
\section{Appendix}
\label{sec:appendix}

\subsection{Proofs}
\label{sec:appendix_proofs}

Restatement of Proposition \ref{lemma:asymptotic_solution}
\begin{proposition}
    % \label{lemma:asymptotic_solution}
    Given the objective in Eq.~\ref{eq:gen_objective}, if $\mathcal{D}^1 = P(X,S)$ (infinite sample regime) and $\theta_0$ in Algorithm \ref{alg:meta_algo} is the weakest admissible regularization, then $\tau^*=\tau_{\theta_0}$, which also minimizes pinball loss over all admissible regularizations
     $\mathbb{E}_{\mathcal{D}}\big[\ell_{1-\alpha}(q_{\tau^*}(X),S)) \big] \le \mathbb{E}_{\mathcal{D}}\big[\ell_{1-\alpha}(q_{\tau_{\theta}}(X),S)) \big], \forall \theta \in \Theta$ such that $ \theta \ge \theta_0$.
\end{proposition}

\begin{proof}
    We first show that in the infinite sample regime the $\textsc{mcr}$ is zero $\forall \theta \in \Theta$, making all $\theta$ equivalent according to the $\textsc{mcr}$ criteria. Then we show that Algorithm \ref{alg:meta_algo} would choose $\theta^* = \theta_0$ and since $\theta_0$ is the lowest regularization it achieves the smallest expected pinball loss.

    Given access to the real distribution  $\mathcal{D}_1 = P(X,S)$ for any  $\theta \in \Theta$ we get a finite set partition $\mathcal{G}_{\tau_{\theta}}$ such that the $1-\alpha$ quantile estimate $q_{\tau_{\theta}}(X)$ is the exact group conditional quantile of the non-conformity score distribution for the group that contains the instance $X$.
    \begin{equation}
        q_{\tau_{\theta}}(X) = F_{S|G=g_{\tau_{\theta}}(X)}^{-1}(1-\alpha)
    \end{equation}
    where $g_{\tau_{\theta}}(X) \in \mathcal{G}_{\tau_{\theta}}, \forall X \in \mathcal{X}$. Then, in this asymptotic regime the group conditional miscoverage (Definition \ref{def:conditional_miscoverage}) $MC_{\alpha}(q_{\tau_{\theta}},g_{\tau_{\theta}};g_j) = 0$  $\forall g \in \mathcal{G}_{\tau_{\theta}}$ , $\forall g \in \mathcal{G}_{\tau_{\theta}}$ and $\forall \theta \in \Theta$. Then $\textsc{mcr}_{\alpha}$ ($\tau_{\theta}$) as defined in Eq. \ref{eq:miscoverage_ratio} is 0 $\forall \theta \in \Theta$. 

    Since Algorithm \ref{alg:meta_algo} terminates on the first $\theta$ that achieves the minimum $\textsc{mcr}$ then $\theta^* = \theta_0$. Since $\theta_0$ is the weakest regularization, and we assume infinite sample regime to learn $\tau_{\theta} \forall \theta \in \Theta$  then $\mathbb{E}_{\mathcal{D}}\big[\ell_{1-\alpha}(q_{\tau^*}(X),S)) \big] \le \mathbb{E}_{\mathcal{D}}\big[\ell_{1-\alpha}(q_{\tau_{\theta}}(X),S)) \big], \forall \theta \in \Theta$ such that $ \theta \ge \theta_0$.
    
\end{proof}
    % timeout=11200,
    % n_trials=100,

% "lambda_l1": trial.suggest_loguniform("lambda_l1", 1e-8, 10.0),
%             "lambda_l2": trial.suggest_loguniform("lambda_l2", 1e-8, 10.0),
%             "num_leaves": trial.suggest_int("num_leaves", 2, 100),
%             "feature_fraction": trial.suggest_float("feature_fraction", 0.4, 1.0),
%             "bagging_fraction": trial.suggest_float("bagging_fraction", 0.4, 1.0),
%             "bagging_freq": trial.suggest_int("bagging_freq", 1, 7),
%             "boosting_type": trial.suggest_categorical(
%                 "boosting_type", ["gbdt", "dart"]
%             ),
%             "max_depth": trial.suggest_int(
%                 "max_depth", int(np.minimum(1, max_depth)), max_depth
%             ),
%             "learning_rate": trial.suggest_loguniform("learning_rate", 1e-8, 1e-1),
%             "num_boost_round": trial.suggest_int("num_boost_round", 1, 100),
%             "min_child_samples": trial.suggest_int(
%                 "min_child_samples",
%                 min_child_samples,
%                 int(np.maximum(200, min_child_samples)),
%             ),
%             "random_state": trial.suggest_int("random_state", 40, 50)

{
\subsection{Experimental Details}
\subsubsection{Learning Decision Tree Based Regions}

We learn a decision tree that approximates the non-conformity score quantile by optimizing Eq. \ref{eq:quantile_regression_tree_objective}. To do so, we first learn a surrogate model $h$ that minimizes the pinball loss $1-\alpha$ of the non-conformity scores. 

