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Optimal Robust Recourse with $L^p$-Bounded Model Change

Phone Kyaw, Kshitij Kayastha, Shahin Jabbari

TL;DR

This work addresses robust recourse under model changes by introducing provably optimal algorithms for generalized linear models when model shifts are measured with $L^p$ norms ($p\ge 1$, $p\neq \infty$). By linearizing locally and solving a tractable set of convex subproblems, the authors obtain exact optimal recourse in polynomial time in the feature dimension, significantly reducing the price of recourse compared to $L^\infty$-based approaches. Empirical results on real datasets show substantial cost reductions, improved sparsity, and resilience to post-processing, across linear and non-linear models. The findings have practical implications for deploying robust, cost-effective recourse in dynamic environments where models are regularly updated.

Abstract

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 $L^{\infty}$ norm. However, using the $L^{\infty}$ norm can lead to recourse solutions with a high price. To address this shortcoming, we consider more constrained model changes defined by the $L^p$ norm, where $p\geq 1$ but $p\neq \infty$, 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.

Optimal Robust Recourse with $L^p$-Bounded Model Change

TL;DR

This work addresses robust recourse under model changes by introducing provably optimal algorithms for generalized linear models when model shifts are measured with norms (, ). By linearizing locally and solving a tractable set of convex subproblems, the authors obtain exact optimal recourse in polynomial time in the feature dimension, significantly reducing the price of recourse compared to -based approaches. Empirical results on real datasets show substantial cost reductions, improved sparsity, and resilience to post-processing, across linear and non-linear models. The findings have practical implications for deploying robust, cost-effective recourse in dynamic environments where models are regularly updated.

Abstract

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 norm. However, using the norm can lead to recourse solutions with a high price. To address this shortcoming, we consider more constrained model changes defined by the norm, where but , 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.

Paper Structure

This paper contains 17 sections, 3 theorems, 7 equations, 12 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Algorithms and Analysis
  5. Experiments
  6. Analysis of Price of Recourse
  7. Trade-off Between Implementation Cost and Validity
  8. Sparsity
  9. Feasibility
  10. Conclusion and Discussion
  11. Omitted Proofs from Section \ref{['sec:alg-analysis']}
  12. Omitted Details from Section \ref{['sec:exp']}
  13. Additional Experimental Details
  14. Linearization Using SmoothGrad SmilkovTKVW17
  15. Additional Details About the Trade-off Between Validity and Implementation Cost
  16. ...and 2 more sections

Key Result

Theorem 1

If $f_{\theta_0}$ is a generalized linear model, then Algorithm alg:l1-l1 returns a recourse $x$ that minimizes Equation eq:xr for $p\geq 1$ and $p\ne\infty$ in time polynomial in the number of dimensions $d$.

Figures (12)

  • Figure 1: The frontier of the trade-off between validity and implementation cost on the Small Business Administration dataset and logistic regression models with $\alpha=0.5$. Each subfigure corresponds to a different measure of validity. In each subfigure, curves show the trade-off for different algorithms.
  • Figure 2: The frontier of the trade-off between validity and implementation cost on the German Credit dataset and logistic regression models with $\alpha=0.5$. Each subfigure corresponds to a different measure of validity. In each subfigure, curves show the trade-off for different algorithms.
  • Figure 3: Number of changed features for the German and Small Business Datasets for logistic regression models. Left and right columns correspond to $\alpha=0.1$ and $\alpha=0.5$, respectively. The top row corresponds to the German Credit dataset, while the bottom row corresponds to the Small Business Administration dataset. In each subfigure, bars depict the number of changed features for each of the algorithms at different $\lambda$ values.
  • Figure 4: The frontier of the trade-off between validity and implementation cost on the Small Business Administration dataset and logistic regression models after post-processing. The left and right columns correspond to $\alpha=0.1$ and $\alpha=0.5$. In each subfigure, curves show the trade-off for different algorithms. For each algorithm, solid and dashed lines depict the performance before and after hardmax post-processing is applied.
  • Figure 5: The frontier of the trade-off between validity and implementation cost on the German Credit dataset with $\alpha=0.1$. The left and right columns correspond to logistic regression and neural network models. Each row corresponds to a different measure of validity. In each subfigure, curves show the trade-off for different algorithms.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof : Sketch of the Proof
  • Theorem 2: KayasthaGJ24
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof