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Probabilistically Robust Recourse: Navigating the Trade-offs between Costs and Robustness in Algorithmic Recourse

Martin Pawelczyk, Teresa Datta, Johannes van-den-Heuvel, Gjergji Kasneci, Himabindu Lakkaraju

TL;DR

This work proposes 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.

Abstract

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. robustness trade-offs. More specifically, our framework Probabilistically ROBust rEcourse (\texttt{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. different classes of underlying models (e.g., linear models, tree based models etc.), and leverage these results to efficiently optimize the proposed objective. Experimental evaluation with multiple real world datasets demonstrates the efficacy of the proposed framework.

Probabilistically Robust Recourse: Navigating the Trade-offs between Costs and Robustness in Algorithmic Recourse

TL;DR

This work proposes 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.

Abstract

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. robustness trade-offs. More specifically, our framework Probabilistically ROBust rEcourse (\texttt{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. different classes of underlying models (e.g., linear models, tree based models etc.), and leverage these results to efficiently optimize the proposed objective. Experimental evaluation with multiple real world datasets demonstrates the efficacy of the proposed framework.
Paper Structure (13 sections, 6 theorems, 7 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 6 theorems, 7 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Algorithmic Recourse: General Formulation
  5. Defining the Recourse Invalidation Rate
  6. Our Framework: Probabilistically Robust Recourse
  7. Recourse invalidation rate aware objective
  8. Optimizing the recourse invalidation rate aware objective
  9. Additional Theoretical Results
  10. Experimental Evaluation
  11. Validating our Theoretical Bounds
  12. Evaluating the PROBE Framework
  13. Conclusion

Key Result

Theorem 1

A first-order approximation $\tilde{\Delta}$ to the recourse invalidation rate $\Delta$ in equation eq:def_invalidation_prob under Gaussian distributed noise in human responses $\bm{\varepsilon} \sim \mathcal{N(\mathbf{0}, \sigma^2\mathbf{I}})$ is given by: where $\Phi$ is the CDF of the univariate standard normal distribution $\mathcal{N}(0,1)$, $f(\check{\mathbf{x}}_{E})$ denotes the logit scor

Figures (6)

  • Figure 1: Pictorial representation of the recourses (counterfactuals) output by various state-of-the-art recourse methods and our framework. The blue line is the decision boundary, and the shaded areas correspond to the regions of recourse invalidation. Fig. \ref{['fig:standard_recourse']} shows the recourse output by approaches such as wachter2017counterfactual where both the recourse cost as well as robustness are low. Fig. \ref{['fig:adv_recourse']} shows the recourse output by approaches such as dominguezolmedo2021adversarial where both the recourse cost and robustness are high. Fig. \ref{['fig:prob_recourse']} shows the recourse output by our framework PROBE in response to user input requesting an intermediate level of recourse robustness.
  • Figure 2: Practical view on navigating the cost/robustness tradeoff for a credit loan example.
  • Figure 3: Navigating between high and low invalidation recourses. The circles around PROBE's recourses have radius $2\sigma$, i.e., this is the region where 95% of recourse inaccuracies fall when $\sigma^2=0.05$. For instance, on the left we set an invalidation target of $r=0.35$, i.e., 35% of the recourse responses would fail under spherical inaccuracies $\bm{\varepsilon} \sim \mathcal{N}(\mathbf{0}, 0.05 \cdot \mathbf{I})$.
  • Figure 4: Comparing PROBE to adversarially robust recourse methods using pareto plots that show the tradeoff between average costs and average invalidation rate (towards bottom left indicates a better performance). For PROBE, the invalidation target is $r \in \{0.35, 0.3, 0.25, 0.20, 0.15\}$, and we generated recourses by setting $\sigma^2, \epsilon \in \{0.005, 0.01, 0.015\}$. The latter are used for ARAR and ROAR.
  • Figure 5: Verifying the theoretical upper bound from Proposition \ref{['lemma:sparsity']} on the logistic regression model. The red boxplots show the empirical recourse invalidation rates for AR(-LIME), Wachter, GS, DICE, ARAR ($\epsilon=0.01$), ROAR ($\epsilon=0.01$) and PROBE ($r=0.35, \sigma^2=0.01$). The blue boxplots show the distribution of upper bounds evaluated by plugging in the corresponding quantities (i.e., $\sigma^2$, $\omega$, etc.) into the bound. The results show no violations of our theoretical bounds. See appendix \ref{['appendix:additional_experiments']} for the full set of results.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Definition 1: Recourse Invalidation Rate
  • Theorem 1: Closed-Form Recourse Invalidation Rate
  • Proposition 1: Exact Recourse IR
  • Corollary 1
  • Proposition 2: General Cost of Recourse
  • Proposition 3: Cost-Robustness Tradeoff
  • Proposition 4: Upper Bound