A Optimizing Over the Returned Counterfactuals (x) follows the form of the objective in equation 1, G(x, A

In this appendix, we discuss a technique to optimize over the counterfactuals found by counterfactual explanation methods, such as [6]. We restate lemma 3.1 and provide a proof. This problem is identical to a well-studied class of bi-level optimization problems in deep learning. In these problems, we must compute the derivative of a function with respect to some parameter (here) that includes an inner argmin, which itself depends on the parameter. We follow [44] to complete the proof.