Related workk

content/experiments/experiments.tex

@@ -2,16 +2,237 @@
 \section{Data Sets}\label{sec:experiments:data_sets}
 \addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{מערך נתונים}}
 
+\paragraph{KEEL benchmark.}
+We evaluate on 12 binary imbalanced datasets from the KEEL repository~\cite{TODO} (Knowledge Extraction based on Evolutionary Learning). All labels are clean. \Cref{tab:datasets:keel} summarises the datasets; imbalance ratios range from 1.8 to 11.6, covering mild to moderate imbalance regimes commonly used in the literature.
+
+\begin{table}[htbp]
+\centering
+\caption{KEEL benchmark datasets used in experiments. IR = negative/positive count ratio.}
+\label{tab:datasets:keel}
+\small
+\begin{tabular}{lrrrrl}
+\toprule
+Dataset & $n$ & Pos & Neg & IR & Source \\
+\midrule
+Glass0       &  214 &  70 & 144 &  2.1 & Glass identification (UCI) \\
+Glass1       &  214 &  76 & 138 &  1.8 & Glass identification (UCI) \\
+Glass2       &  214 &  17 & 197 & 11.6 & Glass identification (UCI) \\
+Glass6       &  214 &  29 & 185 &  6.4 & Glass identification (UCI) \\
+Haberman     &  306 &  81 & 225 &  2.8 & Breast cancer survival, 1958--1970 \\
+Iris0        &  150 &  50 & 100 &  2.0 & \emph{Iris setosa} vs.\ others \\
+New-Thyroid1 &  215 &  35 & 180 &  5.1 & Thyroid function classification \\
+Pima         &  768 & 268 & 500 &  1.9 & Pima Indian diabetes (UCI) \\
+Vehicle2     &  846 & 218 & 628 &  2.9 & Vehicle silhouette (UCI) \\
+Vowel0       &  988 &  90 & 898 & 10.0 & Vowel recognition, class 0 vs.\ rest \\
+Wisconsin    &  683 & 239 & 444 &  1.9 & Wisconsin breast cancer (UCI) \\
+Yeast3       & 1484 & 163 &1321 &  8.1 & Yeast protein localisation (UCI) \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+\paragraph{CIFAR-10N.}
+CIFAR-10N~\cite{TODO} augments the 50{,}000-image CIFAR-10 training set ($32{\times}32$ RGB) with real human-annotated noisy labels collected from Amazon Mechanical Turk~\cite{TODO}. We form a binary task (class~1, automobile, vs.\ rest; IR~$\approx 9$:1) and train under three noise conditions:
+\emph{clean} (original CIFAR-10 labels, 0\% noise),
+\emph{aggre} (majority-vote aggregate of crowdworker annotations, $\approx$9\% noise), and
+\emph{worse} (single worst-annotator labels, $\approx$40\% noise).
+Architecture: a three-block convolutional network trained from scratch.
+
+\paragraph{Large tabular datasets.}
+We include three large-scale tabular datasets spanning diverse imbalanced real-world settings. All labels are considered clean (no injected or crowd-sourced noise).
+
+\begin{itemize}
+  \item \textbf{Bank Marketing}~\cite{TODO} --- UCI repository; Portuguese bank telephone marketing campaigns (Moro et al., 2014). $n = 41{,}188$ samples, 48 features after one-hot encoding. Positive class: client subscribed to term deposit. IR~$\approx 8$:1 ($\approx$11.3\% positive). No label noise.
+
+  \item \textbf{Credit Card Fraud}~\cite{TODO} --- Kaggle / ULB (Worldline + MLG, 2013). $n = 284{,}807$ transactions, 30 features (V1--V28 are PCA-transformed; Time and Amount rescaled). Extreme imbalance: 492 confirmed frauds, IR~$\approx 577$:1. No label noise.
