In this appendix, we discuss a technique to optimize over the counterfactuals found by counterfactual explanation methods, such as [6]. We restate lemma 3.1 and provide a proof. Lemma 3.1 Assuming the counterfactual algorithm A (x) follows the form of the objective in equation 1, @@xcf G(x,A (x)) = 0, and m is the number of parameters in the model, we can write the derivative of counterfactual algorithm A with respect to model parameters as the Jacobian, @ @ A (x)= @2G(x,A (x)) @x2cf 1 G(x,xcf) (7) This problem is identical to a well-studied class of bi-level optimization problems in deep learning. In these problems, we must compute the derivative of a function with respect to some parameter (here) that includes an inner argmin, which itself depends on the parameter. We follow [44] to complete the proof.

Machine learning based predictions are increasingly used in sensitive decision-making applications that directly affect our lives. This has led to extensive research into ensuring the fairness of classifiers. Beyond just fair classification, emerging legislation now mandates that when a classifier delivers a negative decision, it must also offer actionable steps an individual can take to reverse that outcome. This concept is known as algorithmic recourse. Nevertheless, many researchers have expressed concerns about the fairness guarantees within the recourse process itself. In this work, we provide a holistic theoretical characterization of unfairness in algorithmic recourse, formally linking fairness guarantees in recourse and classification, and highlighting limitations of the standard equal cost paradigm. We then introduce a novel fairness framework based on social burden, along with a practical algorithm (MISOB), broadly applicable under real-world conditions. Empirical results on real-world datasets show that MISOB reduces the social burden across all groups without compromising overall classifier accuracy.

Decision makers are increasingly relying on machine learning in sensitive situations. In such settings, algorithmic recourse aims to provide individuals with actionable and minimally costly steps to reverse unfavorable AI-driven decisions. While existing research predominantly focuses on single-individual (i.e., seeker) and single-model (i.e., provider) scenarios, real-world applications often involve multiple interacting stakeholders. Optimizing outcomes for seekers under an individual welfare approach overlooks the inherently multi-agent nature of real-world systems, where individuals interact and compete for limited resources. To address this, we introduce a novel framework for multi-agent algorithmic recourse that accounts for multiple recourse seekers and recourse providers. We model this many-to-many interaction as a capacitated weighted bipartite matching problem, where matches are guided by both recourse cost and provider capacity. Edge weights, reflecting recourse costs, are optimized for social welfare while quantifying the welfare gap between individual welfare and this collectively feasible outcome. We propose a three-layer optimization framework: (1) basic capacitated matching, (2) optimal capacity redistribution to minimize the welfare gap, and (3) cost-aware optimization balancing welfare maximization with capacity adjustment costs. Experimental validation on synthetic and real-world datasets demonstrates that our framework enables the many-to-many algorithmic recourse to achieve near-optimal welfare with minimum modification in system settings. This work extends algorithmic recourse from individual recommendations to system-level design, providing a tractable path toward higher social welfare while maintaining individual actionability.

Two-stage adaptive robust optimization is a powerful approach for planning under uncertainty that aims to balance costs of "here-and-now" first-stage decisions with those of "wait-and-see" recourse decisions made after uncertainty is realized. To embed robustness against uncertainty, modelers typically assume a simple polyhedral or ellipsoidal set over which contingencies may be realized. However, these simple uncertainty sets tend to yield highly conservative decision-making when uncertainties are high-dimensional. In this work, we introduce AGRO, a column-and-constraint generation algorithm that performs adversarial generation for two-stage adaptive robust optimization using a variational autoencoder. AGRO identifies realistic and cost-maximizing contingencies by optimizing over spherical uncertainty sets in a latent space using a projected gradient ascent approach that differentiates the optimal recourse cost with respect to the latent variable. To demonstrate the cost- and time-efficiency of our approach experimentally, we apply AGRO to an adaptive robust capacity expansion problem for a regional power system and show that AGRO is able to reduce costs by up to 7.8% and runtimes by up to 77% in comparison to the conventional column-and-constraint generation algorithm.

In this appendix, we discuss a technique to optimize over the counterfactuals found by counterfactual explanation methods, such as [6]. We restate lemma 3.1 and provide a proof. This problem is identical to a well-studied class of bi-level optimization problems in deep learning. In these problems, we must compute the derivative of a function with respect to some parameter (here) that includes an inner argmin, which itself depends on the parameter. We follow [44] to complete the proof.

As machine learning models are increasingly being employed to make consequential decisions in real-world settings, it becomes critical to ensure that individuals who are adversely impacted (e.g., loan denied) by the predictions of these models are provided with a means for recourse. While several approaches have been proposed to construct recourses for affected individuals, the recourses output by these methods either achieve low costs (i.e., ease-of-implementation) or robustness to small perturbations (i.e., noisy implementations of recourses), but not both due to the inherent trade-offs between the recourse costs and robustness. Furthermore, prior approaches do not provide end users with any agency over navigating the aforementioned trade-offs. In this work, we address the above challenges by proposing the first algorithmic framework which enables users to effectively manage the recourse cost vs. More specifically, our framework Probabilistically ROBust rEcourse (PROBE) lets users choose the probability with which a recourse could get invalidated (recourse invalidation rate) if small changes are made to the recourse i.e., the recourse is implemented somewhat noisily. To this end, we propose a novel objective function which simultaneously minimizes the gap between the achieved (resulting) and desired recourse invalidation rates, minimizes recourse costs, and also ensures that the resulting recourse achieves a positive model prediction. We develop novel theoretical results to characterize the recourse invalidation rates corresponding to any given instance w.r.t. Experimental evaluation with multiple real world datasets demonstrates the efficacy of the proposed framework. Machine learning (ML) models are increasingly being deployed to make a variety of consequential decisions in domains such as finance, healthcare, and policy. Consequently, there is a growing emphasis on designing tools and techniques which can provide recourse to individuals who have been adversely impacted by the predictions of these models (Voigt & Von dem Bussche, 2017).

