@@ -269,7 +269,7 @@ The first two terms in Equation~\ref{eq:eccco} correspond to the counterfactual
The entire procedure for Generating ECCCos is described in Algorithm~\ref{alg:eccco}. For the sake of simplicity and without loss of generality, we limit our attention to generating a single counterfactual $\mathbf{x}^\prime=f(\mathbf{z}^\prime)$ where in contrast to Equation~\ref{eq:eccco}$\mathbf{z}^\prime$ denotes a $1$-dimensional array containing a single counterfactual state. That state is initialized by passing the factual $\mathbf{x}$ through the encoder $f^{-1}$ which in our case corresponds to a simple feature transformer, rather than the encoder part of VAE as in REVISE~\citep{joshi2019realistic}. Next, we generate a buffer of $N_{\mathcal{B}}$ conditional samples $\hat{\mathbf{x}}_{\theta}|\mathbf{y}^*$ using SGLD (Equation~\ref{eq:sgld}) and conformalise the model $M_{\theta}$ through Split Conformal Prediction on training data $\mathcal{D}$.
Finally, we search counterfactuals through gradient descent. Let $\mathcal{L}(\mathbf{z}^\prime,\mathbf{y}^*,\hat{\mathbf{x}}_{\theta, t})$ denote our loss function defined in Equation~\ref{eq:eccco}. Then in each iteration, we first randomly draw $n_{\mathcal{B}}$ samples from the buffer $\mathcal{B}$ before updating the counterfactual state $\mathbf{z}^\prime$ by moving in the negative direction of that loss function. The search terminates once the convergence criterium is met or the maximum number of iterations $T$ has been exhausted. Note that the choice of convergence criterium has important implications on the final counterfactual (for more detail on this see Appendix~\ref{app:eccco}).
Finally, we search counterfactuals through gradient descent. Let $\mathcal{L}(\mathbf{z}^\prime,\mathbf{y}^*,\hat{\mathbf{x}}_{\theta, t}; \Lambda, \alpha)$ denote our loss function defined in Equation~\ref{eq:eccco}. Then in each iteration, we first randomly draw $n_{\mathcal{B}}$ samples from the buffer $\mathcal{B}$ before updating the counterfactual state $\mathbf{z}^\prime$ by moving in the negative direction of that loss function. The search terminates once the convergence criterium is met or the maximum number of iterations $T$ has been exhausted. Note that the choice of convergence criterium has important implications on the final counterfactual (for more detail on this see Appendix~\ref{app:eccco}).
Figure~\ref{fig:eccco-mnist} presents ECCCos for the MNIST example from Section~\ref{background} for various black-box models of increasing complexity from left to right: a simple Multi-Layer Perceptron (MLP); an Ensemble of MLPs, each of the same architecture as the single MLP; a Joint Energy Model (JEM) based on the same MLP architecture; and finally, an Ensemble of these JEMs. Since Deep Ensembles have an improved capacity for predictive uncertainty quantification and JEMs are explicitly trained to learn plausible representations of the input data, it is intuitive to see that the plausibility of counterfactuals visibly improves from left to right. This provides some first anecdotal evidence that ECCCos achieve plausibility while maintaining faithfulness to the Black Box.
