# Counterfactual Explanations for Model Ensembles Using Entropic Risk Measures

arXiv: https://arxiv.org/abs/2503.07934 · HTML: https://mlsanity.com/paper/2503.07934/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 66 references that are in the map.
- First posted on arXiv 11 Mar 2025.

## Builds on (most cited first, 30 of 33)

- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [On the Robustness of Interpretability Methods](https://mlsanity.com/paper/1806.08049/) (2018)
- [A survey of algorithmic recourse: definitions, formulations, solutions, and prospects](https://mlsanity.com/paper/2010.04050/) (2020)
- [The Hidden Assumptions Behind Counterfactual Explanations and Principal Reasons](https://mlsanity.com/paper/1912.04930/) (2019)
- [Towards Robust and Reliable Algorithmic Recourse](https://mlsanity.com/paper/2102.13620/) (2021)
- [On Counterfactual Explanations under Predictive Multiplicity](https://mlsanity.com/paper/2006.13132/) (2020)
- [On the Adversarial Robustness of Causal Algorithmic Recourse](https://mlsanity.com/paper/2112.11313/) (2021)
- [Robust Counterfactual Explanations for Tree-Based Ensembles](https://mlsanity.com/paper/2207.02739/) (2022)
- [Rashomon Capacity: A Metric for Predictive Multiplicity in Classification](https://mlsanity.com/paper/2206.01295/) (2022)
- [Robust Counterfactual Explanations for Neural Networks With Probabilistic Guarantees](https://mlsanity.com/paper/2305.11997/) (2023)
- [Predictive Multiplicity in Probabilistic Classification](https://mlsanity.com/paper/2206.01131/) (2022)
- [Probabilistically Robust Recourse: Navigating the Trade-offs between Costs and Robustness in Algorithmic Recourse](https://mlsanity.com/paper/2203.06768/) (2022)
- [A Survey on the Robustness of Feature Importance and Counterfactual Explanations](https://mlsanity.com/paper/2111.00358/) (2021)
- [Consistent Counterfactuals for Deep Models](https://mlsanity.com/paper/2110.03109/) (2021)
- [Optimal Counterfactual Explanations in Tree Ensembles](https://mlsanity.com/paper/2106.06631/) (2021)
- [CERTIFAI: Counterfactual Explanations for Robustness, Transparency, Interpretability, and Fairness of Artificial Intelligence models](https://mlsanity.com/paper/1905.07857/) (2019)
- [Issues with post-hoc counterfactual explanations: a discussion](https://mlsanity.com/paper/1906.04774/) (2019)
- [Robust Counterfactual Explanations in Machine Learning: A Survey](https://mlsanity.com/paper/2402.01928/) (2024)
- [Formalising the Robustness of Counterfactual Explanations for Neural Networks](https://mlsanity.com/paper/2208.14878/) (2022)
- [When Ensembling Smaller Models is More Efficient than Single Large Models](https://mlsanity.com/paper/2005.00570/) (2020)
- [Provably Robust and Plausible Counterfactual Explanations for Neural Networks via Robust Optimisation](https://mlsanity.com/paper/2309.12545/) (2023)
- [Finding Regions of Counterfactual Explanations via Robust Optimization](https://mlsanity.com/paper/2301.11113/) (2023)
- [Robust Bayesian Recourse](https://mlsanity.com/paper/2206.10833/) (2022)
- [Uncertainty Estimation and Out-of-Distribution Detection for Counterfactual Explanations: Pitfalls and Solutions](https://mlsanity.com/paper/2107.09734/) (2021)
- [Recourse under Model Multiplicity via Argumentative Ensembling (Technical Report)](https://mlsanity.com/paper/2312.15097/) (2023)
- [Model Reconstruction Using Counterfactual Explanations: A Perspective From Polytope Theory](https://mlsanity.com/paper/2405.05369/) (2024)
- [Fair lending needs explainable models for responsible recommendation](https://mlsanity.com/paper/1809.04684/) (2018)
- [Counterfactual Explanations with Probabilistic Guarantees on their Robustness to Model Change](https://mlsanity.com/paper/2408.04842/) (2024)
- [Global Counterfactual Explanations: Investigations, Implementations and Improvements](https://mlsanity.com/paper/2204.06917/) (2022)
- [Coverage-Validity-Aware Algorithmic Recourse](https://mlsanity.com/paper/2311.11349/) (2023)

## Built on by (most cited first, 5 of 5)

- [Provably Robust Bayesian Counterfactual Explanations under Model Changes](https://mlsanity.com/paper/2601.16659/) (2026)
- [Axiomatic Foundations of Counterfactual Explanations](https://mlsanity.com/paper/2602.04028/) (2026)
- [The Role of Causality in Algorithmic Recourse](https://mlsanity.com/paper/2607.28497/) (2026)
- [Density-Guided Robust Counterfactual Explanations on Tabular Data under Model Multiplicity](https://mlsanity.com/paper/2605.30901/) (2026)
- [COFFEQA: Counterfactual-Guided Few-Shot Distillation for LLM Question Answering](https://mlsanity.com/paper/2610.12837/) (2026)

## Abstract (arXiv)

Counterfactual explanations indicate the smallest change in input that can translate to a different outcome for a machine learning model. Counterfactuals have generated immense interest in high-stakes applications such as finance, education, hiring, etc. In several use-cases, the decision-making process often relies on an ensemble of models rather than just one. Despite significant research on counterfactuals for one model, the problem of generating a single counterfactual explanation for an ensemble of models has received limited interest. Each individual model might lead to a different counterfactual, whereas trying to find a counterfactual accepted by all models might significantly increase cost (effort). We propose a novel strategy to find the counterfactual for an ensemble of models using the perspective of entropic risk measure. Entropic risk is a convex risk measure that satisfies several desirable properties. We incorporate our proposed risk measure into a novel constrained optimization to generate counterfactuals for ensembles that stay valid for several models. The main significance of our measure is that it provides a knob that allows for the generation of counterfactuals that stay valid under an adjustable fraction of the models. We also show that a limiting case of our entropic-risk-based strategy yields a counterfactual valid for all models in the ensemble (worst-case min-max approach). We study the trade-off between the cost (effort) for the counterfactual and its validity for an ensemble by varying degrees of risk aversion, as determined by our risk parameter knob. We validate our performance on real-world datasets.