From c8896508b6bc381cc5936c879a043675001ee650 Mon Sep 17 00:00:00 2001
From: pat-alt <altmeyerpat@gmail.com>
Date: Mon, 27 Mar 2023 16:01:48 +0100
Subject: [PATCH] slowly slowly

---
 paper/paper.pdf | Bin 300896 -> 303597 bytes
 paper/paper.tex |  13 ++++++++++++-
 2 files changed, 12 insertions(+), 1 deletion(-)

diff --git a/paper/paper.pdf b/paper/paper.pdf
index c7415e8824b3eed5bcaed01f656d614c1476b78b..7d0ed8f859f68dec063dbf93b95a347f3a51d7c4 100644
GIT binary patch
delta 83471
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diff --git a/paper/paper.tex b/paper/paper.tex
index 507a2136..771e4b67 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -249,7 +249,18 @@ Observe from Equation~\ref{eq:scp} that Conformal Prediction works on an instanc
 
 \subsection{Conformal Counterfactual Explanations}
 
-The fact that conformal classifiers produce set-valued predictions introduces a challenge: it is not immediately obvious how to use such classifiers in the context of gradient-based counterfactual search. Put differently, it is not clear how to use prediction sets in Equation~\ref{eq:general}. Fortunately, \citet{stutz2022learning} have recently proposed a framework for Conformal Training that also hinges on differentiability. To evaluate the performance of conformal classifiers during training, they introduce a custom loss function as well as a smooth set size penalty. Their key idea lies in forming soft assignment scores for each class to be included in the prediction set: $C_{\theta,y}(X_i;\alpha):=\sigma\left((s(X_i,y)-\alpha) T^{-1}\right)$ for $y\in\{1,...,K\}$ where $\sigma$ is the sigmoid function and $T$ is a temperature hyper-parameter. 
+The fact that conformal classifiers produce set-valued predictions introduces a challenge: it is not immediately obvious how to use such classifiers in the context of gradient-based counterfactual search. Put differently, it is not clear how to use prediction sets in Equation~\ref{eq:general}. Fortunately, \citet{stutz2022learning} have recently proposed a framework for Conformal Training that also hinges on differentiability. Specifically, they show how Stochastic Gradient Descent can be used to train classifiers not only for the discriminative task but also for additional objectives related to Conformal Prediction. One such objective is \textit{efficiency}: for a given target error rate $alpha$, the efficiency of a conformal classifier improves as its average prediction set size decreases. To this end, the authors introduce a smooth set size penalty,
+
+\begin{equation}\label{eq:setsize}
+  \begin{aligned}
+    \Omega(C_{\theta}(x;\alpha))&=\max \left(0, \sum_{y\in\mathcal{Y}}C_{\theta,y}(X_i;\alpha) - \kappa \right)
+  \end{aligned}
+\end{equation}
+
+where $\kappa \in \{0,1\}$ is a hyper-parameter and $C_{\theta,y}(X_i;\alpha)$ can be interpreted as the probability of label $y$ being included in the prediction set. Formally, it is defined as $C_{\theta,y}(X_i;\alpha):=\sigma\left((s(X_i,y)-\alpha) T^{-1}\right)$ for $y\in\{1,...,K\}$ where $\sigma$ is the sigmoid function and $T$ is a hyper-parameter used for temperature scaling \citep{stutz2022learning}.
+
+We propose to include this penalty in the counterfactual search objective (Equation~\ref{eq:general}) 
+
 
 
 \section{Experiments}
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