Learning Models for Actionable Recourse

1 Introduction . In recent years , there has been a growing interest in using machine learning to inform consequential decisions in legal and financial decision-making—e.g. , deciding whether to give an applicant a loan [ Hardt et al. , 2016 ] , bail to a defendant [ Lakkaraju and Rudin , 2017 ] , or parole to a prisoner [ Zeng et al. , 2017 ] . Because these decisions have an impact on the lives of the concerned individuals , it is critical to explain why the model made its prediction . Explanations are important not only to ensure that there are no issues with the way the prediction is made ( e.g. , making sure the decision is free of racial/gender bias [ Hardt et al. , 2016 ] and does not suffer from causal issues [ Bastani et al. , 2017 ] ) , but also to give the affected individual a justification for the decision . Thus , there has been a great deal of recent interest in explainable machine learning [ Lou et al. , 2012 , Wang and Rudin , 2015 , Ribeiro et al. , 2016b , Lakkaraju et al. , 2019 ] . We focus on counterfactual explanations [ Wachter et al. , 2017 ] , which specify how features can be changed to obtain a different model prediction . These explanations can be used to provide individuals who are negatively impacted by model outcomes with actionable recourses—i.e. , actions they can take to receive a positive outcome [ Ustun et al. , 2019 ] . For instance , for an applicant denied a loan , an actionable recourse might be : “ to get a loan , increase your income by $ 1,000. ” Such a recourse must satisfy two properties to be actionable : ( i ) it only changes features that the individual can realistically modify—e.g. , it can not change race or gender , but can change income , and ( ii ) the magnitude of the change must be reasonable . An actionable recourse like this may be considered necessary in some settings because it provides an individual with agency over a consequential decision that affects them . Prior research has addressed the need for recourses through post-hoc algorithms for computing individual recourses corresponding to certain kinds of models—e.g. , using integer linear programming ⇤Work started at Harvard University . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . in the context of linear models [ Ustun et al. , 2019 ] , or using gradient descent on the input for differentiable models [ Wachter et al. , 2017 ] . However , these approaches do not guarantee that actionable recourses exist ; at best , Ustun et al . [ 2019 ] guarantee that they find one if it exists , and only for linear models . In other words , many affected individuals may not even be prescribed any actions that they can take to change their outcome . We aim to provide tools for guaranteeing the existence of actionable recourses for domains in which recourses may be necessary . We propose a novel algorithm for learning models in a way that is designed to ensure the existence of actionable recourses so that affected individuals receive recourse with a high probability . To the best of our knowledge , our approach is the first to train models for which recourse is likely to exist with a high probability . It builds on adversarial training [ Goodfellow et al. , 2015a ] , which is designed to ensure that models are robust to adversarial examples . At a high level , given a binary classification model f✓ : X ! Y , where Y = { 0 , 1 } , an adversarial example is a perturbation 2 to an input x 2 X such that f✓ ( x+ ) 6= f✓ ( x ) . In a typical adversarial training setting , we choose to be “ small ” in some sense ( e.g. , in terms of L1 norm of 2 ) and aim to guarantee that adversarial examples do not exist—i.e. , f ( x+ ) = f ( x ) for all 2 . In contrast , in our setting , we intuitively want to ensure that adversarial examples do exist . In particular , given an input x for which f ( x ) = 0 , we want there to exist recourse 2 such that f ( x + ) = 1 , where in our case , is the given set of all permissible recourses for the application domain . Thus , we adapt existing adversarial training algorithms to ensure the existence of recourse . As an added benefit , these algorithms can compute recourse significantly faster than existing techniques for general differentiable models [ Wachter et al. , 2017 ] . While adversarial training heuristically encourages recourse to exist , it does not provide any theoretical guarantees . We build on PAC confidence sets [ Park et al. , 2020 ] to guarantee that recourse exists with high probability ( assuming the test distribution is the same as the training distribution ) . We evaluate our approach on four real world datasets that cover lending , recidivism , bail , and credit outcomes . Our results demonstrate that our approach is very effective at improving recourse rates ( i.e. , the probability that individuals are given recourse ) without noticeably reducing accuracy . In addition , we show that we achieve this improvement in recourse rates without noticeably harming the quality of recourses or the brittleness of the underlying model . Firstly , we empirically demonstrate that our approach improves the rate at which models provide recourses that are grounded in reality : We find that our approach encourages the existence of recourses that both obey causal constraints driven by real-world causal relationships and are in-distribution to the original data . Secondly , we show that our approach encourages the existence of robust recourses ( i.e. , recourses that result in positive outcomes even when