% max_depth=2, min_child_samples=50

\paragraph{Step 1: Learn Surrogate Model $h$}
In our experiments $h$ is an LGBM quantile regressor that we learn using Optuna \cite{akiba2019optuna} with the following hyperparameters over 5-fold validation where the final set of parameters for $h^*$ is chosen based on best average pinball loss plus one standard deviation.
\begin{itemize}
    \item Optimizer Configuration: \textsc{n\_trials} = 200, \textsc{timeout} = 11200. 
    \item LGBM Model Parameters Exploration: $\textsc{lambda\_1} \sim \text{loguniform}(1e-8, 10.0)$,$\textsc{lambda\_2} \sim \text{loguniform}(1e-8, 10.0)$, $\textsc{learning\_rate} \sim \text{loguniform}(1e-8, 10.0)$, $\text{bagging\_fraction} \in [0.4,1.0]$, 
    $\text{bagging\_freq} \in [1,7]$,
    $\text{num\_leaves} \in [2,100]$,  $\text{num\_boost\_round} \in [1, 100]$, $\text{min\_child\_samples} \in [50,200]$, $\text{max\_depth} = 2$ , 
\end{itemize}


\paragraph{Step 2: Learn The Decision Tree Model $\tau$}
To learn $\tau$ we optimize the mean square error distance w.r.t. the prediction of the quantile LGBM regressor $h^*$ learned in the previous step. As stated in Section \ref{sec:experiments} in Algorithm \ref{alg:meta_algo} we consider trees up to a maximum depth of 5 and at least 50 samples per leaf. The regularization parameter $\theta$ is the cost complexity pruning variable. We set $\theta_0 = 1e-5$ and a step size $\Delta_{\theta_t} = 9 \times \theta_t$.
}
\subsection{Additional Experiments}
\label{sec:appendix_results}

Figure \ref{fig:concrete_graph} shows the decision trees that were obtain for the different datasets. We observe that the discovered regions have different prediction interval widths indicating that the model's prediction uncertainty is significantly different. Figure \ref{fig:joint_plots_extended} shows the scatter and joint distribution between the prediction interval widths and coverage of the discovered groups. It extends Figure \ref{fig:joint_plots} in the main manuscript including all datasets and the groups discovered by the \textsc{RF-G} approach proposed by \cite{amoukou2023adaptive}.  Table \ref{tab:summary_lasso_methods} shows the same comparison presented in Table \ref{tab:summary_methods} but for a $\textsc{lasso}$ base model. {We observe that the number of discovered groups by the proposed method $\textsc{mcr\_dtree}$ is larger than those of a $\textsc{lgbm}$ regression model for the same dataset (Table \ref{tab:summary_methods}). In most cases, the $\textsc{lgbm}$ model is equal or better than $\textsc{lasso}$ in terms of r2 score, and therefore reduces the unexplained variance of the target $Y|X$. This leads to less regions of different uncertainty and tighter prediction sets.  }


\begin{figure}[h!]
\centering

\subfloat[Housing]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/housing_graph.png}
\label{fig:housing_graph}} 
\subfloat[Concrete]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/concrete_graph.png}
\label{fig:concrete_graph}}


\subfloat[Energy]{
\includegraphics[width=0.45\columnwidth]{UAI2024/figures/energy_graph.png}
\label{fig:energy_graph}} 
\subfloat[Power]{
\includegraphics[width=0.45\columnwidth]{UAI2024/figures/power_graph.png}
\label{fig:power_graph}} 

\subfloat[Kin8nm]{
\includegraphics[width=0.45\columnwidth]{UAI2024/figures/kin8mn_graph.png}
\label{fig:kin8mn_graph}} 
\subfloat[Protein]{
\includegraphics[width=0.45\columnwidth]{UAI2024/figures/protein_graph.png}
\label{fig:protein_graph}} 

\begin{center}
\caption{Example of decision trees identified for each regression dataset. (\ref{fig:housing_graph}) In the Housing dataset groups are defined based on the features corresponding to average number of rooms per dwelling (RM) and weighted distances to five Boston employment centers (DIS). (\ref{fig:concrete_graph}) In the Concrete dataset the groups are defined based on the Cement and Fine Aggregate components ($kg$ in a $m^3$ mixture). (\ref{fig:energy_graph}) the groups in the Energy dataset are defined based on Glazing Area Distribution (X8), Glazing Area (X7) and Wall Area (X3). (\ref{fig:power_graph}) In the Power dataset groups are defined based on Ambient Temperature (AT), Exhaust Vacuum (V) and Relative Humidity (RH).  (\ref{fig:kin8mn_graph}) In the kin8nm dataset the groups are defined by the measurements on sensors from links 3, 5 and 6 from the robot arm. (\ref{fig:protein_graph}) In the protein dataset the groups are defined by the features corresponding to fractional area of exposed non polar residue (F3) and fractional area of exposed non polar part of residue (F4). }
\label{fig:tree_graphs}
\end{center}
\end{figure}


\begin{figure}[h!]
\centering

\subfloat[Housing]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/housing_jp_rfg.png}
\label{fig:housing_jp}} 
\subfloat[Concrete]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/concrete_jp_rfg.png}
\label{fig:concrete_jp}}
\subfloat[Energy]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/energy_jp_rfg.png}
\label{fig:energy_jp}} 

