ElliCE: Efficient and Provably Robust Algorithmic Recourse via the Rashomon Sets

TL;DR

ElliCE tackles the unreliability of counterfactual recourse under model multiplicity by approximating the Rashomon set of near-optimal models with an ellipsoid $\\hat{\\mathcal{R}}(\\epsilon)$ defined by a Hessian $H$. It reduces robust recourse to a convex QCQP where the inner worst-case model has a closed-form objective $\\hat{\\boldsymbol{\\theta}}^\\top \\mathbf{x}_c - \\sqrt{2\\epsilon \\mathbf{x}_c^\\top H^{-1}\\mathbf{x}_c}$, enabling efficient computation of counterfactuals that are valid for all models in the ellipsoid. The framework provides formal guarantees—validity, uniqueness, stability, and alignment with principal curvature directions—and supports actionability via immutable feature handling, range constraints, and sparsity controls. Empirically, ElliCE achieves higher robustness and up to $10^3\times$ speedups over baselines across nine high-stakes tabular datasets and both linear and MLP models, while maintaining plausibility when counterfactuals lie on the data manifold. This yields stable, actionable recourse under model evolution, with practical applicability to finance, healthcare, and beyond, though limitations remain regarding global Rashomon-set coverage for deep nets and needs for broader fairness analyses.

Abstract

Machine learning models now influence decisions that directly affect people's lives, making it important to understand not only their predictions, but also how individuals could act to obtain better results. Algorithmic recourse provides actionable input modifications to achieve more favorable outcomes, typically relying on counterfactual explanations to suggest such changes. However, when the Rashomon set - the set of near-optimal models - is large, standard counterfactual explanations can become unreliable, as a recourse action valid for one model may fail under another. We introduce ElliCE, a novel framework for robust algorithmic recourse that optimizes counterfactuals over an ellipsoidal approximation of the Rashomon set. The resulting explanations are provably valid over this ellipsoid, with theoretical guarantees on uniqueness, stability, and alignment with key feature directions. Empirically, ElliCE generates counterfactuals that are not only more robust but also more flexible, adapting to user-specified feature constraints while being substantially faster than existing baselines. This provides a principled and practical solution for reliable recourse under model uncertainty, ensuring stable recommendations for users even as models evolve.

Figures (22)

• Figure 1: Robustness evaluation of ElliCE against baselines. The plot displays the robustness metric (y-axis) as a function of the target robustness level $\varepsilon_{\text{target}}$ for the evaluators (x-axis). ElliCE consistently outperforms all baselines across all robustness levels. See Appendix \ref{['appendix:experiments']} for more figures. For ElliCE+R for MLPs, we apply additional regularization to the Hessian, using $\lambda = 0.1$ instead of $0.001$.
• Figure 2: (a,b) Sensitivity of ElliCE's robustness (y-axis) to its internal $\epsilon$ hyperparameter (x-axis). Robustness is evaluated against Random Retrain (a) and an Ellipsoidal Rashomon set approximation defined with a fixed $\varepsilon_{\text{target}}$ (b). (c, d) Robustness vs. $\ell_2$ proximity trade-off (c) and plausibility (d) of counterfactuals generated by ElliCE and baselines on Diabetes dataset.
• Figure 3: Robustness evaluation of ElliCE against baselines on MLPs using data-supported generation across all datasets.
• Figure 4: Length evaluation of ElliCE against baselines on MLPs using data-supported generation across all datasets.
• Figure 5: LOF evaluation of ElliCE against baselines on MLPs using data-supported generation across all datasets.

Theorems & Definitions (24)

• Theorem 1: Closed-form solution
• Corollary 1
• Theorem 2: Uniqueness
• Theorem 3: Stability
• Theorem 4: Alignment with Important Feature Directions
• Theorem 5: Robustness-Proximity Trade-off
• Theorem \ref{th:closed_form_solution}: Closed-form solution
• proof
• Theorem \ref{thm:uniqueness}: Uniqueness
• proof