more work on paper

paper/paper.tex

@@ -107,7 +107,7 @@ When people talk about Black Box Models, this is usually the type of model they
 
 In the context of CE, the idea that no two explanations are the same arises almost naturally. Even the baseline approach proposed by \citet{wachter2017counterfactual} can yield a diverse set of explanations if counterfactuals are initialised randomly. This multiplicity of explanations has not only been acknowledged in the literature but positively embraced: since individuals seeking Algorithmic Recourse (AR) have unique preferences,~\citet{mothilal2020explaining}, for example, have prescribed \textit{diversity} as an explicit goal for counterfactuals. More generally, the literature on CE and AR has brought forward a myriad of desiderata for explanations, which we will discuss in more detail in the following section.
 
-\section{Adversarial Example or Valid Explanation?}\label{background}
+\section{From Adversarial Examples to Plausible Explanations}\label{background}
 
 Most state-of-the-art approaches to generating Counterfactual Explanations (CE) rely on gradient descent to optimize different flavours of the same counterfactual search objective,
 
@@ -119,21 +119,23 @@ Most state-of-the-art approaches to generating Counterfactual Explanations (CE)
 
 where $\text{yloss}$ denotes the primary loss function already introduced above and $\text{cost}$ is either a single penalty or a collection of penalties that are used to impose constraints through regularization. Following the convention in \citet{altmeyer2023endogenous} we use $\mathbf{s}^\prime=\{ s_k\}_K$ to denote the vector $K$-dimensional array of counterfactual states. This is to explicitly account for the fact that we can generate multiple counterfactuals, as with DiCE \citep{mothilal2020explaining}, and may choose to traverse a latent representation $\mathcal{Z}$ of the feature space $\mathcal{X}$, as we will discuss further below. 
 
-Solutions to Equation~\ref{eq:general} are considered valid as soon as the predicted label matches the target label. A stripped-down counterfactual explanation is therefore little different from an adversarial example. In Figure~\ref{fig:adv}, for example, generic counterfactual search as in \citet{wachter2017counterfactual} has been applied to MNIST data.
+Solutions to Equation~\ref{eq:general} are considered valid as soon as the predicted label matches the target label. A stripped-down counterfactual explanation is therefore little different from an adversarial example. In Figure~\ref{fig:adv}, for example, we have the baseline approach proposed in \citet{wachter2017counterfactual} to MNIST data (centre panel). This approach solves Equation~\ref{eq:general} through gradient-descent in the feature space with a penalty for the distance between the factual $x$ and the counterfactual $x^{\prime}$. The underlying classifier $M$ is a simple Multi-Layer Perceptron (MLP) with good test accuracy. For the generated counterfactual $x^{\prime}$ the model predicts the target label with high confidence (centre panel in Figure~\ref{fig:adv}). The explanation is valid by definition, even though it looks a lot like an Adversarial Example \citep{goodfellow2014explaining}. \citet{schut2021generating} make the connection between Adversarial Examples and Counterfactual Explanations explicit and propose using a Jacobian-Based Saliency Map Attack to solve Equation~\ref{eq:general}. They demonstrate that this approach yields realistic and sparse counterfactuals for Bayesian, adversarially robust classifiers. Applying their approach to our simple MNIST classifier does not yield a realistic counterfactual but this one, too, is valid (right panel in Figure~\ref{fig:adv}). 
 
-To properly serve both AI practitioners and individuals affected by AI decision-making systems, Counterfactual Explanations should have certain desirable properties, sometimes referred to as \textit{desiderata}. Besides diversity, which we already introduced above, some of the most prominent desiderata include sparsity, proximity \citep{wachter2017counterfactual}, actionability~\citep{ustun2019actionable}, plausibility \citep{joshi2019realistic,poyiadzi2020face,schut2021generating}, robustness \citep{upadhyay2021robust,pawelczyk2022probabilistically,altmeyer2023endogenous} and causality~\citep{karimi2021algorithmic}. Researchers have come up with various ways to meet these desiderata, which have been surveyed in~\citep{verma2020counterfactual} and~\citep{karimi2020survey}. 
+The crucial difference between Adversarial Examples (AE) and Counterfactual Explanations is one of intent. While an AE is intended to go unnoticed, a CE should have certain desirable properties. The literature has made this explicit by introducing various so-called \textit{desiderata}. To properly serve both AI practitioners and individuals affected by AI decision-making systems, counterfactuals should be sparse, proximate~\citep{wachter2017counterfactual}, actionable~\citep{ustun2019actionable}, diverse~\citep{mothilal2020explaining}, plausible~\citep{joshi2019realistic,poyiadzi2020face,schut2021generating}, robust~\citep{upadhyay2021robust,pawelczyk2022probabilistically,altmeyer2023endogenous} and causal~\citep{karimi2021algorithmic} among other things. Researchers have come up with various ways to meet these desiderata, which have been surveyed in~\citep{verma2020counterfactual} and~\citep{karimi2020survey}. 
+
+To fulfil this latter goal, researchers have come up with a myriad of ways. @joshi2019realistic were among the first to suggest that instead of searching counterfactuals in the feature space, we can instead traverse a latent embedding learned by a surrogate generative model. Similarly, @poyiadzi2020face use density ... Finally, @karimi2021algorithmic argues that counterfactuals should comply with the causal model that generates them [CHECK IF WE CAN PHASE THIS LIKE THIS]. Other related approaches include ... All of these different approaches have a common goal: they aim to ensure that the generated counterfactuals comply with the (learned) data-generating process (DGB). 
 
 \begin{figure}
   \centering
   \begin{minipage}[t]{0.45\textwidth}
     \centering
     \includegraphics[width=\textwidth]{../www/you_may_not_like_it.png}
-    \caption{You may not like it, but this is what stripped-down counterfactuals look like. Counterfactuals for turning an 8 (eight) into a 3 (three): original image (left); counterfactual produced using \citet{wachter2017counterfactual} (center); and a counterfactual produced using JSMA-based approach introduced by \citep{schut2021generating}.}\label{fig:adv}
+    \caption{You may not like it, but this is what stripped-down counterfactuals look like. Counterfactuals for turning an 8 (eight) into a 3 (three): original image (left); counterfactual produced using \citet{wachter2017counterfactual} (centre); and a counterfactual produced using JSMA-based approach introduced by \citep{schut2021generating}.}\label{fig:adv}
   \end{minipage}\hfill
   \begin{minipage}[t]{0.45\textwidth}
     \centering
     \includegraphics[width=\textwidth]{../www/surrogate_gone_wrong.png}
-    \caption{Using surrogates can improve plausibility, but also increases vulnerability. Counterfactuals for turning an 8 (eight) into a 3 (three): original image (left); counterfactual produced using REVISE \citep{joshi2019realistic} with a well-specified surrogate (center); and a counterfactual produced using REVISE \citep{joshi2019realistic} with a poorly specified surrogate (right).}\label{fig:vae}
+    \caption{Using surrogates can improve plausibility, but also increases vulnerability. Counterfactuals for turning an 8 (eight) into a 3 (three): original image (left); counterfactual produced using REVISE \citep{joshi2019realistic} with a well-specified surrogate (centre); and a counterfactual produced using REVISE \citep{joshi2019realistic} with a poorly specified surrogate (right).}\label{fig:vae}
   \end{minipage}
 \end{figure}