+
+  \item \textbf{Home Credit}~\cite{TODO} --- Kaggle Home Credit Default Risk competition (2018). $n = 307{,}511$ loan applications, $\approx$120 numerical features from the primary application table. Positive class: loan default. IR~$\approx 11.4$:1 ($\approx$8\% default rate). Labels are proxy labels (repayment outcome); no injected noise.
+\end{itemize}
+
 \section{Performance Measures}\label{sec:experiments:performance_measures}
 \addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{מדדי ביצועים}}
 
-To be continued\footnotemark. 
+We report the \emph{area under the ROC curve} (AUC) averaged over independent training episodes (see \Cref{sec:experiments:configurations}). AUC measures overall ranking quality and is a standard metric for imbalanced classification. All results are presented as mean $\pm$ standard deviation over episodes.
-
-\footnotetext{foot note text}
 
 \section{Configurations and Parameter Settings}
 \label{sec:experiments:configurations}
 \addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{קונפיגורציה ופרמטרים}}
 
+We compare \textsc{Roll}-AoC (the primary proposed loss, optimising the area under the ROC curve via KDE) and \textsc{Roll}-TPR (a variant targeting TPR at a fixed operating point) against the following baselines:
+
+\begin{itemize}
+  \item \textbf{BCE-W} --- binary cross-entropy with class-frequency inverse weighting.
+  \item \textbf{Focal} --- focal loss~\cite{TODO} with focusing parameter $\gamma = 2$.
+  \item \textbf{Asym.} --- asymmetric loss~\cite{TODO}.
+  \item \textbf{GCE} --- generalised cross-entropy~\cite{TODO} with $q = 0.7$.
+  \item \textbf{LibAUC} --- PESG-based AUC maximisation from the LibAUC library~\cite{TODO}.
+  \item \textbf{MAE} --- mean absolute error (symmetric noise-robust loss).
+\end{itemize}
+
+All methods use the same network architecture, optimiser (Adam for non-LibAUC methods), and early stopping. Each configuration is run for $n = 5$ independent episodes on KEEL and large datasets, and $n = 3$ on CIFAR-10N. Reported AUC is evaluated on the held-out test split.
+
 \section{Results}\label{sec:experiments:results}
 \addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{תוצאות}}
+
+\subsection{KEEL Benchmark}\label{sec:results:keel}
+
+\Cref{tab:results:keel} reports mean test AUC across the 12 KEEL datasets. No single method dominates the benchmark: on easy datasets (Iris0, Wisconsin, Vowel0, Vehicle2, Glass6) nearly all methods converge to near-perfect AUC, leaving little room for differentiation. On the harder, more imbalanced datasets the picture is more varied. \textsc{Roll}-AoC competes well but does not claim the top position on most individual datasets; this is consistent with no-free-lunch intuitions for heterogeneous benchmarks.