Beliefs and values are increasingly being incorporated into our AI systems through alignment processes, such as carefully curating data collection principles or regularizing the loss function used for training. However, the meta-alignment problem is that these human beliefs are diverse and not aligned across populations; furthermore, the implicit strength of each belief may not be well calibrated even among humans, especially when trying to generalize across contexts. Specifically, in high regret situations, we observe that contextual counterfactuals and recourse costs are particularly important in updating a decision maker's beliefs and the strengths to which such beliefs are held. Therefore, we argue that including counterfactuals is key to an accurate calibration of beliefs during alignment. To do this, we first segment belief diversity into two categories: subjectivity (across individuals within a population) and epistemic uncertainty (within an individual across different contexts). By leveraging our notion of epistemic uncertainty, we introduce `the belief calibration cycle' framework to more holistically calibrate this diversity of beliefs with context-driven counterfactual reasoning by using a multi-objective optimization. We empirically apply our framework for finding a Pareto frontier of clustered optimal belief strengths that generalize across different contexts, demonstrating its efficacy on a toy dataset for credit decisions.

Algorithmic Recourse refers to the capability of an algorithm to provide explanations and make recommendations in response to an appeal or challenge raised by an individual who has been affected negatively by its decision Wachter et al. (2017); Ustun et al. (2019); Karimi et al. (2020); Venkatasubramanian and Alfano (2020). This concept is particularly important in areas such as finance, healthcare, and criminal justice where decisions made by algorithms can have significant impacts on people's lives Chou et al. (2022). Recently, there has been an explosion of proposals for counterfactual explainers in the emerging field of algorithmic recourse Guidotti (2022); Stepin et al. (2021); Karimi et al. (2021); Verma et al. (2020). Ensuring fairness and robustness in algorithmic decision-making processes is crucial to guarantee fair and just outcomes for all involved. In the context of algorithmic recourse, robustness refers to the ability of an algorithm to withstand unreliability, manipulation, or deception by malicious actors, while still providing fair and accurate recourse recommendations Slack et al. (2021); Upadhyay et al. (2021); Dominguez-Olmedo et al. (2022); Pawelczyk et al. (2022). There are four types of unreliabilities in counterfactual explanations Mishra et al. (2021): Robustness to input perturbations: Examining recourse behavior in response to slight input changes while the classifier remains unchanged Dominguez-Olmedo et al. (2022).

The goal of algorithmic recourse is to reverse unfavorable decisions (e.g., from loan denial to approval) under automated decision making by suggesting actionable feature changes (e.g., reduce the number of credit cards). To generate low-cost recourse the majority of methods work under the assumption that the features are independently manipulable (IMF). To address the feature dependency issue the recourse problem is usually studied through the causal recourse paradigm. However, it is well known that strong assumptions, as encoded in causal models and structural equations, hinder the applicability of these methods in complex domains where causal dependency structures are ambiguous. In this work, we develop \texttt{DEAR} (DisEntangling Algorithmic Recourse), a novel and practical recourse framework that bridges the gap between the IMF and the strong causal assumptions. \texttt{DEAR} generates recourses by disentangling the latent representation of co-varying features from a subset of promising recourse features to capture the main practical recourse desiderata. Our experiments on real-world data corroborate our theoretically motivated recourse model and highlight our framework's ability to provide reliable, low-cost recourse in the presence of feature dependencies.

While machine learning and ranking-based systems are in widespread use for sensitive decision-making processes (e.g., determining job candidates, assigning credit scores), they are rife with concerns over unintended biases in their outcomes, which makes algorithmic fairness (e.g., demographic parity, equal opportunity) an objective of interest. 'Algorithmic recourse' offers feasible recovery actions to change unwanted outcomes through the modification of attributes. We introduce the notion of ranked group-level recourse fairness, and develop a 'recourse-aware ranking' solution that satisfies ranked recourse fairness constraints while minimizing the cost of suggested modifications. Our solution suggests interventions that can reorder the ranked list of database records and mitigate group-level unfairness; specifically, disproportionate representation of sub-groups and recourse cost imbalance. This re-ranking identifies the minimum modifications to data points, with these attribute modifications weighted according to their ease of recourse. We then present an efficient block-based extension that enables re-ranking at any granularity (e.g., multiple brackets of bank loan interest rates, multiple pages of search engine results). Evaluation on real datasets shows that, while existing methods may even exacerbate recourse unfairness, our solution -- RAGUEL -- significantly improves recourse-aware fairness. RAGUEL outperforms alternatives at improving recourse fairness, through a combined process of counterfactual generation and re-ranking, whilst remaining efficient for large-scale datasets.