\subfloat[Power]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/power_jp_rfg.png}
\label{fig:power_JP}} 
\subfloat[Kin8mn]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/kin8mn_jp_rfg.png}
\label{fig:kin8mn_jp}} 
\subfloat[Protein]{
\includegraphics[width=0.3\columnwidth]{UAI2024/figures/protein_jp_rfg.png}
\label{fig:protein_jp}} 

\begin{center}
\caption{Scatter and distribution plot of the prediction interval widths (x-axis) versus coverage (y-axis) of the groups discovered by the proposed $\textsc{mcr\_dtree}$, $\textsc{pb\_dtree}$ and $\textsc{rf-g}$ methods across 6 datasets. Here we plot all the groups obtained across 5-Fold realizations. The size of the groups points represents the group size (number of samples). The target coverage is 0.9, we observe that $\textsc{mcr\_dtree}$ tends to identify a smaller number of groups of varying sizes, with group-conditional coverages concentrated around the 0.9 objective. Moreover, the identified groups show diversity in the range of interval widths. $\textsc{pb\_dtree}$ detects a significant larger number of (smaller) groups,  with a larger variance in terms of group-conditional coverage.}
\label{fig:joint_plots_extended}
\end{center}
% \vspace{-0.2in}
\end{figure}




\begin{table}[h!]
\centering
\small
\scalebox{0.9}{
\begin{tabular}{l|r|rrr|c}
% \footnotesize
\toprule
 &\multicolumn{1}{c}{$\textsc{mcr}$ } &  \multicolumn{3}{c}{coverage} & \multicolumn{1}{c}{num } \\
model &  & average &   max group &   min group  &     groups  \\
                                              &          &       &       &         \\
\midrule
\multicolumn{6}{l}{Housing: nsamples = 506, nfeatures = 13 | $\textsc{lasso}$-Regressor R2 = 0.69 $\pm$  0.04}\\
\midrule

$\textsc{lcp-rf-g}$ &  2.71$\pm$0.77 &   0.8$\pm$0.06 &  0.91$\pm$0.08 &  0.75$\pm$0.07 &   2.6$\pm$0.55 \\
$\textsc{rf-g}$ &  0.42$\pm$0.38 &  \textbf{0.91$\pm$0.03} &  0.96$\pm$0.03 &  0.81$\pm$0.15 &   3.2$\pm$0.45 \\
$\textsc{pb-kmeans}$ &   1.47$\pm$0.49 &  0.86$\pm$0.03 &  0.98$\pm$0.03 &  0.44$\pm$0.43 &  14.2$\pm$15.02 \\
$\textsc{mcr-kmeans}$ & 1.35$\pm$0.74 &  0.88$\pm$0.04 &  0.97$\pm$0.03 &  0.69$\pm$0.38 &   7.4$\pm$11.52 \\
$\textsc{pb\_dtree}$  &  0.32$\pm$0.21 &  0.88$\pm$0.03 &  0.98$\pm$0.05 &  0.83$\pm$0.05 &   4.0$\pm$1.87 \\
% 5 &  0.33$\pm$0.27 &  0.89$\pm$0.04 &  0.98$\pm$0.03 &  0.84$\pm$0.04 &   3.2$\pm$1.64 \\
$\textsc{mcr\_dtree}$  &  \textbf{0.25$\pm$0.39} &  0.89$\pm$0.04 &  \textbf{0.95$\pm$0.04} &  \textbf{0.84$\pm$0.07} &   3.6$\pm$2.07 \\

\midrule
\multicolumn{6}{l}{Concrete: nsamples = 1030, nfeatures = 8 | $\textsc{lasso}$-Regressor R2 = 0.60 $\pm$  0.05}\\
\midrule


$\textsc{lcp-rf-g}$ &                       1.37$\pm$1.12 &  0.83$\pm$0.02 &  \textbf{0.96$\pm$0.04} &   0.7$\pm$0.05 &   5.4$\pm$0.55 \\
$\textsc{rf-g}$ &  0.29 $\pm$0.15 &  0.91$\pm$0.02 &  0.98$\pm$0.03 &   0.8$\pm$0.08 &   5.0$\pm$0.71 \\
$\textsc{pb-kmeans}$  &     0.89$\pm$0.48 &   \textbf{0.9$\pm$0.05} &    1.0$\pm$0.0 &  0.26$\pm$0.37 &  37.2$\pm$16.93 \\
$\textsc{mcr-kmeans}$  &        0.43$\pm$0.43 &  0.92$\pm$0.02 &  0.97$\pm$0.03 &    0.7$\pm$0.3 &  15.8$\pm$18.98 \\
$\textsc{pb\_dtree}$ &                       0.25$\pm$0.14 &   \textbf{0.9$\pm$0.03} &    1.0$\pm$0.0 &   0.8$\pm$0.07 &   7.0$\pm$2.24 \\
% 5 &                        0.18$\pm$0.1 &   0.9$\pm$0.02 &    1.0$\pm$0.0 &  0.83$\pm$0.02 &   5.8$\pm$3.03 \\
$\textsc{mcr\_dtree}$ &                       \textbf{0.15$\pm$0.09} &   \textbf{0.9$\pm$0.03} &    1.0$\pm$0.0 &  \textbf{0.84$\pm$0.04} &   6.8$\pm$2.39 \\

\midrule
\multicolumn{6}{l}{Energy: nsamples = 768, nfeatures = 8 | $\textsc{lasso}$-Regressor R2 = 0.91 $\pm$  0.005}\\
\midrule

