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 \emph{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 \emph{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 \emph{Stability} as defined by our measure will remain valid after potential ``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：生成鲁棒的反事实解释——即模型更新或轻微变化后仍保持有效——正引起广泛兴趣。现有文献在寻找鲁棒反事实时，通常假设原始模型$m$与新模型$M$在参数空间内有界，即$\|\text{Params}(M){-}\text{Params}(m)\|{<}\Delta$。然而，模型在参数空间发生显著变化时，其预测结果或对给定数据集的精度可能几乎不变。本文引入一种称为“自然发生”模型变化的数学抽象，允许参数空间任意变化，但要求数据流形上的预测变化有限。随后，我们提出一种度量——称之为“稳定性”——用于量化可微模型（如神经网络）反事实对潜在模型变化的鲁棒性。我们的主要贡献在于证明：根据所提度量，具有足够高“稳定性”值的反事实在潜在“自然发生”模型变化后仍能以高概率保持有效（基于独立高斯函数Lipschitz性质的浓度界）。由于该量化依赖于数据点周围的局部Lipschitz常数（该常数并非总是可得），我们还探讨了所提度量的实际松弛形式，并通过实验展示如何将其整合到神经网络鲁棒反事实的搜索过程中，从而使反事实接近真实数据、具现实性且能在模型变化后保持有效。本工作与模型多重性（即Rashomon效应）也具有有趣关联。