Density-Guided Robust Counterfactual Explanations on Tabular Data under Model Multiplicity

Counterfactual explanations (CEs) are essential for actionable recourse, yet their reliability is often compromised in low-density regions, where classifiers exhibit high variance. Unlike existing methods that rely on expensive ensemble intersections to define stability, we propose DensityFlow, a generative framework that constructs robust CEs by adhering to the high-confidence data manifold. Specifically, we model the counterfactual generation as continuous-time dynamics parameterized by Neural ODE, guided by a differentiable density score to actively avoid uncertain, low-density areas. This density score is learned via Noise Contrastive Estimation, effectively leveraging a $(K{+}1)$-way discriminator to estimate density ratios. For black-box settings, we introduce a local proxy distillation mechanism that aligns a lightweight surrogate with the target model strictly within the trajectory of CE generation, enabling efficient gradient-based optimization with minimal queries. Experiments demonstrate that \textit{DensityFlow} achieves superior validity under model multiplicity while significantly reducing query costs compared to ensemble-based baselines. Our implementation is available at \url{https://github.com/G-AILab/DensityFlow}.

Machine learning systems are increasingly used to support decisions in areas such as finance, healthcare, and public services. When such a system rejects a person or assigns an unfavorable outcome, a common way to explain the decision is to provide a “what-if” suggestion: for example, what would need to change for the person to receive a better outcome. However, these suggestions can be unreliable. A small change may work for one trained model but fail for another equally reasonable model, especially when the suggested example lies far from the kinds of cases seen in the training data. This paper proposes DensityFlow, a method for generating more reliable what-if explanations for tabular data. The main idea is to guide the explanation toward regions that are well supported by the training data, instead of allowing it to move into rare or unrealistic regions where different models may disagree. DensityFlow also reduces the need to repeatedly query black-box prediction systems by learning a small local approximation only around the explanations being generated. Experiments on synthetic and real-world datasets show that DensityFlow produces explanations that are more consistently accepted by different models while keeping the required changes reasonably small.