Model Reconstruction Using Counterfactual Explanations: A Perspective From Polytope Theory

TL;DR

This work analyzes how model reconstruction using counterfactuals can be improved by further leveraging the fact that the counterfactuals also lie quite close to the decision boundary and proposes a strategy for model reconstruction that is called Counterfactual Clamping Attack (CCA), which trains a surrogate model using a unique loss function that treats counterfactuals differently than ordinary instances.

Abstract

Counterfactual explanations provide ways of achieving a favorable model outcome with minimum input perturbation. However, counterfactual explanations can also be leveraged to reconstruct the model by strategically training a surrogate model to give similar predictions as the original (target) model. In this work, we analyze how model reconstruction using counterfactuals can be improved by further leveraging the fact that the counterfactuals also lie quite close to the decision boundary. Our main contribution is to derive novel theoretical relationships between the error in model reconstruction and the number of counterfactual queries required using polytope theory. Our theoretical analysis leads us to propose a strategy for model reconstruction that we call Counterfactual Clamping Attack (CCA) which trains a surrogate model using a unique loss function that treats counterfactuals differently than ordinary instances. Our approach also alleviates the related problem of decision boundary shift that arises in existing model reconstruction approaches when counterfactuals are treated as ordinary instances. Experimental results demonstrate that our strategy improves fidelity between the target and surrogate model predictions on several datasets.

Figures (21)

• Figure 1: Decision boundary shift when counterfactuals are treated as ordinary labeled points.
• Figure 2: Problem setting
• Figure 3: Polytope approximation of a convex decision boundary using the closest counterfactuals.
• Figure 4: Approximating a concave region needs denser queries w.r.t. a convex region.
• Figure 5: $\mathcal{N}_\epsilon$ grid and inverse counterfactual regions. Thick solid lines indicate the decision boundary pieces (${\mathbb{H}}_i$'s). White color depicts the accepted region. Pale-colored are the inverse counterfactual regions of the ${\mathbb{H}}_i$'s with the matching color. In this case $k(\epsilon)=7$ and $v^*(\epsilon)$ is the area of lower amber region.

Theorems & Definitions (35)

• Definition 2.1: Counterfactual Generating Mechanism
• Definition 2.2: Closest Counterfactual
• Definition 2.3: Inverse Counterfactual Region
• Definition 2.4: Fidelity ulrichModelExtract
• Definition 2.5: Hypersurface, manifoldsLee
• Definition 2.6: Touching Hypersurfaces
• Lemma 2.6
• Lemma 3.0
• Theorem 3.1
• Remark 3.2: Relaxing the Convexity Assumption