW 4 major objections 8 minor 24 references

Directional Influence Function: Estimating Training Data Influence in Constrained Learning

T0 review · 4 major / 8 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Classical influence functions break under constraints; a directional version that linearizes the optimality variational inequality recovers leave-one-out solution changes.

desk verdict 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. read the letter →

arxiv 2607.23388 v3 pith:B2WH56FO submitted 2026-07-25 cs.LG cs.AI

Xin Wang , R. Tyrrell Rockafellar , Xuegang (Jeff) Ban This is my paper · ORCID
classification cs.LGcs.AI MSC 90C3168T0549J40
keywords influencefunctionconstrainedlearningvariationalinequalitydataattributiondirectionalderivativefairnessconstraintssensitivityanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 8 minor

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.

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 (4)
  1. [§4.2, Proposition 9] 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.
  2. [§5.1, Fig. 3 (and §4.2, Def. of D via critical cone)] 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
  3. [§5.2, constrained CNN] 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
  4. [§5.2, Fig. 4] 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.
minor comments (8)
  1. [§4.3, QP (18)] 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.
  2. [§4.3, Theorem 10] 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.
  3. [Appendix B.1, Lemma 21] 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).
  4. [§4.2, Theorem 8] 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.
  5. [§4.2, Proposition 11] 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.
  6. [§2, footnote 1] 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.
  7. [Figures 1 and 3] 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.
  8. [General] 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).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DIF is derived from linearized VI sensitivity analysis and validated against independent LOO retraining.

full rationale

The paper defines DIF as the directional derivative of the constrained solution map θ(ε), then shows via standard variational-analysis arguments (linearized VI, critical cone, Reduction Lemma) that this derivative equals the θ-component of the unique solution of an auxiliary VI, which is equivalently the primal solution of a QP. Existence rests on LICQ/SOSC; when no constraints are active, DIF reduces to classical IF. None of these steps is definitional of the empirical target: the LOO solution/loss changes used in Section 5 are recomputed by independent retraining and are not inputs to the DIF estimator. Citations to Rockafellar–Wets and Dontchev–Rockafellar supply textbook lemmas on normal/critical cones; they do not encode the paper’s influence claim. No parameter is fitted to LOO and then reported as a prediction. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The central claim rests on standard twice-differentiability, two classical constraint qualifications (LICQ, SOSC), and the identification of the directional derivative of the solution map with the solution of a linearized VI on the critical cone. No numeric constants are fitted to produce the estimator itself. The only invented object is the DIF estimator and its auxiliary QP. Experimental hyperparameters (group count, α, network width) affect validation, not the derivation.

free parameters (2)
  • fairness slack τ and group-partition α in CNN experiments = α=0.5, m=3 (τ not numerically specified in main text)
    Chosen for the MNIST fairness setup (m=3, α=0.5); they shape the active-set geometry on which DIF is tested but are not fitted to make DIF match LOO.
  • penalty coefficients ρ, k in the penalty-IF baseline
    Hand-chosen soft-constraint strengths used only for the baseline comparator, not for DIF.
assumptions (5)
  • domain assumption All loss and constraint functions are twice continuously differentiable (C²).
    Stated at the end of §1; required for Hessians and mixed derivatives in the linearized VI.
  • domain assumption LICQ: active constraint gradients are linearly independent at the reference solution.
    Assumption 1 of Thm. 8; standard CQ guaranteeing unique multipliers and well-behaved critical cone.
  • domain assumption SOSC: Lagrangian Hessian positive definite on the subspace orthogonal to binding constraint gradients.
    Assumption 2 of Thm. 8; used in Lemma 17–18 to prove the auxiliary VI is single-valued and the directional derivative exists.
  • standard math Reduction Lemma / critical-cone normal-cone equivalence for convex sets (Dontchev–Rockafellar).
    Lemma 16; bridges the original VI to the auxiliary VI on the critical cone K.
  • ad hoc to paper Equality constraints can be rewritten as opposing inequalities without changing the analysis.
    Footnote in §2; simplifies the PCL formulation but is not proved for the DIF construction when equalities are present as equalities.
invented entities (2)
  • Directional Influence Function (DIF)
    purpose: Directional derivative of the constrained solution map θ(ε) along the data-removal direction, used as a feasible influence estimator.
    Definition 2; the paper’s central constructed object. Independent evidence is the LOO-matching experiments, which are internal to the paper.
  • Auxiliary VI / QP (18) for computing DIF
    purpose: Computable first-order system whose solution is claimed equal to DIF.
    Prop. 5, Thm. 10; the practical algorithm. No external validation beyond the paper’s own LOO trials.

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Cite this review

Pith. "Pith review of Directional Influence Function: Estimating Training Data Influence in Constrained Learning." pith.science (2026). https://pith.science/paper/B2WH56FO

@misc{pith2026260723388,
  author       = {Pith},
  title        = {Pith review of: Directional Influence Function: Estimating Training Data Influence in Constrained Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2WH56FO}},
  note         = {Machine review of arXiv:2607.23388}
}
read the original abstract

As constrained learning becomes increasingly common, models are trained under explicit feasibility requirements to enforce fairness, safety, robustness, regulariza- tion, and physics or logic constraints. Understanding how training samples in- fluence the model solution (e.g., learned parameters) is crucial for interpretability and robustness. The classical influence function (IF) estimates sample contribu- tions via local sensitivity analysis, measuring how the solution changes when a specific training sample is perturbed or removed. However, IF becomes unreli- able in constrained settings: data perturbations can reshape both the objective and the feasible region, leading to estimates that violate feasibility. In response, we propose the Directional Influence Function (DIF), a novel estimator that explicitly incorporates these constraints into influence estimation. DIF formulates the opti- mality conditions of constrained learning as a variational inequality (VI) and ana- lyzes how perturbing training data affects this VI. We validate DIF on constrained linear regression and demonstrate that it recovers leave-one-out retraining results, whereas IF and penalty-based IF exhibit significant bias. We further apply DIF to fairness-constrained CNNs, where DIF accurately predicts test loss changes under data removal and aligns closely with actual retraining. Our results establish DIF as an efficient and reliable tool for data attribution in constrained learning.

Figures

Figures reproduced from arXiv: 2607.23388 by the authors.

Figure 1
Figure 1. Loss landscape of the toy example when ε = 0 and ε = − 1 3 . The left plot illustrates the non-existence of the derivative of θ(ε) at ε = 0 as dθ(ε) dε [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The relationship between (2) and (18). Theorem 10 (Auxiliary problem). Let (ω ⋆ , ζ⋆ ) denote the optimal primal-dual solution to the QP (18). Then (w ⋆ , ζ⋆ ) also satisfies the VI (15). In particular, the QP (18) and the auxiliary VI (15) admit the same solution pair under substitution (w ∗ , ζ∗ ) = (∆ˆθ, ∆λˆ). Proof. See Appendix B.6. Proposition 11. If no constraints are active at the solution ¯θ, i.e., IActive … view at source ↗
Figure 3
Figure 3. Comparison of solution changes on a constrained linear regression task. (a) IF and the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Actual vs. DIF predicted loss differences on a misclassified test sample. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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