+
+\begin{table}[htbp]
+\centering
+\caption{Mean test AUC ($\pm$\,std, $n=5$ episodes) on KEEL benchmark datasets. Best per row in \textbf{bold}.}
+\label{tab:results:keel}
+\resizebox{\textwidth}{!}{%
+\begin{tabular}{lcccccccc}
+\toprule
+Dataset & \textsc{Roll}-AoC & \textsc{Roll}-TPR & BCE-W & Focal & Asym. & GCE & LibAUC & MAE \\
+\midrule
+Glass0 & 0.796{\tiny{$\pm$0.046}} & 0.826{\tiny{$\pm$0.032}} & 0.839{\tiny{$\pm$0.033}} & 0.802{\tiny{$\pm$0.020}} & 0.814{\tiny{$\pm$0.034}} & 0.818{\tiny{$\pm$0.032}} & \textbf{0.843{\tiny{$\pm$0.035}}} & 0.826{\tiny{$\pm$0.037}} \\
+Glass1 & 0.722{\tiny{$\pm$0.033}} & 0.698{\tiny{$\pm$0.042}} & 0.710{\tiny{$\pm$0.018}} & 0.712{\tiny{$\pm$0.036}} & 0.744{\tiny{$\pm$0.044}} & 0.742{\tiny{$\pm$0.039}} & 0.723{\tiny{$\pm$0.054}} & \textbf{0.769{\tiny{$\pm$0.031}}} \\
+Glass2 & 0.723{\tiny{$\pm$0.062}} & 0.701{\tiny{$\pm$0.055}} & 0.661{\tiny{$\pm$0.069}} & 0.677{\tiny{$\pm$0.072}} & \textbf{0.750{\tiny{$\pm$0.075}}} & 0.689{\tiny{$\pm$0.023}} & 0.719{\tiny{$\pm$0.084}} & 0.679{\tiny{$\pm$0.018}} \\
+Glass6 & 0.975{\tiny{$\pm$0.014}} & 0.975{\tiny{$\pm$0.027}} & 0.972{\tiny{$\pm$0.025}} & 0.974{\tiny{$\pm$0.020}} & 0.982{\tiny{$\pm$0.013}} & 0.971{\tiny{$\pm$0.020}} & 0.969{\tiny{$\pm$0.012}} & \textbf{0.985{\tiny{$\pm$0.006}}} \\
+Haberman & 0.696{\tiny{$\pm$0.045}} & 0.675{\tiny{$\pm$0.063}} & 0.736{\tiny{$\pm$0.012}} & \textbf{0.742{\tiny{$\pm$0.025}}} & 0.741{\tiny{$\pm$0.018}} & 0.737{\tiny{$\pm$0.016}} & 0.616{\tiny{$\pm$0.116}} & 0.671{\tiny{$\pm$0.102}} \\
+Iris0 & \textbf{1.000{\tiny{$\pm$0.000}}} & 0.981{\tiny{$\pm$0.027}} & 0.900{\tiny{$\pm$0.224}} & \textbf{1.000{\tiny{$\pm$0.000}}} & \textbf{1.000{\tiny{$\pm$0.000}}} & \textbf{1.000{\tiny{$\pm$0.000}}} & \textbf{1.000{\tiny{$\pm$0.000}}} & 0.900{\tiny{$\pm$0.224}} \\
+New-Thyroid1 & 0.989{\tiny{$\pm$0.010}} & 0.845{\tiny{$\pm$0.241}} & 0.992{\tiny{$\pm$0.005}} & 0.977{\tiny{$\pm$0.033}} & 0.988{\tiny{$\pm$0.007}} & 0.987{\tiny{$\pm$0.011}} & 0.985{\tiny{$\pm$0.012}} & \textbf{0.994{\tiny{$\pm$0.003}}} \\
+Pima & 0.778{\tiny{$\pm$0.024}} & 0.755{\tiny{$\pm$0.014}} & \textbf{0.787{\tiny{$\pm$0.014}}} & 0.775{\tiny{$\pm$0.020}} & 0.771{\tiny{$\pm$0.019}} & 0.785{\tiny{$\pm$0.005}} & 0.761{\tiny{$\pm$0.015}} & 0.778{\tiny{$\pm$0.005}} \\
+Vehicle2 & 0.982{\tiny{$\pm$0.003}} & 0.966{\tiny{$\pm$0.020}} & \textbf{0.984{\tiny{$\pm$0.004}}} & 0.982{\tiny{$\pm$0.003}} & 0.980{\tiny{$\pm$0.004}} & 0.982{\tiny{$\pm$0.004}} & 0.982{\tiny{$\pm$0.003}} & 0.983{\tiny{$\pm$0.004}} \\
+Vowel0 & 0.993{\tiny{$\pm$0.005}} & 0.987{\tiny{$\pm$0.002}} & 0.995{\tiny{$\pm$0.004}} & 0.996{\tiny{$\pm$0.003}} & 0.995{\tiny{$\pm$0.003}} & 0.997{\tiny{$\pm$0.003}} & 0.991{\tiny{$\pm$0.007}} & \textbf{0.998{\tiny{$\pm$0.001}}} \\
