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\title{Conformal Counterfactual Explanations}


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\author{%
  Patrick Altmeyer\thanks{Use footnote for providing further information
    about author (webpage, alternative address)---\emph{not} for acknowledging
    funding agencies.} \\
  Faculty of Electrical Engineering, Mathematics and Computer Science\\
  Delft University of Technology\\
  2628 XE Delft, The Netherlands \\
  \texttt{p.altmeyer@tudelft.nl} \\
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\begin{document}

\maketitle

\begin{abstract}
  We propose Conformal Counterfactual Explanations: an effortless and rigorous way to produce plausible and conformal Counterfactual Explanations for Black Box Models using Conformal Prediction. To address the need for plausible explanations, existing work has primarily relied on surrogate models to learn the data-generating process. This effectively reallocates the task of learning realistic representations of the data from the model itself to the surrogate. Consequently, the generated explanations may look plausible to humans but not necessarily conform with the behaviour of the Black Box Model. We formalise this notion through the introduction of new evaluation measures. In order to still address the need for plausibility, we build on a recent approach that works by minimizing predictive model uncertainty. Using differentiable Conformal Prediction, we relax the previous assumption that the Black Box Model can produce predictive uncertainty estimates.
\end{abstract}

\section{Introduction}\label{intro}

Counterfactual Explanations are a powerful, flexible and intuitive way to not only explain Black Box Models but also enable affected individuals to challenge them through the means of Algorithmic Recourse. Instead of opening the black box, Counterfactual Explanations work under the premise of strategically perturbing model inputs to understand model behaviour \citep{wachter2017counterfactual}. Intuitively speaking, we generate explanations in this context by asking simple what-if questions of the following nature: `Our credit risk model currently predicts that this individual's credit profile is too risky to offer them a loan. What if they reduced their monthly expenditures by 10\%? Will our model then predict that the individual is credit-worthy'? 

This is typically implemented by defining a target outcome $t \in \mathcal{Y}$ for some individual $x \in \mathcal{X}$, for which the model $M:\mathcal{X}\mapsto\mathcal{Y}$ initially predicts a different outcome: $M(x)\ne t$. Counterfactuals are then searched by minimizing a loss function that compares the predicted model output to the target outcome: $\text{yloss}(M(x),t)$. Since Counterfactual Explanations (CE) work directly with the Black Box Model, they always have full local fidelity by construction. Fidelity is defined as the degree to which explanations approximate the predictions of the Black Box Model. This arguably one of the most important evaluation metrics for model explanations, since any explanation that explains a prediction not actually made by the model is useless \citep{molnar2020interpretable}. 

In situations where full fidelity is a requirement, CE therefore offers a more appropriate solution to Explainable Artificial Intelligence (XAI) than other popular approaches like LIME \citep{ribeiro2016why} and SHAP \citep{lundberg2017unified}, which involve local surrogate models. But even full fidelity is not a sufficient condition for ensuring that an explanation adequately describes the behaviour of a model. That is because two very distinct explanations can both lead to the same model prediction, especially when dealing with heavily parameterized models:

\begin{quotation}
  […] deep neural networks are typically very underspecified by the available data, and […] parameters [therefore] correspond to a diverse variety of compelling explanations for the data.
  --- \citet{wilson2020case}
\end{quotation}

When people talk about Black Box Models, this is usually the type of model they have in mind. 

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{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,

\begin{equation} \label{eq:general}
\begin{aligned}
\mathbf{s}^\prime &amp;= \arg \min_{\mathbf{s}^\prime \in \mathcal{S}} \left\{  {\text{yloss}(M(f(\mathbf{s}^\prime)),y^*)}+ \lambda {\text{cost}(f(\mathbf{s}^\prime)) }  \right\} 
\end{aligned} 
\end{equation}

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, 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}). 

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}. 

Finding ways to generate \textit{plausible} counterfactuals has been one of the primary concerns. To this end, \citet{joshi2019realistic} were among the first to suggest that instead of searching counterfactuals in the feature space $\mathcal{X}$, we can instead traverse a latent embedding $\mathcal{Z}$ that implicitly codifies the data generating process (DGP) of $x\sim\mathcal{X}$. To learn the latent embedding, they introduce a surrogate model. In particular, they propose to use the latent embedding of a Variational Autoencoder (VAE) trained to generate samples $x^*\sim \mathcal{G}(z)$ where $\mathcal{G}$ denotes the decoder part of the VAE. Provided the surrogate model is well-trained, their proposed approach can yield compelling counterfactual explanations like the one in the centre panel of Figure~ref{fig:vae}. 

Others have proposed similar approaches. \citet{dombrowski2021diffeomorphic} traverse the base space of a normalizing flow to solve Equation~\ref{eq:general}, essentially relying on a different surrogate model for the generative task. \citet{poyiadzi2020face} use density estimators ($\hat{p}: \mathcal{X} \mapsto [0,1]$) to constrain the counterfactual paths. \citet{karimi2021algorithmic} argue that counterfactuals should comply with the causal model that generates the data. All of these different approaches share a common goal: ensuring that the generated counterfactuals comply with the true and unobserved DGP. To summarize this broad objective, we propose the following definition:

\begin{definition}[Plausible Counterfactuals]
  \label{def:plausible}
  Let $\mathcal{X}|t$ denote the true conditional distribution of samples in the target class. Then for $x^{\prime}$ to be considered a plausible counterfactual, we need: $x^{\prime} \sim \mathcal{X}|t$.
\end{definition}

Note that Definition~\ref{def:plausible} subsumes the notion of plausible counterfactual paths, since we can simply apply it to each counterfactual state along the path.

\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} (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 (centre); and a counterfactual produced using REVISE \citep{joshi2019realistic} with a poorly specified surrogate (right).}\label{fig:vae}
  \end{minipage}
\end{figure}

\medskip

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\appendix


\section{Appendix}


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\end{document}</pre></body></html>