Counterfactual Plans under Distributional Ambiguity

Counterfactual explanations are attracting significant attention due to the flourishing applications of machine learning models in consequential domains . A counterfactual plan consists of multiple possibilities to modify a given instance so that the model ’ s prediction will be altered . As the predictive model can be updated subject to the future arrival of new data , a counterfactual plan may become ineffective , or infeasible , with respect to the future values of the model parameters . In this work , we study the counterfactual plans under model uncertainty , in which the distribution of the model parameters is partially prescribed using only the first- and second-moment information . First , we propose an uncertainty quantification tool to compute the lower and upper bounds of the probability of validity for any given counterfactual plan . We then provide corrective methods to adjust the counterfactual plan to improve the validity measure . The numerical experiments validate our bounds and demonstrate that our correction increases the robustness of the counterfactual plans in different real-world datasets . 1 INTRODUCTION . Machine learning models , thanks to their superior predictive performance , are blooming with increasing applications in consequential decision-making tasks . Along with the potential to help making better decisions , current machine learning models are also raising concerns about their explainability and transparency , especially in domains where humans are at stake . These domains span from loan approvals ( Siddiqi , 2012 ) , university admission ( Waters & Miikkulainen , 2014 ) to job hiring ( Ajunwa et al. , 2016 ) . In these applications , it is instructive to understand why a particular algorithmic decision is made , and counterfactual explanations act as a useful toolkit to comprehend ( black-box ) machine learning models ( Wachter et al. , 2017 ) . Counterfactual explanation is also known in the field of interpretable machine learning as contrastive explanation ( Miller , 2018 ; Karimi et al. , 2020b ) or recourse ( Ustun et al. , 2019 ) . A counterfactual explanation suggests how an instance should be modified so as to receive an alternate algorithmic outcome . As such , it could be used as a suggestion for improvement purposes . For example , a student is rejected from graduate study , and the university can provide one or multiple counterfactuals to guide the applicant for admission in the following year . A concrete example may be of the form “ get a GRE score of at least 325 ” or “ get a 6-month research experience ” . In practice , providing a counterfactual plan consisting of multiple examples is highly desirable because a single counterfactual to every applicant with the same covariates may be unsatisfactory in practice ( Wachter et al. , 2017 ) . Indeed , the covariates can barely capture the intrinsic behaviors , constraints and unrevealed preferences of the person they represent so that the users with the same features may have different preferences to modify their input . As a consequence , a pre-emptive design choice is to provide a “ menu ” of possible recourses , and let the applicant choose the recourse that fits them best . Viewed in this way , a counterfactual plan has the potential to increase satisfaction and build trust among the stakeholders of any machine learning applications . Constructing a counterfactual plan , however , is not a straightforward task because of the many competing criteria in the design process . By definition , the plan should be valid : by committing to any counterfactual in the plan , the application should be able to flip his current unfavorable outcome to a favorable one . However , each possibility in the plan should be in the proximity of the covariates of the applicant so that the modification is actionable . Further , the plan should consist of a diverse range of recourses to accommodate the different tastes and preferences of the population . Russell ( 2019 ) propose a mixed integer programming method to generate a counterfactual plan for a linear classifier , in which the diversity is