There is an emerging interest in generating robust counterfactual explanations that would remain valid if the model is updated or changed even slightly. Towards finding robust counterfactuals, existing literature often assumes that the original model $m$ and the new model $M$ are bounded in the parameter space, i.e., $\|\text{Params}(M){-}\text{Params}(m)\|{<}\Delta$. However, models can often change significantly in the parameter space with little to no change in their predictions or accuracy on the given dataset. In this work, we introduce a mathematical abstraction termed $\textit{naturally-occurring}$ model change, which allows for arbitrary changes in the parameter space such that the change in predictions on points that lie on the data manifold is limited. Next, we propose a measure -- that we call $\textit{Stability}$ -- to quantify the robustness of counterfactuals to potential model changes for differentiable models, e.g., neural networks. Our main contribution is to show that counterfactuals with sufficiently high value of $\textit{Stability}$ as defined by our measure will remain valid after potential $\textit{naturally-occurring}$ model changes with high probability (leveraging concentration bounds for Lipschitz function of independent Gaussians). Since our quantification depends on the local Lipschitz constant around a data point which is not always available, we also examine practical relaxations of our proposed measure and demonstrate experimentally how they can be incorporated to find robust counterfactuals for neural networks that are close, realistic, and remain valid after potential model changes. This work also has interesting connections with model multiplicity, also known as, the Rashomon effect.

翻译：生成鲁棒反事实解释（即在模型发生更新或轻微变化时仍保持有效性）正引发学界日益浓厚的兴趣。现有文献在寻找鲁棒反事实解释时，通常假设原始模型 $m$ 与新模型 $M$ 在参数空间中有界，即 $\|\text{Params}(M){-}\text{Params}(m)\|{<}\Delta$。然而，模型在参数空间中可能发生显著变化，同时对其在给定数据集上的预测或准确率影响甚微。本文引入一种称为"自然发生"模型变化的数学抽象概念，该概念允许参数空间出现任意变化，但限制数据流形上点的预测变化幅度。随后，我们提出一种衡量指标——称为"稳定性"（Stability）——用于量化可微模型（如神经网络）中反事实解释对潜在模型变化的鲁棒性。我们的主要贡献在于证明：根据该指标定义的具有足够高"稳定性"值的反事实解释，在经历潜在的"自然发生"模型变化后，仍能高概率保持有效（利用独立高斯分布Lipschitz函数的浓度界）。由于该量化依赖于数据点周围的局部Lipschitz常数（该常数并非始终可得），我们还考察了所提度量指标的实用松弛形式，并通过实验证明如何将其融入方法以寻找神经网络的鲁棒反事实解释——这些解释需具备邻近性、真实性，且在潜在模型变化后仍保持有效性。本研究还与模型多重性（即Rashomon效应）存在有趣关联。