Robust Algorithmic Recourse Under Model Multiplicity With Probabilistic Guarantees

Faisal Hamman;Erfaun Noorani;Saumitra Mishra;Daniele Magazzeni;Sanghamitra Dutta
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Abstract

There is an emerging interest in generating robust algorithmic recourse that would remain valid if the model is updated or changed even slightly. Towards finding robust algorithmic recourse (or counterfactual explanations), 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)\|{\lt }\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 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 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 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 estimators of our proposed measure and derive a fundamental lower bound on the sample size required to have a precise estimate. We explore methods of using stability measures to generate robust counterfactuals 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.
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模型多重性下的稳健算法求助与概率保证
人们对产生稳健算法追索权的兴趣日渐浓厚,即使模型更新或稍有改变,这种追索权仍然有效。为了找到稳健的算法追索(或反事实解释),现有文献通常假定原始模型 m 和新模型 M 在参数空间中是有界的,即 $\|\text {Params}(M){-}\text {Params}(m)\|{\lt }\Delta $ 。 然而,模型通常会在参数空间中发生显著变化,而其对给定数据集的预测或准确性却几乎没有变化。在这项工作中,我们引入了一种数学抽象,称为自然发生的模型变化,它允许参数空间的任意变化,从而限制了对位于数据流形上的点的预测变化。接下来,我们提出了一种测量方法--我们称之为 "稳定性"--来量化反事实对可微分模型(如神经网络)潜在模型变化的稳健性。我们的主要贡献在于证明,根据我们的测量方法,具有足够高稳定性值的反事实在潜在的 "自然发生的 "模型变化后将以很高的概率保持有效(利用独立高斯的 Lipschitz 函数的集中边界)。由于我们的量化方法取决于数据点周围的局部 Lipschitz 常量,而该常量并不总是可用的,因此我们还研究了我们提出的测量方法的估计值,并得出了精确估计所需的样本量的基本下限。我们探讨了使用稳定性测量方法生成稳健的反事实的方法,这些反事实是接近的、现实的,并且在潜在的模型变化后仍然有效。这项工作还与模型多重性(又称罗生门效应)有着有趣的联系。
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