$\textsc{lcp-rf-g}$ &  0.38$\pm$0.19 &  0.88$\pm$0.05 &  \textbf{0.98$\pm$0.03} &   0.8$\pm$0.08 &   4.8$\pm$0.45 \\
$\textsc{rf-g}$ &  0.12$\pm$0.12 &  \textbf{0.94$\pm$0.02} &    1.0$\pm$0.0 &  0.87$\pm$0.06 &   5.0$\pm$0.71 \\
$\textsc{pb-kmeans}$ & 1.07$\pm$0.77 &  0.87$\pm$0.04 &  0.99$\pm$0.02 &   0.18$\pm$0.4 &  38.2$\pm$19.15 \\
$\textsc{mcr-kmeans}$ &  0.32$\pm$0.41 &  \textbf{0.94$\pm$0.03} &  \textbf{0.98$\pm$0.04} &  0.83$\pm$0.13 &  13.0$\pm$11.92 \\
$\textsc{pb\_dtree}$ &  0.12$\pm$0.16 &  \textbf{0.94$\pm$0.02} &  0.99$\pm$0.03 &  0.84$\pm$0.11 &   9.0$\pm$3.46 \\
$\textsc{mcr\_dtree}$ &  \textbf{0.05$\pm$0.09} &  \textbf{0.94$\pm$0.02} &  \textbf{0.98$\pm$0.02} &  \textbf{0.89$\pm$0.03} &   6.0$\pm$3.24 \\
% 6 &  0.06$\pm$0.08 &  0.94$\pm$0.02 &  0.97$\pm$0.03 &  0.88$\pm$0.02 &   6.2$\pm$3.56 \\
\midrule
\multicolumn{6}{l}{Power: nsamples = 9568, nfeatures = 4 | $\textsc{lasso}$-Regressor R2 = 0.93 $\pm$  0.003}\\
\midrule
$\textsc{lcp-rf-g}$ &  2.04$\pm$1.26 &  0.82$\pm$0.05 &  0.86$\pm$0.08 &  0.78$\pm$0.05 &   6.0$\pm$2.24 \\
$\textsc{rf-g}$ &  0.83$\pm$0.57 &    \textbf{0.9$\pm$0.0} &  \textbf{0.93$\pm$0.02} &  0.87$\pm$0.01 &   5.2$\pm$0.84 \\
$\textsc{pb-kmeans}$ &  0.73$\pm$0.27 &  0.91$\pm$0.01 &  0.99$\pm$0.02 &  0.78$\pm$0.05 &  37.2$\pm$5.22 \\
$\textsc{mcr-kmeans}$ &   0.46$\pm$0.15 &    \textbf{0.9$\pm$0.0} &  \textbf{0.93$\pm$0.03} &  \textbf{0.88$\pm$0.03} &   6.0$\pm$7.28 \\
$\textsc{pb\_dtree}$ &  0.08$\pm$0.05 &   \textbf{0.9$\pm$0.01} &  0.94$\pm$0.03 &  0.87$\pm$0.02 &   6.4$\pm$4.16 \\
$\textsc{mcr\_dtree}$ &  \textbf{0.06$\pm$0.05} &    \textbf{0.9$\pm$0.0} &  0.94$\pm$0.01 &  \textbf{0.88$\pm$0.02} &   7.4$\pm$3.71 \\
% 6 &  0.08$\pm$0.09 &   0.9$\pm$0.01 &  0.99$\pm$0.03 &  0.87$\pm$0.03 &   6.0$\pm$3.67 \\

\midrule
\multicolumn{6}{l}{Protein: : nsamples = 45730, nfeatures = 9 | $\textsc{lasso}$-Regressor R2 = 0.28 $\pm$  0.01}\\
\midrule
$\textsc{lcp-rf-g}$ & 0.89$\pm$0.56 &     0.87$\pm$0.03 & \textbf{0.92$\pm$0.02} & 0.75$\pm$0.04 &    5.8$\pm$1.6 \\
$\textsc{rf-g}$ & 0.44$\pm$0.37 &     \textbf{0.9$\pm$0.0} & 0.95$\pm$0.05 & 0.87$\pm$0.02 &    6.00$\pm$1.59 \\
$\textsc{pb-kmeans}$ &  0.71$\pm$0.75 &  \textbf{0.9$\pm$0.0} &    1.0$\pm$0.0 &   0.65$\pm$0.21 &   42.6$\pm$7.86 \\
$\textsc{mcr-kmeans}$ &  0.52$\pm$0.21 &  \textbf{0.9$\pm$0.0} &  0.96$\pm$0.05 &  0.76$\pm$0.24 &  16.2$\pm$12.91 \\
$\textsc{pb\_dtree}$ &   0.44$\pm$0.37 &  \textbf{0.9$\pm$0.0} &    1.0$\pm$0.0 &  0.83$\pm$0.02 &   15.6$\pm$0.89 \\
% 3 &  0.44$\pm$0.37 &  0.9$\pm$0.0 &  0.92$\pm$0.03 &  0.88$\pm$0.04 &    4.2$\pm$2.49 \\
$\textsc{mcr\_dtree}$ &   \textbf{0.2$\pm$0.08} &  \textbf{0.9$\pm$0.0} &  0.93$\pm$0.03 &  \textbf{0.89$\pm$0.01} &    5.6$\pm$2.19 \\




\midrule
\multicolumn{6}{l}{kin8mn: : nsamples = 8192, nfeatures = 8 | $\textsc{lasso}$-Regressor R2 = 0.40 $\pm$  0.007}\\
\midrule