changed in small ways ) . Lastly , we show that these improvements in recourse rates do not render the underlying classifier brittle.2 Related work . Beyond Wachter et al . [ 2017 ] , Ustun et al . [ 2019 ] , other approaches have been proposed for generating recourses [ Zhang et al. , 2018 , Hendricks et al. , 2018 , Mothilal et al. , 2020 , Looveren and Klaise , 2019 , Poyiadzi et al. , 2020 , Karimi et al. , 2020a , b , c ] . However , all these works focus on how to compute recourse for given predictive models ; in contrast , our goal is to learn predictive models that provide recourses at high rates . Any of these methods can be used in conjunction with ours . Our work also builds on adversarial training [ Szegedy et al. , 2014 , Goodfellow et al. , 2015b , Bastani et al. , 2016 , Shaham et al. , 2018 ] . While recent work in model explainability has leveraged adversarial training to improve robustness of explanations [ Lakkaraju et al. , 2020b ] , their goal is to reduce the rate of adversarial examples , whereas ours is to increase the rate of recourses . 2 Problem Formulation . Preliminaries . Consider a binary classifier f✓ : X ! Y , where x 2 X ✓ RnX are the features , y 2 Y = { 0 , 1 } are the labels , and ✓ 2 ⇥ ✓ Rn⇥ are the parameters . We assume that f✓ has the form f✓ ( x ) = ( g✓ ( x ) ✓0 ) , where g✓ : X ! [ 0 , 1 ] is a scoring function and ✓0 2 R is a decision threshold . We assume that g✓ is differentiable in x—i.e. , rxg✓ ( x ) exists almost everywhere.3 2Our code is available at https : //github.com/alexisjihyeross/adversarial_recourse . 3Note that we have focused on real-valued features . We discuss how our algorithm can be extended to handling categorical features in Section B.4 . Recourse . We seek to ensure that individuals given negative outcomes by f✓ are also given recourse . Definition 2.1 . Given a classifier f✓ : X ! Y and a set ✓ RnX , an input x 2 X has recourse if there exists 2 such that f ( x+ ) = 1 . We use XR✓ to denote the set of inputs for which recourse exists . The specific design of the set of permissible recourses ✓ RnX is domain specific . We assume that 0 2 ; then , recourse automatically exists for positive outcomes f✓ ( x ) = 1 by taking = 0 . In addition , we assume that is a polytope—i.e. , = { 2 RnX | A + b 0 } for some A 2 Rk·nX and b 2 Rk . That is , can be expressed as a set of affine constraints . This assumption is required for computational tractability . A standard choice is = { | k k1 max } , which says that the recourse can suggest changes to any feature by a bounded amount . We can apply this constraint after an affine transformation of that appropriately scales different covariates . In addition , we often want to restrict features—e.g. , to avoid suggesting that an individual change their age , we can constrain i = 0 , or to avoid suggesting that an individual decrease their income to qualify for a loan , we can constrain i 0 . In principle , can also be tailored to individuals—e.g. , disallow suggesting increased education for individuals who can not afford to do so . Finally , should be designed large enough so it includes a plausible recourse for every individual , yet small enough to ensure that the recourses do not overburden the individuals . All of these considerations are domain-specific ; we describe our choices for datasets in our experiments in Section 4 . Probably Approximately has REcourse ( PARE ) . Our goal is to ensure that recourse exists for most individuals . Given a confidence level ✏ 2 R > 0 , we say that ✓ approximately has recourse if Pp ( x ) ⇥ x 2 XR✓ ⇤ 1 ✏ , i.e. , recourse exists for f✓ with probability at least 1 ✏ w.r.t . the distribution p ( x ) over individuals . Then , our goal is to design an algorithm for estimating the model parameters ✓ so that f✓ approximately has recourse . To do so , our algorithm takes as input a held-out calibration dataset Z ✓ X ⇥Y of examples ( x , y ) ⇠ p , where p ( x , y ) is the distribution over labeled examples , and outputs model parameters ✓̂ ( Z ) . Then , as in the probably approximately correct ( PAC ) learning framework [ Valiant , 1984 ] , our algorithm might additionally fail due to the randomness in Z . Thus , given a second confidence level ↵ 2 R > 0 , we say that ✓̂ Probably Approximately has REcourse ( PARE ) if Pp ( Z ) h ✓̂ ( Z ) approximately has recourse i 1 ↵ , where p ( Z ) = Q ( x , y ) 2Z p ( x , y ) . In other words , our algorithm produces a model that approximately has recourse with probability at least 1 over p ( Z ) . Constructing a PARE classifier . Note that we can trivially obtain a PARE classifier f✓ by choosing the decision threshold ✓0 = 0 , in which case f✓ ( x ) = 1 for all x 2 X . However , this approach is undesirable since it assigns a positive outcome to all individuals . Instead , we want to maximize the performance of f✓ ( e.g. , in terms of accuracy , F1 score , etc . ) subject to a constraint that f✓ is PARE . Thus , we divide the problem of constructing a PARE classifier into two parts ( i ) increasing recourse rate : we train g✓ in a way that heuristically increases the rate at which inputs x 2 X have recourse ( for any ✓0 ) , and ( ii ) guaranteeing recourse : we choose ✓0 to guarantee that the resulting f✓ is PARE .

Summary: The author(s) proposed an adversarial training-based classification algorithm that guarantees affected individuals receive recourse with a high probability. The paper considers an interesting topic. The numerical experiments demonstrate the effectiveness of the proposed algorithm.