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Directional Influence Function: Estimating Training Data Influence in Constrained Learning

Xin Wang, R. Tyrrell Rockafellar, Xuegang, Ban

Year: 2026Venue: arXiv preprintArea: cs.LGType: PreprintEmbeddings: 6

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%

Last extracted: 8/1/2026, 10:34:41 AM

Summary

The paper proposes the Directional Influence Function (DIF), a novel estimator for measuring training data influence in constrained learning settings. Unlike classical influence functions (IF) which become unreliable when constraints reshape the feasible region, DIF formulates optimality conditions as a variational inequality (VI) to accurately estimate sample contributions. The method is validated on constrained linear regression and fairness-constrained CNNs, demonstrating high accuracy in predicting test loss changes compared to leave-one-out retraining.

Entities (8)

R. Tyrrell Rockafellar · person · 99%Xin Wang · person · 99%Xuegang Ban · person · 99%Directional Influence Function · method · 98%Influence Function · method · 95%Variational Inequality · mathematical-concept · 92%Constrained Linear Regression · task · 90%Fairness-constrained CNNs · model · 90%

Relation Signals (7)

Directional Influence Function → proposedby → Xin Wang

confidence 95% · Authors: Xin Wang... propose the Directional Influence Function (DIF)

Directional Influence Function → proposedby → R. Tyrrell Rockafellar

confidence 95% · Authors: Xin Wang, R. Tyrrell Rockafellar... propose the Directional Influence Function (DIF)

Directional Influence Function → proposedby → Xuegang Ban

confidence 95% · Authors: Xin Wang, R. Tyrrell Rockafellar, Xuegang (Jeff)Ban... propose the Directional Influence Function (DIF)

Directional Influence Function → uses → Variational Inequality

confidence 92% · DIF formulates the optimality conditions of constrained learning as a variational inequality (VI)

Directional Influence Function → appliedto → Constrained Linear Regression

confidence 90% · We validate DIF on constrained linear regression

Directional Influence Function → appliedto → Fairness-constrained CNNs

confidence 90% · We further apply DIF to fairness-constrained CNNs

Influence Function → comparedwith → Directional Influence Function

confidence 85% · whereas IF and penalty-based IF exhibit significant bias... DIF... recovers leave-one-out retraining results

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Abstract

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.

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Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Machine Learning arXiv:2607.23388v2 (cs) This paper has been withdrawn by Xin Wang [Submitted on 25 Jul 2026 (v1), last revised 28 Jul 2026 (this version, v2)] Title:Directional Influence Function: Estimating Training Data Influence in Constrained Learning Authors:Xin Wang, R. Tyrrell Rockafellar, Xuegang (Jeff)Ban View a PDF of the paper titled Directional Influence Function: Estimating Training Data Influence in Constrained Learning, by Xin Wang and 2 other authors No PDF available, click to view other formats 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. Comments: Need revision Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI) Cite as: arXiv:2607.23388 [cs.LG] (or arXiv:2607.23388v2 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2607.23388 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xin Wang [view email] [v1] Sat, 25 Jul 2026 22:50:09 UTC (170 KB) [v2] Tue, 28 Jul 2026 03:42:07 UTC (1 KB) (withdrawn) Full-text links: Access Paper: View a PDF of the paper titled Directional Influence Function: Estimating Training Data Influence in Constrained Learning, by Xin Wang and 2 other authorsWithdrawn No license for this version due to withdrawn Current browse context: cs.LG < prev | next > new | recent | 2026-07 Change to browse by: cs cs.AI References & Citations NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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