$\textsc{lcp-rf-g}$ &  1.68$\pm$0.29 &  0.79$\pm$0.01 &  0.81$\pm$0.01 &  0.77$\pm$0.01 &     3.0$\pm$0.0 \\
$\textsc{rf-g}$ &  \textbf{0.21$\pm$0.04} &   \textbf{0.9$\pm$0.01} &  \textbf{0.91$\pm$0.01} &   \textbf{0.88$\pm$0.0} &    3.2$\pm$0.45 \\
$\textsc{pb-kmeans}$ &  0.67$\pm$0.16 &  0.92$\pm$0.01 &  0.99$\pm$0.01 &  0.76$\pm$0.04 &  39.4$\pm$14.06 \\
$\textsc{mcr-kmeans}$ &  0.44$\pm$0.37 &   \textbf{0.9$\pm$0.01} &  0.93$\pm$0.04 &  0.87$\pm$0.05 &  11.6$\pm$21.47  \\
$\textsc{pb\_dtree}$ &  0.41$\pm$0.36 &  0.89$\pm$0.01 &  0.98$\pm$0.04 &  0.82$\pm$0.07 &   14.2$\pm$3.03 \\
$\textsc{mcr\_dtree}$ &  0.24$\pm$0.18 &   \textbf{0.9$\pm$0.01} &  0.94$\pm$0.04 &  \textbf{0.88$\pm$0.02} &    6.4$\pm$5.37 \\
% 6 &  0.35$\pm$0.22 &   0.9$\pm$0.01 &  0.97$\pm$0.03 &  0.86$\pm$0.01 &    9.2$\pm$2.68 \\


\bottomrule
\end{tabular}
}
\caption{Comparison between the group discovery partition methods. We show MCR, marginal, minimum, and maximum coverage group coverage on the identified partition. We also report the number of groups per approach. Standard deviations are computed across 5 data splits. The proposed $\textsc{mcr\_dtree}$ is consistently better in terms of $\textsc{mcr}$,  with values consistently below $1$, indicating that the discovered groups improve worst-group under-coverage w.r.t. to single threshold SCP. Every dataset uses a $\textsc{lasso}$ regressor as the base model. We highlight the lowest $\textsc{mcr}$ and the smallest average coverage above the objective (0.9). For methods that achieved the marginal coverage objective we highlight the max and min group coverage closest to the 0.9 objective.}
\label{tab:summary_lasso_methods}
% \vspace{-0.1in}
\end{table}




% \begin{table}[h!]
% \centering
% \footnotesize
% \scalebox{0.7}{
% \begin{tabular}{l|r|rrr|c}
% % \footnotesize
% \toprule
%  &\multicolumn{1}{c}{MCR } &  \multicolumn{3}{c}{coverage} & \multicolumn{1}{c}{num } \\
% model &  & average &   max group &   min group  &     groups  \\
%                                               &          &       &       &         \\
% \midrule
% \multicolumn{6}{l}{Housing: nsamples = 506, nfeatures = 13 | $\textsc{lgbm}$-Regressor R2 = 0.64 $\pm$  0.03}\\
% \midrule

% $\textsc{lcp-rf-g}$ &  1.45$\pm$ 1.14 &   .8$\pm$ .04 &  .91$\pm$ .07 &  .64$\pm$ .15 &  3.6$\pm$ .55 \\
% $\textsc{rf-g}$ &   .77$\pm$ .6 &  .93$\pm$ .03 &  .99$\pm$ .01 &  .86$\pm$ .06 &  3.6$\pm$ .55 \\
% $\textsc{pb-kmeans}$  &   .81$\pm$ .3 &  .92$\pm$ .02 &  .97$\pm$ .04 &  .68$\pm$ .33 &  8.4$\pm$ 8.65 \\
% $\textsc{mcr-kmeans}$ &     .75$\pm$ .12  &  \textbf{.91$\pm$ .05} &  \textbf{.95$\pm$ .05} &  .84$\pm$ .13 &    2.2$\pm$ 1.64\\
% $\textsc{pb\_dtree}$  &  .68$\pm$ .31 &  .89$\pm$ .02 &  .94$\pm$ .03 &  .83$\pm$ .04 &  3.4$\pm$ .55 \\
% $\textsc{mcr\_dtree}$  & \textbf{.65$\pm$ .17} &  .92$\pm$ .03 &  \textbf{.95$\pm$ .04} &  \textbf{.88$\pm$ .07} &   2.2$\pm$ 1.3 \\

% \midrule
% \multicolumn{6}{l}{Concrete: nsamples = 1030, nfeatures = 8 | $\textsc{lgbm}$-Regressor R2 = 0.82 $\pm$  0.026}\\
% \midrule


% $\textsc{lcp-rf-g}$ &  1.84$\pm$ 1.66 &  .83$\pm$ .01 &  .94$\pm$ .05 &  .69$\pm$ .11 &  4.6$\pm$ .55 \\
% $\textsc{rf-g}$ &  .82$\pm$ .68 &   \textbf{.9$\pm$ .05} &  .97$\pm$ .02 &  .81$\pm$ .11 &  4.6$\pm$ .55 \\
% $\textsc{pb-kmeans}$  & .66$\pm$ .48 &  .91$\pm$ .05 &  .97$\pm$ .05 &  .83$\pm$ .07 &  7.0$\pm$ 3.24 \\
% $\textsc{mcr-kmeans}$ &  .88$\pm$ .27 &  .91$\pm$ .05 &  \textbf{.92$\pm$ .06} &  \textbf{.88$\pm$ .05} &  4.2$\pm$ 7.16 \\
% $\textsc{pb\_dtree}$  &  .94$\pm$ .57 &  .89$\pm$ .04 &  .98$\pm$ .02 &  .77$\pm$ .07 &  6.6$\pm$ .55 \\
% $\textsc{mcr\_dtree}$  &   \textbf{.55$\pm$ .72} &   \textbf{.9$\pm$ .04} &  \textbf{.92$\pm$ .06} &  \textbf{.88$\pm$ .04} &  2.4$\pm$ 2.61 \\

