COVARIANCE-ROBUST MINIMAX PROBABILITY MA-CHINES FOR ALGORITHMIC RECOURSE

Abstract

Algorithmic recourse is rising as a prominent technique to promote the explainability and transparency of the predictive model in ethical machine learning. Existing approaches to algorithmic recourse often assume an invariant predictive model; however, this model, in reality, is usually updated temporally upon the input of new data. Thus, a recourse that is valid respective to the present model may become invalid for the future model. To resolve this issue, we propose a pipeline to generate a model-agnostic recourse that is robust to model shifts. Our pipeline first estimates a linear surrogate of the nonlinear (black-box) model using covariance-robust minimax probability machines (MPM); then, the recourse is generated with respect to this robust linear surrogate. We show that the covariance-robust MPM recovers popular regularization schemes, including 2 -regularization and class-reweighting. We also show that our covariance-robust MPM pushes the decision boundary in an intuitive manner, which facilitates an interpretable generation of a robust recourse. The numerical results demonstrate the usefulness and robustness of our pipeline.

1. INTRODUCTION

The recent prevalence of machine learning (ML) in supporting consequential decisions involving humans such as loan approval (Moscato et al., 2021) , job hiring (Cohen et al., 2019; Schumann et al., 2020) , and criminal justice (Brayne & Christin, 2021) urges the need of transparent ML systems with explanations and feedback to users (Doshi-Velez & Kim, 2017; Miller, 2019) . One popular and emerging approach to providing feedback is the algorithmic recourse (Ustun et al., 2019) . A recourse suggests how the input instance should be modified to alter the outcome of a predictive model. Consider a specific scenario in which an individual is rejected from receiving a loan by a financial institution's ML model. Recently, it has become a legal necessity to provide explanations and recommendations to the individual so that they can improve their situation and obtain a loan in the future (GDPR, Voigt & Von dem Bussche (2017)). For example, an explanation can be "increase the income to $5000" or "reduce the debt/asset ratio to below 20%". Leveraging the recourses, financial institutions can assess the reliability of their ML predictive models and increase user engagement through actionable feedback and acceptance guarantee if they fulfill the requirements. To construct plausible and meaningful recourses, one must assess and strike a balance between conflicting criteria. They can be: (1) validity, a recourse should effectively reverse the unfavorable prediction of the model into a favorable one, (2) proximity, recourse should be close to the original input instance to alleviate the efforts required, and thus to encourage the adoption of the recourse, (3) actionability, prescribed modifications should follow causal laws of our society (Ustun et al., 2019; Karimi et al., 2021) ; for example, one can not modify their race or decrease their age. Various techniques were proposed to devise algorithmic recourses for a given predictive model, extensive surveys are provided in (Karimi et al., 2020a; Stepin et al., 2021; Pawelczyk et al., 2021; Verma et al., 2020) . Wachter et al. (2017) introduced the definition of counterfactual explanations and proposed a gradient-based approach to find the nearest instance that yields a favorable outcome. Ustun et al. (2019) proposed a mixed integer programming formulation (AR) that can find recourses for a linear classifier with a flexible design of the actionability constraints. Alternatively, Karimi et al. (2021; 2020b) investigated the nearest recourse through the lens of minimal intervention to take causal relationships between features into account. Recent works including Russell (2019) and Mothilal et al. (2020) also studied the problem of generating a menu of diverse recourses to provide multiple possibilities that users might choose. The aforementioned methods rely on an assumption of an invariant predictive model. Nevertheless, machine learning models are usually re-trained or re-calibrated as new data arrive. Thus, a valid recourse at present may become invalid in the future, leading to an exemplary case where a rejected applicant may spend efforts to improve their income and reapply for a loan, but then is rejected (again) simply because the ML model has been updated. This leads to a potential inefficiency due to the waste of resources and loss of trust in the recommendation and in the ML system (Rudin, 2019) . Despite the promising results, existing methods are often restricted to the linear classifiers setting to be able to introduce actionability or robustness (Ustun et al., 2019; Russell, 2019; Upadhyay et al., 2021; Rawal et al., 2020) . For non-linear classifiers, a linear surrogate method such as LIME (Ribeiro et al., 2016) is used to approximate the local decision boundary of the black-box classifiers; the recourse is then generated respectively to the (linear) surrogate model instead of the nonlinear model. LIME is well-known for explaining predictions of black-box ML models by fitting a reweighted linear regression model to the perturbed samples around an input instance. In the recourse literature, LIME is the most common linear surrogate for the local decision boundary of the black-box models (Ustun et al., 2019; Upadhyay et al., 2021) . Unfortunately, the LIME surrogate has several limitations. Firstly, Laugel et al. ( 2018) and White & Garcez (2019) showed that LIME may not be faithful to the underlying models because LIME might be influenced by input features at a global scale rather than a local scale. Secondly, explanations generated by perturbation-based methods are also well-known to be sensitive to the original input and the synthesized perturbations (Alvarez-Melis & Jaakkola, 2018; Ghorbani et al., 2019; Slack et al., 2020; 2021; Agarwal et al., 2021; Laugel et al., 2018) . Several works have been proposed to overcome these issues. Laugel et al. (2018) and Vlassopoulos et al. (2020) proposed alternative sampling procedures that generate sample instances in the neighborhood of the closest counterfactual to fit a local surrogate. White & Garcez (2019) integrated counterfactual explanation to local surrogate models to introduce a novel fidelity measure of an explanation. Later, Garreau & von Luxburg (2020) and Agarwal et al. (2021) analyzed theoretically the stabilityfoot_0 of LIME, especially in the low sampling size regime. Zhao et al. (2021) leveraged Bayesian reasoning to improve the consistency in repeated explanations of a single prediction. Nevertheless, the impact and effectiveness of these surrogates on the recourse generation are still unknown. Contributions. We revisit the recourse generation scheme through surrogate models. We propose a novel model-agnostic pipeline that facilitates the generation of robust and actionable recourses. The core innovation in our pipeline is the use of the covariance-robust minimax probability machines (MPM) as a linear surrogate of the nonlinear black-box ML model. Additionally, we contribute • to the field of MPM and robust classifier: We propose and analyze in detail the covariance-robust MPMs in which the set of possible perturbations of the covariance matrices are prescribed using distances on the space of positive semidefinite matrices. Motivated by the statistical distances between Gaussian distributions, we show that the covariance-robustness induces and connects to two prominent regularization schemes of the nominal MPM: if the distance is motivated by the Bures distance, we recover the 2 -regularization, if the distance is motivated by the Fisher-Rao distance, we recover class reweighting schemes.



Throughout, "robustness" is used in the algorithmic recourse setting with respect to the model shifts(Rawal et al., 2020). "Robustness" is also used to indicate the sensitivity of LIME to the sampling distribution. To avoid confusion, in what follows, we use "stability" to refer to the aforementioned sensitivity of LIME.



Studying this phenomenon, Rawal et al. (2020) described several types of model shifts related to the correction, temporal, and geospatial shifts from data. They pointed out that the recourses, even constructed with state-of-the-art algorithms, are vulnerable to distributional shifts in the model's parameters. Pawelczyk et al. (2020) study counterfactual explanations under predictive multiplicity and its relation to the difference in the way two classifiers treat predicted individuals. Black et al. (2021) then show that the constructed recourses might be invalid even for the model retrained with different initial conditions such as weight initialization and leave-one-out variations in data. Recently, Upadhyay et al. (2021) leveraged robust optimization to propose ROAR -a framework for generating recourses that are robust to shifts in the predictive model, which is assumed to be a linear classifier.

