Distributionally Robust Recourse Action

Recourse actions aim to explain a particular algorithmic decision by showing one or multiple ways in which the instance could be modified to receive an alternate outcome . Existing recourse recommendations often assume that the machine learning models do not change over time . However , this assumption does not always hold in practice because of data distribution shifts , and in this case , the recourse actions may become invalid . To redress this shortcoming , we propose the Distributionally Robust Recourse Action framework , which generates a recourse action that has high probability of being valid under a mixture of model shifts . We show that the robust recourse can be found efficiently using a projected gradient descent algorithm and we discuss several extensions of our framework . Numerical experiments with both synthetic and real-world datasets demonstrate the benefits of our proposed framework . 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 is rising as an attractive tool do diagnose why the machine learning models have made a particular decision for a given instance . Recourse work by providing possible actions to modify a 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 . The counterfactual will offer the reasons for rejection by showing what the application package should have been to get approved . A concrete example of a counterfactual in this case may be “ the monthly salary should be higher by $ 500 ” or “ 20 % of the current debt should be reduced ” . Recourses have a positive , forward-looking meaning : they list out the recourse actions that a person should implement so that they can get a more favorable outcome in the future . If a specific application can provide the negative outcomes with recourse actions , it can improve the user engagement and boost the interpretability at the same time ( Ustun et al. , 2019 ; Karimi et al. , 2021 ) . Explanations thus play a central role in the future development of 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 . At the same time , a framework for generating recourse should minimize the cost to take recourse actions to avoid making a drastic change to the characteristics of the input instance . An algorithm for finding recourse must make change to only features that are actionable , and should leave immutable features ( relatively ) unchanged . For example , we must consider date of birth as an immutable feature ; in contrast , we can consider salary or debt amount as actionable features . Various solutions has been proposed to provide recourses for a model prediction ( Karimi et al. , 2021 ; Stepin et al. , 2021 ; Artelt & Hammer , 2019 ) . 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 nearest counterfactual explanations and focus on structured data . Dandl et al . ( 2020 ) proposed a method which finds 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 internalize 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 turns 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 on the machine learning system is lost ( Rudin , 2019 ; Ribeiro et al. , 2016 ) . To tackle this challenge , Upadhyay et al . ( 2021 ) proposed ROAR , a framework for generating instance level recourses ( counterfactual explanations ) 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 . 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 , in order 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 cap 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 resulting reformulation with convergence guarantees . 3 . We extend our DiRRAc framework along several axis to handle mixture weight uncertainty , to minimize the worst-case component probability of receiving unfavorable outcome , and also to inject the Gaussian parametric information . We first describe the recourse action problem with mixture shift 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 subsequently 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 [ K ] = { 1 , . . . , K } . We use Sd+ ( Sd++ ) to denote the space of symmetric positive semidefinite ( definite , respectively ) matrices . For any A ∈ Rm×m , the trace operator is defined as Tr [ A ] = ∑d i=1Aii . We write Qk ∼ ( µk , Σk ) to denote that the distribution Qk has mean vector µk and covariance matrix Σk . If additionally Qk is Gaussian , we write Qk ∼ N ( µk , Σk ) . With a slight abuse of notation , Q ∼ ( Qk , pk ) k∈ [ K ] means Q is a mixture of K component distributions , the k-th component has weight pk and distribution Qk . 2 RECOURSE ACTION UNDER MIXTURE SHIFTS . We consider a binary classification setting with label Y = { 0 , 1 } , where 0 represents the unfavorable outcome while 1 denotes the favorable one . The covariate space is Rd , and any linear classifier Cθ : Rd → Y characterized by the d-dimensional parameter θ is of the form Cθ ( x ) = { 1 if θ > x ≥ 0 , 0 otherwise . Note that the bias term can be internalized into θ by adding an extra dimension , and thus it is omitted . Suppose that at this moment ( t = 0 ) , the current classifier is parametrized by θ0 , and we are given an input instance x0 ∈ Rd with unfavorable outcome , that is , Cθ0 ( x0 ) = 0 . One period of time from now ( t = 1 ) , the parameters of the predictive model will change stochastically , and are represented by a d-dimensional random vector θ̃ . This paper focuses on finding a recourse action x which is reasonably close to the instance x0 , and at the same time , has a high probability of receiving a favorable outcome in the future . Figure 1 gives a bird ’ s eye view of the setup To measure the closeness between the action x and the input x0 , we assume that the covariate space is endowed with a non-negative , continuous cost function c. In addition , suppose temporarily that θ̃ follows a distribution P̂ . Because maximizing the probability of the favorable outcome is equivalent to minimizing the probability of the unfavorable outcome , the recourse can be found by solving min x P̂ ( Cθ̃ ( x ) = 0 ) s. t. x ∈ X , c ( x , x0 ) ≤ δ . ( 1 ) The parameter δ ≥ 0 in ( 1 ) governs how far a recourse action can be from the input instance x0 . Note that we constrain x in a set X which captures operational constraints , for example , the highest education of a credit applicant should not be decreasing over time . In this paper , we model the random vector θ̃ using a finite mixture of distributions with K components , the mixture weights are p̂ satisfying ∑ k∈ [ K ] p̂k = 1 . Each component in the mixture represents one specific type of data shifts : the weights p̂ reflect the proportion of the shift types while the component distribution P̂k representing the ( conditional ) distribution of the future model parameters in the k-th shift . Further information on mixture distributions and their applications in machine learning can be found in Murphy ( 2012 , §3.5 ) . If each P̂k is a Gaussian distribution N ( θ̂k , Σ̂k ) , then P̂ is a mixture of Gaussian distributions . The objective of problem ( 1 ) can be expressed as P̂ ( Cθ̃ ( x ) = 0 ) = ∑ k∈ [ K ] p̂kP̂k ( Cθ̃ ( x ) = 0 ) = ∑ k∈ [ K ] p̂kΦ ( −x > θ̂k√ x > Σ̂kx ) , where the first equality follows from the law of conditional probability , and Φ is the cumulative distribution function of a standard Gaussian distribution . Under the Gaussian assumption , we can solve ( 1 ) using a projected gradient descent type of algorithm ( Boyd & Vandenberghe , 2004 ) . Remark 2.1 ( Nonlinear models ) . Our analysis focuses on linear classifiers , which is a common setup in the literature ( Upadhyay et al. , 2021 ; Ustun et al. , 2019 ; Rawal et al. , 2021 ; Karimi et al. , 2020 ; Wachter et al. , 2018 ; Ribeiro et al. , 2016 ) . To extend to nonlinear classifiers , we can follow a similar approach as in Rawal & Lakkaraju ( 2020 ) and Upadhyay et al . ( 2021 ) by first using LIME ( Ribeiro et al. , 2016 ) to approximate the nonlinear classifiers locally with an interpretable linear model , then subsequently applying our framework .

Summary: The paper provides a framework for recourse (i.e. counterfactual explanations) that is robust to shifts in the model. They formulate the robustified recourse setup as a min-max optimization problem, where the max is over a neighborhood around the distribution over model parameters. The model parameters are drawn from a mixture of K distributions, so that the neighborhood is specified by Gelbrich distance on each component.   They propose a finite-dimensional version of the robustified optimization problem, which can be optimized using projected gradient descent. They evaluate their approach on the German credit dataset, the Small Business Administration dataset, and the Student performance dataset, each of which demonstrates a different type of data distribution shift.