% \midrule
% \multicolumn{6}{l}{Energy: nsamples = 768, nfeatures = 8 | $\textsc{lgbm}$-Regressor R2 = 0.93 $\pm$  0.05}\\
% \midrule

% $\textsc{lcp-rf-g}$ &.99$\pm$ 1.31 &  .87$\pm$ .06 &  .97$\pm$ .03 &  .65$\pm$ 0.05 &    5.0$\pm$ 1.0 \\
% $\textsc{rf-g}$ &.65$\pm$ .1 &  \textbf{.92$\pm$ .03} &  .99$\pm$ .02 &  0.87$\pm$ .06 &   4.8$\pm$ 1.64  \\
% $\textsc{pb-kmeans}$  &  1.04$\pm$ .34 &  .85$\pm$ .07 &    1.0$\pm$ .0 &  .07$\pm$ .15 &  47.8$\pm$ 1.79 \\
% $\textsc{mcr-kmeans}$ &   .68$\pm$ .3 &  .94$\pm$ .03 &  \textbf{.96$\pm$ .05} &  .78$\pm$ .17 &   1.6$\pm$ 9.5 \\
% $\textsc{pb\_dtree}$  &   .63$\pm$ .5 &  .93$\pm$ .03 &  .97$\pm$ .02 &  .87$\pm$ .07 &   3.6$\pm$ 1.52 \\
% $\textsc{mcr\_dtree}$  &  \textbf{.5$\pm$ .46} &  \textbf{.92$\pm$ .03} &  \textbf{.96$\pm$ .03} &  \textbf{.88$\pm$ .07} &   3.2$\pm$ 1.64 \\

% \midrule
% \multicolumn{6}{l}{Power: nsamples = 9568, nfeatures = 4 | $\textsc{lgbm}$-Regressor R2 = 0.95 $\pm$  0.01}\\
% \midrule
% $\textsc{lcp-rf-g}$ & 3.67$\pm$ 2.26 &  .82$\pm$ .05 &  .86$\pm$ .03 &  .78$\pm$ .07 &    4.4$\pm$ 1.95\\
% $\textsc{rf-g}$ & \textbf{.47$\pm$ .22} &    \textbf{.9$\pm$ .0} &  \textbf{.92$\pm$ .01} &  \textbf{.88$\pm$ .01} &    5.0$\pm$ .71 \\
% $\textsc{pb-kmeans}$  &  .76$\pm$ .18 &  \textbf{.9$\pm$ .01} &  .95$\pm$ .03 &  .85$\pm$ .02 &  15.0$\pm$ 7.55 \\
% $\textsc{mcr-kmeans}$ &   .66$\pm$ .23 &  .91$\pm$ .01 &  .96$\pm$ .03 &  .86$\pm$ .02 &  16.6$\pm$ 10.26 \\
% $\textsc{pb\_dtree}$  &   1.13$\pm$ .6 &   \textbf{.9$\pm$ .0} &  .98$\pm$ .04 &  .76$\pm$ .09 &  17.2$\pm$ 9.26 \\
% $\textsc{mcr\_dtree}$  &  .57$\pm$ .2 &  \textbf{.9$\pm$ .01} &  \textbf{.92$\pm$ .03} &  \textbf{.88$\pm$ .03} &   5.8$\pm$ 8.56 \\


% \midrule
% \multicolumn{6}{l}{Protein: nsamples = 45730, nfeatures = 9 | $\textsc{lgbm}$-Regressor R2 = 0.46 $\pm$  0.04}\\
% \midrule

% $\textsc{lcp-rf-g}$ &&&&&\\
% $\textsc{rf-g}$ & .61$\pm$ .36 &  \textbf{.9$\pm$ .0} &  .95$\pm$ .05 &  .88$\pm$ .03 &   11.0$\pm$ 7.0 \\
% $\textsc{pb-kmeans}$  &  .59$\pm$ .57 &  \textbf{.9$\pm$ .0} &    1.0$\pm$ .0 &  .71$\pm$ .22 &   4.8$\pm$ 5.67 \\
% $\textsc{mcr-kmeans}$ &   .47$\pm$ .3 &  \textbf{.9$\pm$ .0} &  .97$\pm$ .05 &  .87$\pm$ .03 &   11.4$\pm$ 8.26 \\
% $\textsc{pb\_dtree}$  &  .79$\pm$ .27 &  \textbf{.9$\pm$ .0} &    1.0$\pm$ .0 &  .81$\pm$ .01 &   31.2$\pm$ .45 \\
% $\textsc{mcr\_dtree}$  &  \textbf{.17$\pm$ .14} &  \textbf{.9$\pm$ .0} &  \textbf{.91$\pm$ .01} &  \textbf{.89$\pm$ .01} &    4.4$\pm$ .89 \\




% \midrule
% \multicolumn{6}{l}{Kin8mn: nsamples = 8192, nfeatures = 8 | $\textsc{lgbm}$-Regressor R2 = 0.62 $\pm$  0.03}\\
% \midrule

