# Optimal Robust Recourse with Lp-Bounded Model Change

arXiv: https://arxiv.org/abs/2509.21293 · HTML: https://mlsanity.com/paper/2509.21293/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 74 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Builds on (most cited first, 30 of 54)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [Interpretable Counterfactual Explanations Guided by Prototypes](https://mlsanity.com/paper/1907.02584/) (2019)
- [Model-Agnostic Counterfactual Explanations for Consequential Decisions](https://mlsanity.com/paper/1905.11190/) (2019)
- [Predictive Multiplicity in Classification](https://mlsanity.com/paper/1909.06677/) (2019)
- [Algorithms with Predictions](https://mlsanity.com/paper/2006.09123/) (2020)
- [Learning Model-Agnostic Counterfactual Explanations for Tabular Data](https://mlsanity.com/paper/1910.09398/) (2019)
- [Algorithmic recourse under imperfect causal knowledge: a probabilistic approach](https://mlsanity.com/paper/2006.06831/) (2020)
- [Towards Realistic Individual Recourse and Actionable Explanations in Black-Box Decision Making Systems](https://mlsanity.com/paper/1907.09615/) (2019)
- [On the Convergence and Robustness of Adversarial Training](https://mlsanity.com/paper/2112.08304/) (2021)
- [The Hidden Assumptions Behind Counterfactual Explanations and Principal Reasons](https://mlsanity.com/paper/1912.04930/) (2019)
- [Counterfactual Explanations Can Be Manipulated](https://mlsanity.com/paper/2106.02666/) (2021)
- [Towards Robust and Reliable Algorithmic Recourse](https://mlsanity.com/paper/2102.13620/) (2021)
- [Exploring the Whole Rashomon Set of Sparse Decision Trees](https://mlsanity.com/paper/2209.08040/) (2022)
- [On the Adversarial Robustness of Causal Algorithmic Recourse](https://mlsanity.com/paper/2112.11313/) (2021)
- [Equalizing Recourse across Groups](https://mlsanity.com/paper/1909.03166/) (2019)
- [Exploring Counterfactual Explanations Through the Lens of Adversarial Examples: A Theoretical and Empirical Analysis](https://mlsanity.com/paper/2106.09992/) (2021)
- [On the Long-term Impact of Algorithmic Decision Policies: Effort Unfairness and Feature Segregation through Social Learning](https://mlsanity.com/paper/1903.01209/) (2019)
- [Robust Counterfactual Explanations for Tree-Based Ensembles](https://mlsanity.com/paper/2207.02739/) (2022)
- [Robust Counterfactual Explanations for Neural Networks With Probabilistic Guarantees](https://mlsanity.com/paper/2305.11997/) (2023)
- [Probabilistically Robust Recourse: Navigating the Trade-offs between Costs and Robustness in Algorithmic Recourse](https://mlsanity.com/paper/2203.06768/) (2022)
- [Consistent Counterfactuals for Deep Models](https://mlsanity.com/paper/2110.03109/) (2021)
- [Causal Strategic Classification: A Tale of Two Shifts](https://mlsanity.com/paper/2302.06280/) (2023)

## Built on by (most cited first, 3 of 3)

- [ElliCE: Efficient and Provably Robust Algorithmic Recourse via the Rashomon Sets](https://mlsanity.com/paper/2602.07674/) (2026)
- [The Role of Causality in Algorithmic Recourse](https://mlsanity.com/paper/2607.28497/) (2026)
- [Algorithmic Recourse Under Competition](https://mlsanity.com/paper/2609.39877/) (2026)

## Abstract (arXiv)

Recourse provides individuals who received undesirable labels (e.g., denied a loan) from algorithmic decision-making systems with a minimum-cost improvement suggestion to achieve the desired outcome. However, in practice, models often get updated to reflect changes in the data distribution or environment, invalidating the recourse recommendations (i.e., following the recourse will not lead to the desirable outcome). The robust recourse literature addresses this issue by providing a framework for computing recourses whose validity is resilient to slight changes in the model. However, since the optimization problem of computing robust recourse is non-convex (even for linear models), most of the current approaches do not have any theoretical guarantee on the optimality of the recourse. Recent work by Kayastha et. al. provides the first provably optimal algorithm for robust recourse with respect to generalized linear models when the model changes are measured using the L∞ norm. However, using the L∞ norm can lead to recourse solutions with a high price. To address this shortcoming, we consider more constrained model changes defined by the Lp norm, where p≥ 1 but pneq ∞, and provide a new algorithm that provably computes the optimal robust recourse for generalized linear models. Empirically, for both linear and non-linear models, we demonstrate that our algorithm achieves a significantly lower price of recourse (up to several orders of magnitude) compared to prior work and also exhibits a better trade-off between the implementation cost of recourse and its validity. Our empirical analysis also illustrates that our approach provides more sparse recourses compared to prior work and remains resilient to post-processing approaches that guarantee feasibility.