Coverage-Validity-Aware Algorithmic Recourse

TL;DR

This work proposes a novel framework to generate a model-agnostic recourse that exhibits robustness to model shifts and proves that by prescribing different covariance robustness, the proposed framework recovers popular regularizations for MPM, including the $\ell_2$-regularization and class-reweighting.

Abstract

Algorithmic recourse emerges as a prominent technique to promote the explainability, transparency, and ethics of machine learning models. Existing algorithmic recourse approaches often assume an invariant predictive model; however, the predictive model is usually updated upon the arrival of new data. Thus, a recourse that is valid respective to the present model may become invalid for the future model. To resolve this issue, we propose a novel framework to generate a model-agnostic recourse that exhibits robustness to model shifts. Our framework first builds a coverage-validity-aware linear surrogate of the nonlinear (black-box) model; then, the recourse is generated with respect to the linear surrogate. We establish a theoretical connection between our coverage-validity-aware linear surrogate and the minimax probability machines (MPM). We then prove that by prescribing different covariance robustness, the proposed framework recovers popular regularizations for MPM, including the $\ell_2$-regularization and class-reweighting. Furthermore, we show that our surrogate pushes the approximate hyperplane intuitively, facilitating not only robust but also interpretable recourses. The numerical results demonstrate the usefulness and robustness of our framework.

Figures (17)

• Figure 1: An example of recourse failures under model shifts. The present model's boundary is plotted in a dashed curve, separating the input space into the rejection (yellow) and approval (blue) regions. The original input (star) lies in the rejection region of the present model. The recommended recourse (square symbol) is in the approval region of the present model. However, as the boundary shifts to the solid line, the recourse falls into the rejection region under the new boundary.
• Figure 2: The sampler synthesizes new instances around $x_0$ and queries the predicted labels from the classifier $f$ . The moment information $(\widehat{\mu}_y, \widehat{\Sigma}_y)$ estimated from the synthetic pseudo-labeled data (represented by triangles and ellipsoids) serves as inputs to find the (covariance-robust) Coverage-Validity-Aware Surrogate. The surrogate $\theta^\varphi$ (red hyperplane) is the target classifier to generate recourses (red circle).
• Figure 3: An intuitive explanation of the robustification mechanism. Left: CVAS, right: covariance-robust CVAS with $\rho_{-1} > \rho_{+1}$. By increasing the radius $\rho_{-1}$, the worst-case covariance matrix of the class $-1$ is inflated (bigger green ellipsoid) and shifts the surrogate boundary towards the class $+1$. The projection of the input $x_0$ onto the hyperplane will tend to lie deeper into the favorable region and may become more robust to model shifts.
• Figure 4: Four possibilities of the relative position of the surrogate to the data clusters. The black arrow indicates the normal vector of the hyperplane, pointing toward the region associated with the positive label. In case 1, the two mean vectors are classified correctly by the hyperplane. In cases 2 and 3, only one of the mean vectors is classified correctly. Case 4 is trivial since no mean vectors are classified correctly, leading to both trivial coverage and validity $\mathrm{Co}_{\widehat{\Sigma}_{+1}} = \mathrm{Va}_{\widehat{\Sigma}_{-1}} = 0$.
• Figure 5: Comparison of the asymptotic hyperplanes of Quadratic, Bures, and Fisher-Rao surrogates as $\rho_{-1} \to \infty$.

Theorems & Definitions (31)

• Proposition 2.1: Equivalence characterization
• Lemma 2.2: Optimal solution, adapted from ref:lanckriet2001minimax
• Proposition 3.1: Robust surrogate under model shifts
• Proposition 3.2: Gaussian equivalence
• Definition 4.1: Quadratic divergence
• Theorem 4.2: Quadratic surrogate
• Definition 4.3: Bures divergence
• Theorem 4.4: Bures surrogate
• Proposition 4.5: Bures divergence
• Definition 4.6: Fisher-Rao distance