+Wisconsin & 0.997{\tiny{$\pm$0.002}} & 0.800{\tiny{$\pm$0.441}} & \textbf{0.999{\tiny{$\pm$0.000}}} & \textbf{0.999{\tiny{$\pm$0.001}}} & 0.999{\tiny{$\pm$0.001}} & 0.999{\tiny{$\pm$0.001}} & 0.999{\tiny{$\pm$0.000}} & 0.998{\tiny{$\pm$0.001}} \\
+Yeast3 & 0.906{\tiny{$\pm$0.024}} & 0.908{\tiny{$\pm$0.021}} & 0.894{\tiny{$\pm$0.008}} & 0.908{\tiny{$\pm$0.019}} & \textbf{0.935{\tiny{$\pm$0.014}}} & 0.927{\tiny{$\pm$0.006}} & \textbf{0.935{\tiny{$\pm$0.016}}} & 0.915{\tiny{$\pm$0.010}} \\
+\bottomrule
+\end{tabular}%
+}
+\end{table}
+
+\subsection{CIFAR-10N: Label-Noise Robustness}\label{sec:results:cifar10n}
+
+\Cref{tab:results:cifar10n} shows AUC under three noise levels on CIFAR-10N. Under \emph{clean} labels all methods reach AUC $\geq 0.972$, with \textsc{Roll}-AoC at the top (0.980). As noise increases the advantage of \textsc{Roll}-AoC grows: under \emph{aggre} noise it leads by 0.2~pp over the next-best method (GCE, 0.932), and under \emph{worse} noise the margin widens to 1.2~pp over MAE (0.809) and 4.7~pp over BCE-W (0.738). This suggests that the KDE-smoothed loss surface provides implicit robustness to label noise.
+
+\begin{table}[htbp]
+\centering
+\caption{Mean test AUC ($\pm$\,std, $n=3$ episodes) on CIFAR-10N under three label-noise levels. Best per row in \textbf{bold}.}
+\label{tab:results:cifar10n}
+\resizebox{\textwidth}{!}{%
+\begin{tabular}{lccccccc}
+\toprule
+Noise & \textsc{Roll}-AoC & BCE-W & Focal & Asym. & GCE & LibAUC & MAE \\
+\midrule
+Clean & \textbf{0.980{\tiny{$\pm$0.001}}} & 0.977{\tiny{$\pm$0.002}} & 0.977{\tiny{$\pm$0.000}} & 0.972{\tiny{$\pm$0.003}} & 0.976{\tiny{$\pm$0.002}} & 0.979{\tiny{$\pm$0.000}} & 0.976{\tiny{$\pm$0.002}} \\
+Aggre. & \textbf{0.934{\tiny{$\pm$0.002}}} & 0.924{\tiny{$\pm$0.003}} & 0.912{\tiny{$\pm$0.014}} & 0.910{\tiny{$\pm$0.004}} & 0.932{\tiny{$\pm$0.005}} & 0.921{\tiny{$\pm$0.004}} & 0.930{\tiny{$\pm$0.002}} \\
+Worse & \textbf{0.821{\tiny{$\pm$0.001}}} & 0.738{\tiny{$\pm$0.009}} & 0.734{\tiny{$\pm$0.009}} & 0.720{\tiny{$\pm$0.004}} & 0.774{\tiny{$\pm$0.005}} & 0.752{\tiny{$\pm$0.006}} & 0.809{\tiny{$\pm$0.004}} \\
+\bottomrule
+\end{tabular}%
+}
+\end{table}
+
+\subsection{Large Real-World Datasets}\label{sec:results:large}
+
+\Cref{tab:results:large} covers three large-scale datasets. On Credit Card Fraud \textsc{Roll}-TPR is the top performer (0.974) with \textsc{Roll}-AoC close behind (0.968), and both outperform all non-ROLL baselines. On Home Credit \textsc{Roll}-AoC edges LibAUC (0.712 vs.\ 0.711). On Bank Marketing, however, GCE achieves the highest AUC (0.746) while \textsc{Roll}-AoC and \textsc{Roll}-TPR underperform (0.511 and 0.438 respectively); the cause of this degradation on this dataset is an open question and a direction for future investigation.
