In order to prove that the objective function in Eqn. 1 is non-normal, non-negative, non-monotone, and submodular, we need to prove the following: any one of the terms in the objective is non-normal all the terms in the objective are non-negative any one of the terms in the objective is non-monotone all the terms in the objective are submodular Non-normality Let us consider the term f This metric can never be negative by definition. This is clearly a diminishing returns function i.e., more additional instances in the data are covered Before we prove Theorem 2.2, we will first discuss how several previously proposed methods which Eqn.1 can be reduced to the objectives employed by prior approaches which provide instance level Subsuming other objective functions: The objective optimized by Wachter et al. is Higher values of recourse accuracy are desired; lower values of mean fcost are desired. Explanation vs. Recourse Accuracy for COMP AS (left), Credit (middle), and Bail (right) datasets A.2.2 User Study We manually constructed a two level recourse set (as our black box model) for the bail application. We deliberately ensured that this black box was biased against individuals who are not Caucasian. This two level recourse set (black box) is shown in Figure 4. We used AR-LIME as a comparison point in our user study.

Over the past decade, machine learning (ML) models are being increasingly deployed to make a variety of consequential decisions ranging from hiring decisions to loan approvals.

As predictive models are increasingly being deployed in high-stakes decision-making, there has been a lot of interest in developing algorithms which can provide recourses to affected individuals. While developing such tools is important, it is even more critical to analyse and interpret a predictive model, and vet it thoroughly to ensure that the recourses it offers are meaningful and non-discriminatory before it is deployed in the real world. To this end, we propose a novel model agnostic framework called Actionable Recourse Summaries (AReS) to construct global counterfactual explanations which provide an interpretable and accurate summary of recourses for the entire population. We formulate a novel objective which simultaneously optimizes for correctness of the recourses and interpretability of the explanations, while minimizing overall recourse costs across the entire population. More specifically, our objective enables us to learn, with optimality guarantees on recourse correctness, a small number of compact rule sets each of which capture recourses for well defined subpopulations within the data. We also demonstrate theoretically that several of the prior approaches proposed to generate recourses for individuals are special cases of our framework. Experimental evaluation with real world datasets and user studies demonstrate that our framework can provide decision makers with a comprehensive overview of recourses corresponding to any black box model, and consequently help detect undesirable model biases and discrimination.