% $\textsc{lcp-rf-g}$ &   2.32$\pm$ 1.1 &  .8$\pm$ .02 &  .84$\pm$ .02 &  .75$\pm$ .04 &   4.6$\pm$ 1.34 \\
% $\textsc{rf-g}$ &  \textbf{.32$\pm$ .18} &   \textbf{.9$\pm$ .0} &  .93$\pm$ .01 &  .87$\pm$ .01 &   5.2$\pm$ .45 \\
% $\textsc{pb-kmeans}$  &  .96$\pm$ 0.67 &  .92$\pm$ .0 &    1.0$\pm$ .0 &  .72$\pm$ .03 &  41.0$\pm$ 8.57 \\
% $\textsc{mcr-kmeans}$ & .76$\pm$ .16    &  .91$\pm$ .02 &  .94$\pm$ .05 &  .82$\pm$ .11 &  20.6$\pm$ 7.06 \\
% $\textsc{pb\_dtree}$  &    .73$\pm$ .39 & \textbf{.9$\pm$ .01} &  .97$\pm$ .03 &   .8$\pm$ .07 &  16.4$\pm$ 6.58 \\
% $\textsc{mcr\_dtree}$  &    .4$\pm$ .2 &  \textbf{.9$\pm$ .01} &  \textbf{.91$\pm$ .02} &  \textbf{.89$\pm$ .02} &   3.0$\pm$ 1.41 \\

% \midrule
% \multicolumn{6}{c}{Naval: nsamples = 11934, nfeatures = 16 | $\textsc{lgbm}$-Regressor R2 = 0.94 $\pm$  0.01 }\\
% \midrule
% $\textsc{lcp-rf-g}$ &   0.7$\pm$ 0.12 &   0.8$\pm$ 0.01 &  0.86$\pm$ 0.03 &  0.73$\pm$ 0.03 &     9.0$\pm$ 0.0 \\
% $\textsc{rf-g}$ &  0.15$\pm$ 0.06 &  0.91$\pm$ 0.01 &  0.96$\pm$ 0.03 &  0.83$\pm$ 0.06 &  19.0$\pm$ 17.32 \\
% $\textsc{pb-kmeans}$  &  0.18$\pm$ 0.03 &  0.92$\pm$ 0.01 &   1.0$\pm$ 0.01 &  0.81$\pm$ 0.02 &  42.6$\pm$ 10.19 \\
% $\textsc{mcr-kmeans}$  &  0.24$\pm$ 0.09 &   0.91$\pm$ 0.0 &  0.98$\pm$ 0.02 &  0.82$\pm$ 0.02 &   28.2$\pm$ 6.53 \\

% $\textsc{pb\_dtree}$  &    0.13$\pm$ 0.05 &  0.9$\pm$ 0.0 &  0.94$\pm$ 0.03 &  0.86$\pm$ 0.04 &    7.4$\pm$ 4.34 \\
% $\textsc{mcr\_dtree}$  &   0.12$\pm$ 0.1 &      0.9$\pm$ 0.0 &  0.91$\pm$ 0.01 &  0.89$\pm$ 0.01 &    3.4$\pm$ 1.34 \\


% \bottomrule
% \end{tabular}
% }
% \caption{Comparison between the group discovery partition methods. We show MCR, marginal, minimum, and maximum coverage group coverage on the identified partition. We also report the number of groups per approach. Standard deviations are computed across 5 data splits. The proposed $\textsc{mcr\_dtree}$ is consistently better in terms of MCR,  with values consistently below $1$, indicating that the discovered groups improve worst-group under-coverage w.r.t. to single threshold SCP. Every dataset uses a $\textsc{lgbm}$ regressor as the base model. {\color{blue} We highlight the lowest MCR and the smallest average coverage above the objective (0.9). For methods that achieved the marginal coverage objective we highlight the max and min group coverage closest to the 0.9 objective. } Additional results are available in the Appendix}
% \label{tab:summary_methods_extended}
% \end{table}





% \begin{table}[h!]
% \centering
% \footnotesize
% \scalebox{0.7}{
% \begin{tabular}{l|r|rrr|c}
% % \footnotesize
% \toprule
%  &\multicolumn{1}{c}{MCR } &  \multicolumn{3}{c}{coverage} & \multicolumn{1}{c}{num } \\
% model &  & average &   max group &   min group  &     groups  \\
%                                               &          &       &       &         \\
% \midrule
% \multicolumn{3}{c}{housing}\\
% \midrule

% LCP-RF-G &  1.45$\pm$1.14 &   .8$\pm$.04 &  .91$\pm$.07 &  .64$\pm$.15 &  3.6$\pm$.55 \\
% SCP-RF-G &   .77$\pm$.6 &  .93$\pm$.03 &  .99$\pm$.01 &  .86$\pm$.06 &  3.6$\pm$.55 \\
% KMEANSPB &   .81$\pm$.3 &  .92$\pm$.02 &  .97$\pm$.04 &  .68$\pm$.33 &  8.4$\pm$8.65 \\
% KMEANSMCR &     .95$\pm$.12  &  .91$\pm$.04 &  .92$\pm$.06 &  .89$\pm$.05 &  1.6$\pm$1.34 \\
% RTPB &  .68$\pm$.31 &  .89$\pm$.02 &  .94$\pm$.03 &  .83$\pm$.04 &  3.4$\pm$.55 \\
% RTMCR &  .65$\pm$.17 &  .92$\pm$.03 &  .95$\pm$.04 &  .88$\pm$.07 &   2.2$\pm$1.3 \\
% \midrule
% \multicolumn{3}{c}{concrete}\\
% \midrule