imposed using a rule-based approach . In Dandl et al . ( 2020 ) , the authors propose a model-agnostic approach using a multi-objective evolutionary algorithm to construct a diverse counterfactual plan . More recently , Mothilal et al . ( 2020 ) use the determinantal point process to measure the diversity of a plan . The authors then formulate an optimization problem to find the counterfactual plan that minimizes the weighted sum of three terms representing validity , proximity , and diversity . A critical drawback of the existing works is the assumption of an invariant predictive model , which often fails to hold in practical settings . In fact , during a turbulent pandemic time , it is difficult to assume that the demographic population of students applying for postgraduate studies remain unchanged . And even in the case that the demography remains unchanged , special pandemic conditions such as hybrid learning mode or travel bans may affect the applicants ’ package , which in turn leads to fluctuations of the covariate distribution in the applicant pool . These shifts in the data are channeled to the shift in the parameters of the predictive model : when the machine learning models are re-trained or re-calibrated with new data , their parameters also change accordingly ( Venkatasubramanian & Alfano , 2020 ) . This raises an emerging concern because the counterfactual plan is usually designed to be valid to only the current model , but that is not enough to guarantee any validity on the future models . Thus , the counterfactual plan carries a promise of a favorable future outcome , nevertheless , this promise is fragile . It is hence reasonable to demand the counterfactual plan to be robust with respect to the shift of the parameters . Pawelczyk et al . ( 2020 ) study the sparsity of counterfactuals and its non-robustness under different fixed models ( predictive multiplicity ) . Rawal et al . ( 2020 ) consider the counterfactual plan problem and describe several types of model shift related to the correction , temporal , and geospatial shift from data . They also study the trade-off between the recourse proximity and its validity regarding to the model updates . Most recently , Upadhyay et al . ( 2021 ) leverage robust optimization to generate a counterfactual that is robust to some constrained perturbations of the model ’ s parameters . However , both works consider only the single counterfactual settings . Contributions . We study the many facets of the counterfactual plans with respect to random future model parameters . We focus on a linear classification setting and we prescribe the random model parameters only through the first- and second-moment information . We contribute concretely 1. a diagnostic tool to assess the validity of a counterfactual plan . It provides a lower and upper bound on the probability of joint validity of a given plan subject to uncertain model parameters . 2. a correction tool to improve the validity of a counterfactual plan , while keeping the modifications to each counterfactual at a minimal level . The corrections are intuitive and admit closed-form expression . 3. a COunterfactual Plan under Ambiguity ( COPA ) framework to construct a counterfactual plan which explicitly takes the model uncertainty into consideration . It minimizes the weighted sum of validity , proximity and diversity terms , and can be solved efficiently using gradient descents . Each of our above contributions is exposed in Section 2 , 3 and 4 , respectively . In Section 5 , we conduct experiments on both synthetic and real-world datasets to demonstrate the efficiency of our corrections and of our COPA framework . All proofs can be found in the appendix . General setup . Consider a covariate space Rd and a linear binary classification setting . Each linear classifier can be parametrized by θ ∈ Rd with decision output Cθ ( x ) = 1 if θ > x ≥ 0 , and 0 otherwise , where 0 represents an unfavorable outcome . Note that we omit the bias term to avoid clutter , taking the bias term into account can be achieved by extending the dimension of x and θ by an extra dimension . A counterfactual plan is a set of J counterfactual