DISTRIBUTIONALLY ROBUST RECOURSE ACTION

Abstract

A recourse action aims to explain a particular algorithmic decision by showing one specific way in which the instance could be modified to receive an alternate outcome. Existing recourse generation methods often assume that the machine learning model does not change over time. However, this assumption does not always hold in practice because of data distribution shifts, and in this case, the recourse action may become invalid. To redress this shortcoming, we propose the Distributionally Robust Recourse Action (DiRRAc) framework, which generates a recourse action that has a high probability of being valid under a mixture of model shifts. We formulate the robustified recourse setup as a min-max optimization problem, where the max problem is specified by Gelbrich distance over an ambiguity set around the distribution of model parameters. Then we suggest a projected gradient descent algorithm to find a robust recourse according to the min-max objective. We show that our DiRRAc framework can be extended to hedge against the misspecification of the mixture weights. Numerical experiments with both synthetic and three real-world datasets demonstrate the benefits of our proposed framework over state-of-the-art recourse methods.

1. INTRODUCTION

Post-hoc explanations of machine learning models are useful for understanding and making reliable predictions in consequential domains such as loan approvals, college admission, and healthcare. Recently, recourse has been rising as an attractive tool to diagnose why machine learning models have made a particular decision for a given instance. A recourse action provides a possible modification of the given instance to receive an alternate decision (Ustun et al., 2019 ). Consider, for example, the case of loan approvals in which a credit application is rejected. Recourse will offer the reasons for rejection by showing what the application package should have been to get approved. A concrete example of a recourse in this case may be "the monthly salary should be higher by $500" or "20% of the current debt should be reduced". A recourse action has a positive, forward-looking meaning: they list out a directive modification that a person should implement so that they can get a more favorable outcome in the future. If a machine learning system can provide the negative outcomes with the corresponding recourse action, it can improve user engagement and boost the interpretability at the same time (Ustun et al., 2019; Karimi et al., 2021b) . Explanations thus play a central role in the future development of human-computer interaction as well as human-centric machine learning. Despite its attractiveness, providing recourse for the negative instances is not a trivial task. For realworld implementation, designing a recourse needs to strike an intricate balance between conflicting criteria. First and foremost, a recourse action should be feasible: if the prescribed action is taken, then the prediction of a machine learning model should be flipped. Further, to avoid making a drastic change to the characteristics of the input instance, a framework for generating recourse should minimize the cost of implementing the recourse action. An algorithm for finding recourse must make changes to only features that are actionable and should leave immutable features (relatively) unchanged. For example, we must consider the date of birth as an immutable feature; in contrast, we can consider salary or debt amount as actionable features. Various solutions have been proposed to provide recourses for a model prediction (Karimi et al., 2021b; Stepin et al., 2021; Artelt & Hammer, 2019; Pawelczyk et al., 2021; 2020; Verma et al., 2020) . For instance, Ustun et al. ( 2019) used an integer programming approach to obtain actionable recourses and also provide a feasibility guarantee for linear models. Karimi et al. (2020) proposed a model-agnostic approach to generate the nearest counterfactual explanations and focus on structured data. Dandl et al. (2020) proposed a method that finds the counterfactual by solving a multi-objective optimization problem. Recently, Russell (2019) and Mothilal et al. ( 2020) focus on finding a set of multiple diverse recourse actions, where the diversity is imposed by a rule-based approach or by internalizing a determinant point process cost in the objective function. These aforementioned approaches make a fundamental assumption that the machine learning model does not change over time. However, the dire reality suggests that this assumption rarely holds. In fact, data shifts are so common nowadays in machine learning that they have sparkled the emerging field of domain generalization and domain adaptation. Organizations usually retrain models as a response to data shifts, and this induces corresponding shifts in the machine learning models parameters, which in turn cause serious concerns for the feasibility of the recourse action in the future (Rawal et al., 2021) . In fact, all of the aforementioned approaches design the action which is feasible only with the current model parameters, and they provide no feasibility guarantee for the future parameters. If a recourse action fails to generate a favorable outcome in the future, then the recourse action may become less beneficial (Venkatasubramanian & Alfano, 2020), the pledge of a brighter outcome is shattered, and the trust in the machine learning system is lost (Rudin, 2019). To tackle this challenge, Upadhyay et al. ( 2021) proposed ROAR, a framework for generating instance-level recourses that are robust to shifts in the underlying predictive model. ROAR used a robust optimization approach that hedges against an uncertainty set containing plausible values of the future model parameters. However, it is well-known that robust optimization solutions can be overly conservative because they may hedge against a pathological parameter in the uncertainty set (Ben-Tal et al., 2017; Roos & den Hertog, 2020) . A promising approach that can promote robustness while at the same time prevent from over-conservatism is the distributionally robust optimization framework (El Ghaoui et al., 2003; Delage & Ye, 2010; Rahimian & Mehrotra, 2019; Bertsimas et al., 2018) . This framework models the future model parameters as random variables whose underlying distribution is unknown but is likely to be contained in an ambiguity set. The solution is designed to counter the worst-case distribution in the ambiguity set in a min-max sense. Distributionally robust optimization is also gaining popularity in many estimation and prediction tasks in machine learning (Namkoong & Duchi, 2017; Kuhn et al., 2019) . Contributions. This paper combines ideas and techniques from two principal branches of explainable artificial intelligence: counterfactual explanations and robustness to resolve the recourse problem under uncertainty. Concretely, our main contributions are the following: 1. We propose the framework of Distributionally Robust Recourse Action (DiRRAc) for designing a recourse action that is robust to mixture shifts of the model parameters. Our DiRRAc maximizes the probability that the action is feasible with respect to a mixture shift of model parameters while at the same time confines the action in the neighborhood of the input instance. Moreover, the DiRRAc model also hedges against the misspecification of the nominal distribution using a min-max form with a mixture ambiguity set prescribed by moment information. 2. We reformulate the DiRRAc problem into a finite-dimensional optimization problem with an explicit objective function. We also provide a projected gradient descent to solve the problem. 3. We extend our DiRRAc framework along several axis to handle mixture weight uncertainty, to minimize the worst-case component probability of receiving the unfavorable outcome, and also to incorporate the Gaussian parametric information. We first describe the recourse action problem with mixture shifts in Section 2. In Section 3, we present our proposed DiRRAc framework, its reformulation and the numerical routine for solving it. The extension to the parametric Gaussian setting will be discussed in Section 4. Section 5 reports the numerical experiments showing the benefits of the DiRRAc framework and its extensions. Notations. For each integer K, we have  Q k ∼ (µ k , Σ k ). If additionally Q k is Gaussian, we write Q k ∼ N (µ k , Σ k ). Writing Q ∼ (Q k , p k ) k∈[K] means Q is a mixture of K components, the k-th component has weight p k and distribution Q k .



[K] = {1, . . . , K}. We use S d + (S d ++ ) to denote the space of symmetric positive semidefinite (definite) matrices. For any A ∈ R m×m , the trace operator is Tr A = d i=1 A ii . If a distribution Q k has mean µ k and covariance matrix Σ k , we write

