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 paper proposes the idea of training machine models that are effectively recourse ready. The authors propose leveraging adversarial training and PAC confidence sets to train models that "theoretically guarantee recourse". The experiments with four datasets demonstrate increased recourse rates while sacrificing accuracy mildly.