explanations { xj } j=1 , ... , J , and we denote { xj } for short . When J = 1 , we have a single counterfactual explanation problem , which is the subject of recent works ( Ustun et al. , 2019 ; Karimi et al. , 2020a ; Upadhyay et al. , 2021 ) . Next , we define the joint validity of a counterfactual plan . Definition 1.1 ( Joint validity ) . A counterfactual plan { xj } is valid with respect to a realization θ if Cθ ( xj ) = 1 for all j = 1 , . . . , J . Notations . We use Sd++ ( Sd+ ) to denote the space of symmetric positive ( semi ) definite matrices . For any A ∈ Rm×m , the trace operator is Tr [ A ] = ∑d i=1Aii . For any integer J , [ J ] , { 1 , . . . , J } . 2 VALIDITY BOUNDS OF COUNTERFACTUAL PLANS . In this section , we propose a diagnostic tool to benchmark the validity of a pre-computed counterfactual plan { xj } . We model the random model parameters θ̃ with a nominal distribution P̂ . Instead of making a strong assumption on a specific parametric form of P̂ such as Gaussian distribution , we only assume that P̂ is known only up to the second moment . More specifically , we assume that under P̂ , θ̃ has a nominal mean vector µ̂ and nominal covariance matrix Σ̂ ∈ Sd++ . Definition 2.1 ( Gelbrich distance ) . The Gelbrich distance between two pairs ( µ1 , Σ1 ) ∈ Rd × Sd+ and ( µ2 , Σ2 ) ∈ Rd × Sd+ is defined as G ( ( µ1 , Σ1 ) , ( µ2 , Σ2 ) ) , √ ‖µ1 − µ2‖22 + Tr [ Σ1 + Σ2 − 2 ( Σ 1 2 2 Σ1Σ 1 2 2 ) 1 2 ] . The Gelbrich distance is closely related to the optimal transport distance between Gaussian distributions . Indeed , G ( ( µ1 , Σ1 ) , ( µ2 , Σ2 ) ) is equal to the type-2 Wasserstein distance between two Gaussian distributions N ( µ1 , Σ1 ) and N ( µ2 , Σ2 ) ( Gelbrich , 1990 ) . It is thus trivial that G is a distance on Rd×Sd+ , and as a consequence , it is symmetric andG ( ( µ1 , Σ1 ) , ( µ2 , Σ2 ) ) = 0 if and only if ( µ1 , Σ1 ) = ( µ2 , Σ2 ) . Using the Gelbrich distance to design the moment ambiguity set for distributionally robust optimization leads to many desirable properties such as computational tractability and performance guarantees ( Kuhn et al. , 2019 ) . Motivated by this idea , we first construct the following uncertainty set U , { ( µ , Σ ) ∈ Rd × Sd+ : G ( ( µ , Σ ) , ( µ̂ , Σ̂ ) ) ≤ ρ } , which is formally a ρ-neighborhood in the mean vector-covariance matrix space around the nominal moment ( µ̂ , Σ̂ ) . The ambiguity set for the distributions of θ̃ is obtained by lifting U to generate a family of probability measures that satisfy the moment conditions B , { Q ∈ P : ∃ ( µ , Σ ) ∈ U such that Q ∼ ( µ , Σ ) } , . where P is a set of all probability measures supported on Rd and Q ∼ ( µ , Σ ) indicates that Q has mean vector µ and covariance matrix Σ . The central question of this section is : If the distribution of θ̃ belongs to B , what is the probability that a given plan { xj } is valid ? To answer this question , we define the event set Θ ( { xj } ) that contains all model parameter values that renders { xj } jointly valid . Under the definition of a linear model , Θ ( { xj } ) is an intersection of J open hyperplanes of the form Θ ( { xj } ) , { θ ∈ Rd : x > j θ ≥ 0 ∀j ∈ [ J ] } . ( 1 ) We name Θ ( { xj } ) the set of favorable parameters . The probability of validity for a plan under a measure Q is Q ( θ̃ ∈ Θ ( { xj } ) ) . We are interested in evaluating the lower and the upper bound probability that the plan { xj } is valid uniformly over all distributions Q ∈ B . This is equivalent to quantifying the following quantities inf Q∈B Q ( θ̃ ∈ Θ ( { xj } ) ) and sup Q∈B Q ( θ̃ ∈ Θ ( { xj } ) ) . In the remainder of this section , we discuss how to evaluate the bounds for these terms . Lower bound . We denote by Θ◦ the interior of the set Θ , that is , Θ◦ ( { xj } ) , { θ ∈ Rd : x > j θ > 0 ∀j } . Note that all the inequalities defining Θ◦ ( { xj } ) are strict inequalities . By definition , we have Θ◦ ( { xj } ) ⊂ Θ ( { xj } ) , and hence infQ∈B Q ( θ̃ ∈ Θ◦ ( { xj } ) ) ≤ infQ∈B Q ( θ̃ ∈ Θ ( { xj } ) ) . Because Θ◦ ( { xj } ) is an open set , we can leverage the generalized Chebyshev lower bound to evaluate the minimum quantity of Q ( θ̃ ∈ Θ◦ ( { xj } ) ) over all distributions with a given mean