% LCP-RF-G &  1.84$\pm$1.66 &  .83$\pm$.01 &  .94$\pm$.05 &  .69$\pm$.11 &  4.6$\pm$.55 \\
% SCP-RF-G &  .82$\pm$.68 &   .9$\pm$.05 &  .97$\pm$.02 &  .81$\pm$.11 &  4.6$\pm$.55 \\
% KMEANSPB & .66$\pm$.48 &  .91$\pm$.05 &  .97$\pm$.05 &  .83$\pm$.07 &  7.0$\pm$3.24 \\
% KMEANSMCR &  .88$\pm$.27 &  .91$\pm$.05 &  .92$\pm$.06 &  .88$\pm$.05 &  4.2$\pm$7.16 \\
% RTPB &  .94$\pm$.57 &  .89$\pm$.04 &  .98$\pm$.02 &  .77$\pm$.07 &  6.6$\pm$.55 \\
% RTMCR &   .55$\pm$.72 &   .9$\pm$.04 &  .92$\pm$.06 &  .88$\pm$.04 &  2.4$\pm$2.61 \\

% \midrule
% \multicolumn{3}{c}{energy}\\
% \midrule

% LCP-RF-G &&&&&\\
% SCP-RF-G &&&&& \\
% KMEANSPB &  1.04$\pm$.34 &  .85$\pm$.07 &    1.0$\pm$.0 &  .07$\pm$.15 &  47.8$\pm$1.79 \\
% KMEANSMCR &   .68$\pm$.3 &  .94$\pm$.03 &  .96$\pm$.05 &  .78$\pm$.17 &   1.6$\pm$9.5 \\
% RTPB &   .63$\pm$.5 &  .93$\pm$.03 &  .97$\pm$.02 &  .87$\pm$.07 &   3.6$\pm$1.52 \\
% RTMCR &  .5$\pm$.46 &  .92$\pm$.03 &  .96$\pm$.03 &  .88$\pm$.07 &   3.2$\pm$1.64 \\



% \midrule
% \multicolumn{3}{c}{power}\\
% \midrule
% LCP-RF-G &&&&&\\
% SCP-RF-G &&&&& \\
% KMEANSPB &  .66$\pm$.18 &  .9$\pm$.01 &  .95$\pm$.03 &  .85$\pm$.02 &  15.0$\pm$7.55 \\
% KMEANSMCR &&&&&    \\
% RTPB &   1.13$\pm$.6 &   .9$\pm$.0 &  .98$\pm$.04 &  .76$\pm$.09 &  17.2$\pm$9.26 \\
% RTMCR &  .77$\pm$.65 &  .9$\pm$.01 &  .92$\pm$.03 &  .88$\pm$.03 &   5.8$\pm$8.56 \\



% \midrule
% \multicolumn{3}{c}{protein}\\
% \midrule


% LCP-RF-G &&&&&\\
% SCP-RF-G &&&&& \\
% KMEANSPB &  .59$\pm$.57 &  .9$\pm$.0 &    1.0$\pm$.0 &  .71$\pm$.22 &   4.8$\pm$5.67 \\
% KMEANSMCR &   .47$\pm$.3 &  .9$\pm$.0 &  .97$\pm$.05 &  .87$\pm$.03 &   11.4$\pm$8.26 \\
% RTPB &  .79$\pm$.27 &  .9$\pm$.0 &    1.0$\pm$.0 &  .81$\pm$.01 &   31.2$\pm$.45 \\
% RTMCR &  .17$\pm$.14 &  .9$\pm$.0 &  .91$\pm$.01 &  .89$\pm$.01 &    4.4$\pm$.89 \\


% \midrule
% \multicolumn{3}{c}{kin8mn}\\
% \midrule

% LCP-RF-G &   2.32$\pm$1.1 &  .8$\pm$.02 &  .84$\pm$.02 &  .75$\pm$.04 &   4.6$\pm$1.34 \\
% SCP-RF-G &  .32$\pm$.18 &   .9$\pm$.0 &  .93$\pm$.01 &  .87$\pm$.01 &   5.2$\pm$.45 \\
% KMEANSPB &  .76$\pm$.16 &  .92$\pm$.0 &    1.0$\pm$.0 &  .72$\pm$.03 &  41.0$\pm$8.57 \\
% KMEANSMCR &&&&& \\
% RTPB &    .73$\pm$.39 & .9$\pm$.01 &  .97$\pm$.03 &   .8$\pm$.07 &  16.4$\pm$6.58 \\
% RTMCR &    .4$\pm$.2 &  .9$\pm$.01 &  .91$\pm$.02 &  .89$\pm$.02 &   3.0$\pm$1.41 \\

% \bottomrule
% \end{tabular}
% }
% \caption{Comparison between the group discovery partition methods. We show MCR, marginal, minimum, and maximum coverage group coverage on the identified partition. We also report the number of groups per approach. Standard deviations are computed across 5 data splits. The proposed $\textsc{mcr\_dtree}$ is consistently better in terms of MCR,  with values consistently below $1$, indicating that the discovered groups improve worst-group under-coverage w.r.t. to single threshold SCP. Every dataset uses a $\textsc{lgbm}$ regressor as the base model.  Additional results are available in the Appendix}
% \label{tab:summary_methods2}
% \end{table}
