# Constraint-Aware Influence Estimation in Deep Constrained Learning

arXiv: https://arxiv.org/abs/2607.23388 · HTML: https://mlsanity.com/paper/2607.23388/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 19 references that are in the map.
- First posted on arXiv 25 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Estimating Training Data Influence by Tracing Gradient Descent](https://mlsanity.com/paper/2002.08484/) (2020)
- [What Neural Networks Memorize and Why: Discovering the Long Tail via Influence Estimation](https://mlsanity.com/paper/2008.03703/) (2020)
- [Probably Approximately Correct Constrained Learning](https://mlsanity.com/paper/2006.05487/) (2020)
- [Diffusion Attribution Score: Evaluating Training Data Influence in Diffusion Models](https://mlsanity.com/paper/2410.18639/) (2024)
- [Resilient Constrained Learning](https://mlsanity.com/paper/2306.02426/) (2023)
- [TimeInf: Time Series Data Contribution via Influence Functions](https://mlsanity.com/paper/2407.15247/) (2024)
- [FairTP: A Prolonged Fairness Framework for Traffic Prediction](https://mlsanity.com/paper/2412.16214/) (2024)
- [Double Duality: Variational Primal-Dual Policy Optimization for Constrained Reinforcement Learning](https://mlsanity.com/paper/2402.10810/) (2024)

## Built on by (most cited first, 1 of 1)

- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)

## Abstract (arXiv)

Constrained learning has been increasingly applied to various domains to ensure explicit feasibility requirements due to fairness, safety, robustness, regularization, and physics or logic constraints. Understanding how training samples influence the solution (e.g., learned parameters) of constrained learning is crucial for interpretability and robustness. The classical influence function may become unreliable 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 new estimator that explicitly incorporates the constraints into influence estimation. DIF formulates the optimality conditions of constrained learning as a variational inequality (VI) and analyzes how perturbing training data affects this VI. We validate DIF in 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.