added energy delta

paper/aaai/author_response.md

@@ -2,10 +2,11 @@
 
 1. Applied to additional commonly used tabular real-world datasets
 2. Energy delta
-3. Better results also for image data
-4. Counterfactual explanations do not scale well to high-dimensional input data
+   1. Better results, in particular for image data
+   2. No longer biased (addressing reviewer concern)
+3. Counterfactual explanations do not scale well to high-dimensional input data
    1. We have added native support for multi-processing and multi-threading
    2. We have run more extensive experiments including fine-tuning hyperparameter choices
-5. We have revisited the mathematical notation.
-6. We have moved the introduction of conformal prediction forward and added more detail in line with reviewer feedback.
-7. We have extended the limitations section. 
\ No newline at end of file
+4. We have revisited the mathematical notation.
+5. We have moved the introduction of conformal prediction forward and added more detail in line with reviewer feedback.
+6. We have extended the limitations section. 
\ No newline at end of file

paper/body.tex

@@ -134,15 +134,15 @@ The parallels between our definitions of plausibility and faithfulness imply tha
 
 where $\mathbf{x}^{\prime}$ denotes the counterfactual and $\mathbf{X}_{\mathbf{y}^+}$ is a subsample of the training data in the target class $\mathbf{y}^+$. By averaging over multiple samples in this manner, we avoid the risk that the nearest neighbour of $\mathbf{x}^{\prime}$ itself is not plausible according to Definition~\ref{def:plausible} (e.g an outlier).
 
-Equation~\ref{eq:impl} gives rise to a similar evaluation metric for unfaithfulness. We merely swap out the subsample of individuals in the target class for a subset $\widehat{\mathbf{X}}_{\mathbf{y}^+}$ of the generated conditional samples:
+Equation~\ref{eq:impl} gives rise to a similar evaluation metric for unfaithfulness. We swap out the subsample of observed individuals in the target class for the set of samples generated through SGLD ($\widehat{\mathbf{X}}_{\mathbf{y}^+}$):
 
 \begin{equation}\label{eq:faith}
   \begin{aligned}
-    \text{unfaith}(\mathbf{x}^{\prime},\widehat{\mathbf{X}}_{\mathbf{y}^+}) = \frac{1}{n_E} \sum_{\mathbf{x} \in \widehat{\mathbf{X}}_{\mathbf{y}^+}} \text{dist}(\mathbf{x}^{\prime},\mathbf{x})
+    \text{unfaith}(\mathbf{x}^{\prime},\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}) = \frac{1}{\lvert \widehat{\mathbf{X}}_{\theta,\mathbf{y}^+} \rvert} \sum_{\mathbf{x} \in \widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}} \text{dist}(\mathbf{x}^{\prime},\mathbf{x})
   \end{aligned}
 \end{equation}
 
-Specifically, we form this subset based on the $n_E$ generated samples with the lowest energy. 
+Our default choice for the $\text{dist}(\cdot)$ in both cases is the L1 Norm.
 
 \section{Energy-Constrained Conformal Counterfactuals}\label{meth}
 
@@ -153,11 +153,19 @@ We begin by stating our proposed objective function, which involves tailored los
 \begin{equation} \label{eq:eccco}
   \begin{aligned}
   \mathbf{Z}^\prime &= \arg \min_{\mathbf{Z}^\prime \in \mathcal{Z}^L} \{  {\text{yloss}(M_{\theta}(f(\mathbf{Z}^\prime)),\mathbf{y}^+)}+ \lambda_{1} {\text{dist}(f(\mathbf{Z}^\prime),\mathbf{x}) } \\
-  &+ \lambda_2 \text{unfaith}(f(\mathbf{Z}^\prime),\widehat{\mathbf{X}}_{\mathbf{y}^+}) + \lambda_3 \Omega(C_{\theta}(f(\mathbf{Z}^\prime);\alpha)) \} 
+  &+ \lambda_2 \Delta\mathcal{E}(\mathbf{Z}^\prime,\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}) + \lambda_3 \Omega(C_{\theta}(f(\mathbf{Z}^\prime);\alpha)) \} 
   \end{aligned} 
 \end{equation}
 
-The first penalty term involving $\lambda_1$ induces proximity like in~\citet{wachter2017counterfactual}. Our default choice for $\text{dist}(\cdot)$ is the L1 Norm due to its sparsity-inducing properties. The second penalty term involving $\lambda_2$ induces faithfulness by constraining the energy of the generated counterfactual where $\text{unfaith}(\cdot)$ corresponds to the metric defined in Equation~\ref{eq:faith}. The third and final penalty term involving $\lambda_3$ ensures that the generated counterfactual is associated with low predictive uncertainty.
+The first penalty term involving $\lambda_1$ induces proximity like in~\citet{wachter2017counterfactual}. Our default choice for $\text{dist}(\cdot)$ is the L1 Norm due to its sparsity-inducing properties. The second penalty term involving $\lambda_2$ induces faithfulness by constraining the energy of the generated counterfactual where we have:
+
+\begin{equation} \label{eq:energy-delta}
+  \begin{aligned}
+    \Delta\mathcal{E}&=\mathcal{E}(f(\mathbf{Z}^\prime)|\mathbf{y}^+)-\mathcal{E}(x|\mathbf{y}^+) &&& x \sim \widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}
+  \end{aligned}
+\end{equation}
+
+In particular, this penalty ensures that the energy of the generated counterfactual is in balance with the energy of the generated conditional samples ($\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}$). The third and final penalty term involving $\lambda_3$ ensures that the generated counterfactual is associated with low predictive uncertainty.
 
