Algorithmic Recourse with Missing Values

TL;DR

This work addresses algorithmic recourse in the presence of missing input values by introducing ARMIN, a framework that integrates multiple imputation into recourse optimization. By defining the action validity over an imputation space and enforcing a probabilistic validity constraint, ARMIN uses sampling and mixed-integer linear optimization to produce valid, low-cost recourse actions for incomplete instances without revealing missing values. The approach is theoretically justified and empirically validated against baselines across several datasets and missing-data mechanisms (MCAR, MAR, MNAR), demonstrating superior action validity and lower costs. The findings advance practical AR in privacy-conscious and real-world settings where feature disclosure is limited, with implications for robustness and fairness in decision-support systems.

Abstract

This paper proposes a new framework of algorithmic recourse (AR) that works even in the presence of missing values. AR aims to provide a recourse action for altering the undesired prediction result given by a classifier. Existing AR methods assume that we can access complete information on the features of an input instance. However, we often encounter missing values in a given instance (e.g., due to privacy concerns), and previous studies have not discussed such a practical situation. In this paper, we first empirically and theoretically show the risk that a naive approach with a single imputation technique fails to obtain good actions regarding their validity, cost, and features to be changed. To alleviate this risk, we formulate the task of obtaining a valid and low-cost action for a given incomplete instance by incorporating the idea of multiple imputation. Then, we provide some theoretical analyses of our task and propose a practical solution based on mixed-integer linear optimization. Experimental results demonstrated the efficacy of our method in the presence of missing values compared to the baselines.

Figures (16)

• Figure 1: Examples of an original instance $\bm{x}$, its imputed instance $\hat{\bm{x}}$, and the decision boundary of a classifier $h$. Here, we drop the feature "Income" in $\bm{x}$ as a missing feature and obtain $\hat{\bm{x}}$ by imputing its value with the empirical mean $66K. For the imputed instance $\hat{\bm{x}}$, we obtain an optimal action $\bm{a}$ using the existing AR method. While the action $\bm{a}$ successfully alters the prediction result of $h$ for the imputed instance $\hat{\bm{x}}$, it fails to do that for the original instance $\bm{x}$.
• Figure 2: Experimental results of our baseline comparison under the MCAR situation, where $D_{\ast} = 2$. The x-axis (resp. y-axis) represents the valid ratio (resp. average cost). Compared to the baselines, our ARMIN attained good balances between the valid ratio and cost for almost all the datasets and classifiers; that is, it achieved higher valid ratios than ImputationAR and lower costs than RubustAR.
• Figure 3: Experimental results of our comparison under the MAR and MNAR situations.
• Figure 4: Experimental results of our trade-off analyses on the GiveMeCredit dataset. (a) Experimental results of the sensitivity analyses of the confidence parameter $\rho$ with $95\%$ confidence intervals. (b) Experimental results of the path analyses under the MAR situation. Here, we selected two examples, and additional examples are shown in \ref{['sec:appendix:experiments']}.
• Figure 5: Experimental results of the XGBoost classifier with missing values $D_{\ast} \in \{ 0, 1, 2, 3, 4 \}$ under the MCAR situation.

Theorems & Definitions (15)

• Remark 1
• Remark 2
• Proposition 1
• Proposition 2
• Proposition 3
• Lemma 1: Closed-form Optimal Action Ustun:FAT*2019Pawelczyk:AISTATS2022
• proof : Proof of \ref{['prop:upper']}
• Proposition 4
• proof
• proof : Proof of \ref{['prop:sample']}