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Generally-Occurring Model Change for Robust Counterfactual Explanations

Ao Xu, Tieru Wu

TL;DR

This work addresses the robustness of counterfactual explanations under changing decision models. It generalizes the prior Naturally-Occurring Model Change concept to Generally-Occurring Model Change (GOMC) and provides probabilistic guarantees that quantify how counterfactuals behave when the underlying model shifts, even beyond Gaussian neighborhood assumptions. A key theoretical result bounds the discrepancy between original and updated predictions in terms of neighborhood data and distributional changes, with a clear link to Lipschitz continuity and subgaussian variability. The paper further applies the framework to a dataset perturbation case study, deriving concrete bounds under gradient-based learning and positioning GOMC as a wider, optimization-theory-backed tool for ensuring robust recourse. Overall, this work enhances reliability and interpretability of counterfactual explanations in dynamic settings with practical implications for robust decision-support systems.

Abstract

With the increasing impact of algorithmic decision-making on human lives, the interpretability of models has become a critical issue in machine learning. Counterfactual explanation is an important method in the field of interpretable machine learning, which can not only help users understand why machine learning models make specific decisions, but also help users understand how to change these decisions. Naturally, it is an important task to study the robustness of counterfactual explanation generation algorithms to model changes. Previous literature has proposed the concept of Naturally-Occurring Model Change, which has given us a deeper understanding of robustness to model change. In this paper, we first further generalize the concept of Naturally-Occurring Model Change, proposing a more general concept of model parameter changes, Generally-Occurring Model Change, which has a wider range of applicability. We also prove the corresponding probabilistic guarantees. In addition, we consider a more specific problem, data set perturbation, and give relevant theoretical results by combining optimization theory.

Generally-Occurring Model Change for Robust Counterfactual Explanations

TL;DR

This work addresses the robustness of counterfactual explanations under changing decision models. It generalizes the prior Naturally-Occurring Model Change concept to Generally-Occurring Model Change (GOMC) and provides probabilistic guarantees that quantify how counterfactuals behave when the underlying model shifts, even beyond Gaussian neighborhood assumptions. A key theoretical result bounds the discrepancy between original and updated predictions in terms of neighborhood data and distributional changes, with a clear link to Lipschitz continuity and subgaussian variability. The paper further applies the framework to a dataset perturbation case study, deriving concrete bounds under gradient-based learning and positioning GOMC as a wider, optimization-theory-backed tool for ensuring robust recourse. Overall, this work enhances reliability and interpretability of counterfactual explanations in dynamic settings with practical implications for robust decision-support systems.

Abstract

With the increasing impact of algorithmic decision-making on human lives, the interpretability of models has become a critical issue in machine learning. Counterfactual explanation is an important method in the field of interpretable machine learning, which can not only help users understand why machine learning models make specific decisions, but also help users understand how to change these decisions. Naturally, it is an important task to study the robustness of counterfactual explanation generation algorithms to model changes. Previous literature has proposed the concept of Naturally-Occurring Model Change, which has given us a deeper understanding of robustness to model change. In this paper, we first further generalize the concept of Naturally-Occurring Model Change, proposing a more general concept of model parameter changes, Generally-Occurring Model Change, which has a wider range of applicability. We also prove the corresponding probabilistic guarantees. In addition, we consider a more specific problem, data set perturbation, and give relevant theoretical results by combining optimization theory.
Paper Structure (18 sections, 5 theorems, 34 equations, 2 figures)

This paper contains 18 sections, 5 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Related Works
  4. Our Paper's Contributions
  5. Preliminaries
  6. Counterfactual Explanations
  7. Gradient Descent
  8. Main Results
  9. Generally-Occurring Model Change
  10. Probabilistic Guarantee
  11. Proof of Theorem \ref{['thm2']}
  12. Case Study: Dataset Perturbation Problem
  13. Problem Setting and Assumptions
  14. Assumption 1.
  15. Assumption 2.
  16. ...and 3 more sections

Key Result

theorem thmcountertheorem

Let $X_1,X_2,\ldots,X_k$ be $k$ i.i.d. random variables with distribution $\mathcal{N}(x,\sigma^2 I_d)$ and $\psi_M^k=\frac{1}{k} \sum_{i=1}^k \left ( m(X_i)-M(X_i) \right )$. Suppose $\left | \mathbb{E}[\psi_M^k|M]-\mathbb{E}[\psi_M^k] \right |<\varepsilon'$. Then, for any $\varepsilon>2\varepsilo where $R_{k,\sigma^2}(x,m)=\frac{1}{k}\sum_{x_i\in N_{x,k} } \left ( m(x_i)-\gamma \cdot \left \| x

Figures (2)

  • Figure 1: Comparison of Three Model Changes. (c) corresponds to the model change defined by Definition \ref{['def:GOMC']}. It builds on (b) by allowing the distribution of M to be not centered at $m(\cdot)$, thus having a larger range of model changes.
  • Figure 2: Data Perturbation: A small portion of the original dataset is modified to create a new dataset. The purple circles in the figure represent the data points from the original dataset, the red circles represent the data points from the new dataset, and the bicolored circles represent the data points that are shared between the two datasets.

Theorems & Definitions (16)

  • definition thmcounterdefinition: Counterfactuals $\bar{x}^*$ induced by norm $\|\cdot\|$
  • definition thmcounterdefinition: Closest Data-Manifold Counterfactuals $\bar{x}^*$ induced by norm $\|\cdot\|$ pmlr-v202-hamman23a
  • definition thmcounterdefinition: $\tau$-expansive
  • definition thmcounterdefinition: $\sigma$-bounded
  • remark thmcounterremark
  • definition thmcounterdefinition: Naturally-Occurring Model Change, pmlr-v202-hamman23a
  • definition thmcounterdefinition: Generally-Occurring Model Change, Main Concept
  • theorem thmcountertheorem: Probability Guarantees under Naturally-Occurring Model Change, pmlr-v202-hamman23a
  • theorem thmcountertheorem: Probability Guarantees under Gernerally-Occurring Model Change, Main Theorem
  • lemma thmcounterlemma: Deviation Bound
  • ...and 6 more