 \begin{figure}
   \centering
@@ -172,11 +180,11 @@ The first penalty term involving $\lambda_1$ induces proximity like in~\citet{wa
     \Ensure $\mathbf{x}^\prime$
     \State Initialize $\mathbf{z}^\prime \gets f^{-1}(\mathbf{x})$ \Comment{Map to counterfactual state space.}
     \State Generate $\left\{\hat{\mathbf{x}}_{\theta,\mathbf{y}^+}\right\}_{n_{\mathcal{B}}} \gets p_{\theta}(\mathbf{x}_{\mathbf{y}^+})$ \Comment{Generate $n_{\mathcal{B}}$ samples using SGLD (Equation~\ref{eq:sgld}).}
-    \State Store $\widehat{\mathbf{X}}_{\mathbf{y}^+} \gets \left\{\hat{\mathbf{x}}_{\theta,\mathbf{y}^+}\right\}_{n_{\mathcal{B}}}$ \Comment{Choose $n_E$ lowest-energy samples.}
+    \State Store $\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+} \gets \left\{\hat{\mathbf{x}}_{\theta,\mathbf{y}^+}\right\}_{n_{\mathcal{B}}}$ \Comment{Choose $n_E$ lowest-energy samples.}
     \State Run \textit{SCP} for $M_{\theta}$ using $\mathcal{D}$ \Comment{Calibrate model through split conformal prediction.}
     \State Initialize $t \gets 0$
     \While{\textit{not converged} or $t < T$} \Comment{For convergence conditions see Appendix~\ref{app:eccco}.}
-    \State $\mathbf{z}^\prime \gets \mathbf{z}^\prime - \eta \nabla_{\mathbf{z}^\prime} \mathcal{L}(\mathbf{z}^\prime,\mathbf{y}^+,\widehat{\mathbf{X}}_{\mathbf{y}^+}; \Lambda, \alpha)$ \Comment{Take gradient step of size $\eta$.}
+    \State $\mathbf{z}^\prime \gets \mathbf{z}^\prime - \eta \nabla_{\mathbf{z}^\prime} \mathcal{L}(\mathbf{z}^\prime,\mathbf{y}^+,\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}; \Lambda, \alpha)$ \Comment{Take gradient step of size $\eta$.}
     \State $t \gets t+1$
     \EndWhile
     \State $\mathbf{x}^\prime \gets f(\mathbf{z}^\prime)$ \Comment{Map back to feature space.}
@@ -187,7 +195,7 @@ Figure~\ref{fig:poc} illustrates how the different components in Equation~\ref{e
 
 While \textit{Wachter} generates a valid counterfactual, it ends up close to the original starting point consistent with its objective. \textit{ECCCo (no EBM)} pushes the counterfactual further into the target domain to minimize predictive uncertainty, but the outcome is still not plausible. The counterfactual produced by \textit{ECCCo (no CP)} is attracted by the generated samples shown in bright yellow. Since the \textit{JEM} has learned the conditional input distribution reasonably well in this case, the counterfactuals are both faithful and plausible. Finally, the outcome for \textit{ECCCo} looks similar, but the additional smooth set size penalty leads to somewhat faster convergence. 
 
-Algorithm~\ref{alg:eccco} describes how exactly \textit{ECCCo} works. For the sake of simplicity and without loss of generality, we limit our attention to generating a single counterfactual $\mathbf{x}^\prime=f(\mathbf{z}^\prime)$. The counterfactual state $\mathbf{z}^\prime$ is initialized by passing the factual $\mathbf{x}$ through a simple feature transformer $f^{-1}$. Next, we generate $n_{\mathcal{B}}$ conditional samples $\hat{\mathbf{x}}_{\theta,\mathbf{y}^+}$ using SGLD (Equation~\ref{eq:sgld}) and store the $n_E$ instances with the lowest energy. We then calibrate the model $M_{\theta}$ through split conformal prediction. Finally, we search counterfactuals through gradient descent where $\mathcal{L}(\mathbf{z}^\prime,\mathbf{y}^+,\widehat{\mathbf{X}}_{\mathbf{y}^+}; \Lambda, \alpha)$ denotes our loss function defined in Equation~\ref{eq:eccco}. 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 which we explain in Appendix~\ref{app:eccco}.
+Algorithm~\ref{alg:eccco} describes how exactly \textit{ECCCo} works. For the sake of simplicity and without loss of generality, we limit our attention to generating a single counterfactual $\mathbf{x}^\prime=f(\mathbf{z}^\prime)$. The counterfactual state $\mathbf{z}^\prime$ is initialized by passing the factual $\mathbf{x}$ through a simple feature transformer $f^{-1}$. Next, we generate $n_{\mathcal{B}}$ conditional samples $\hat{\mathbf{x}}_{\theta,\mathbf{y}^+}$ using SGLD (Equation~\ref{eq:sgld}) and store the $n_E$ instances with the lowest energy. We then calibrate the model $M_{\theta}$ through split conformal prediction. Finally, we search counterfactuals through gradient descent where $\mathcal{L}(\mathbf{z}^\prime,\mathbf{y}^+,\widehat{\mathbf{X}}_{\theta,\mathbf{y}^+}; \Lambda, \alpha)$ denotes our loss function defined in Equation~\ref{eq:eccco}. 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 which we explain in Appendix~\ref{app:eccco}.
 
 \section{Empirical Analysis}\label{emp}