and covariance matrix ( Vandenberghe et al. , 2007 ) . Adding moment uncertainty via the set U is obtained by rejoining two minimization layers . The next theorem presents this result . Theorem 2.2 ( Lower bound ) . For any ρ ∈ R+ , µ̂ ∈ Rd and Σ̂ ∈ Sd+ , let L ? be the optimal value of the following semidefinite program L ? = inf 1− ∑ j∈ [ J ] λj s. t. µ ∈ Rd , Σ ∈ Sd+ , C ∈ Rd×d , M ∈ Sd+ λj ∈ R , zj ∈ Rd , Zj ∈ Sd ∀j ∈ [ J ] −x > j zj ≥ 0 , [ Zj zj z > j λj ] 0 ∀j ∈ [ J ] ∑ j∈ [ J ] [ Zj zj z > j λj ] [ M µ µ > 1 ] , [ Σ C C > Σ̂ ] 0 , [ M − Σ µ µ > 1 ] 0 ‖µ̂‖2 − 2µ̂ > µ+ Tr [ M + Σ̂− 2C ] ≤ ρ2 . ( 2 ) Then we have L ? = inf Q∈B Q ( θ̃ ∈ Θ◦ ( { xj } ) ) ≤ inf Q∈B Q ( θ̃ ∈ Θ ( { xj } ) ) . Upper bound . Because Θ ( { xj } ) is a closed set , we can leverage a duality result to evaluate the maximum quantity of Q ( θ̃ ∈ Θ ( { xj } ) ) over all distributions with a given mean and covariance matrix ( Isii , 1960 ) . Adding moment uncertainty via the set U is obtained by invoking the support function of the moment set . This result is presented in the next theorem Theorem 2.3 ( Upper bound ) . For any ρ ∈ R+ , µ̂ ∈ Rd and Σ̂ ∈ Sd+ , let U ? be the optimal value of the following semidefinite program U ? = inf z0 + γ ( ρ 2 − ‖µ̂‖22 − Tr [ Σ̂ ] ) + q + Tr [ Q ] s. t. γ ∈ R+ , z0 ∈ R , z ∈ Rd , Z ∈ Sd+ , q ∈ R+ , Q ∈ Sd+ , λ ∈ RJ+ [ γI − Z γΣ̂ 12 γΣ̂ 1 2 Q ] 0 , [ γI − Z γµ̂+ z γµ̂ > + z > q ] 0 [ Z z z > z0 ] 0 , [ Z z z > z0 − 1 ] ∑ j∈ [ J ] λj [ 0 12xj 1 2x > j 0 ] . Then we have supQ∈B Q ( θ̃ ∈ Θ ( { xj } ) ) ≤ U ? . Thanks to the choice of the Gelbrich distanceG , both optimization problems in Theorems 2.2 and 2.3 are linear semidefinite programs , and they can be solved efficiently by standard , off-the-shelf solvers such as MOSEK to high dimensions ( MOSEK ApS , 2019 ) . Other choices of distance ( divergence ) are also available : for example , one may opt for the Kullback-Leibler ( KL ) type divergence between Gaussian distribution to prescribe U as in Taskesen et al . ( 2021 ) . Unfortunately , the KL divergence will entail a log-determinant term , and the resulting optimization problems are no longer linear programs and are no longer solvable using MOSEK . Equipped with L ? and U ? , we have the bounds L ? ≤ Q ( { xj } is a valid plan ) ≤ U ? ∀Q ∈ B on the validity of the counterfactual plans { xj } under the distributional ambiguity set B. Complementary information . The previous results show that we can compute the lower bound L ? and upper bound U ? for the probability of validity by solving semidefinite programs . We now show that the two quantities L ? and U ? are complementary to each other in a specific sense . Proposition 2.4 ( Complementary information ) . For any instance , either L ? = 0 or U ? = 1 . More specifically , we have : ( i ) If µ̂ ∈ Θ ( { xj } ) , then U ? = 1 , and ( ii ) If µ̂ 6∈ Θ ( { xj } ) , then L ? = 0 . Because L ? and U ? are bounds for a probability quantity , they are only informative when they are different from 0 and 1 . Proposition 2.4 asserts that the upper bound U ? is trivial when µ̂ ∈ Θ ( { xj } ) , while the lower bound L ? becomes trival if µ̂ 6∈ Θ ( { xj } ) . Next , we leverage these insights to improve the validity of a given counterfactual plan .

Summary: The paper considers the problem of how to provide a collection of counterfactual explanations for a binary linear classifier such that an informed choice can be made on how best to actualise a new input based on individual preferences such that a different classification is likely to be made, even under potentially changing model parameters. It introduces the Counterfactual Plan under Ambiguity (COPA) framework, consisting of a probabilistic validity assessment of a predetermined plan under a given parameter distribution of the classifier, followed by a correction of the predetermined plan to improve validity under that distribution. These are combined in the COPA framework to produce a plan that optimises a weighted combination of the desiderata ‘proximity’, ‘diversity’ and ‘validity’. Experiments show the resulting output satisfies the derived upper and lower bounds on the validity, with good performance on the desiderata.