+
+\begin{table}[htbp]
+\centering
+\caption{Mean test AUC ($\pm$\,std, $n=5$ episodes) on large real-world datasets. Best per row in \textbf{bold}.}
+\label{tab:results:large}
+\resizebox{\textwidth}{!}{%
+\begin{tabular}{lcccccccc}
+\toprule
+Dataset & \textsc{Roll}-AoC & \textsc{Roll}-TPR & BCE-W & Focal & Asym. & GCE & LibAUC & MAE \\
+\midrule
+Bank Marketing & 0.511{\tiny{$\pm$0.047}} & 0.438{\tiny{$\pm$0.044}} & 0.580{\tiny{$\pm$0.040}} & 0.573{\tiny{$\pm$0.044}} & 0.684{\tiny{$\pm$0.033}} & \textbf{0.746{\tiny{$\pm$0.022}}} & 0.652{\tiny{$\pm$0.031}} & 0.707{\tiny{$\pm$0.035}} \\
+Credit Card Fraud & 0.968{\tiny{$\pm$0.001}} & \textbf{0.974{\tiny{$\pm$0.003}}} & 0.949{\tiny{$\pm$0.015}} & 0.942{\tiny{$\pm$0.020}} & 0.940{\tiny{$\pm$0.013}} & 0.948{\tiny{$\pm$0.014}} & 0.966{\tiny{$\pm$0.006}} & 0.954{\tiny{$\pm$0.009}} \\
+Home Credit & \textbf{0.712{\tiny{$\pm$0.001}}} & 0.687{\tiny{$\pm$0.004}} & 0.693{\tiny{$\pm$0.002}} & 0.694{\tiny{$\pm$0.002}} & 0.688{\tiny{$\pm$0.003}} & 0.698{\tiny{$\pm$0.002}} & 0.711{\tiny{$\pm$0.001}} & 0.699{\tiny{$\pm$0.004}} \\
+\bottomrule
+\end{tabular}%
+}
+\end{table}
+
+
+\section{Label-Noise Poisoning Experiment}\label{sec:experiments:poisoning}
+\addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{ניסוי הרעלת תוויות}}
+
+% A controlled experiment that isolates ROLL's label-noise robustness from confounders
+% (dataset size, imbalance ratio).  Protocol: start from a clean dataset, inject
+% \emph{asymmetric} noise --- flip a fraction of positive labels to negative, leave
+% negative labels intact --- and track AUC degradation as the flip rate increases.
+% This mirrors a worst-case practical scenario: rare true positives become false negatives,
+% compounding the imbalance.  Empirical complement to the structural argument in
+% \Cref{sec:noisy-labels}.
+
+\subsection{Protocol}\label{sec:experiments:poisoning:protocol}
+
+% Specify: which dataset(s) were poisoned (one or two from KEEL or large-scale, chosen
+% for mid-range difficulty where differences are visible), the range of flip rates tested
+% (e.g.\ 0\% to 40\% in steps of 10\%), and number of independent episodes per rate.
+% Justify asymmetric noise as the harder, more realistic case compared to symmetric flipping.
+
+\subsection{Results}\label{sec:experiments:poisoning:results}
+
+% Present AUC vs.\ noise-rate curves for all methods.  Key finding: ROLL-AoC degrades
+% more gracefully than pointwise losses (BCE-W, Focal, Asym.) because a distributional
+% objective spreads the influence of each mislabeled sample over the estimated density
+% rather than producing a single large gradient.  Cite structural explanation from
+% \Cref{sec:noisy-labels}; note consistency with CIFAR-10N results in
+% \Cref{sec:results:cifar10n}.
+
+
+\section{Ablation Study}\label{sec:experiments:ablation}
+\addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{ניתוח אבלציה}}
+
+% Isolate the contribution of ROLL's individual design choices to determine which
+% components drive its performance and noise robustness.
+
+\subsection{Effect of KDE Smoothing}\label{sec:experiments:ablation:kde}
+
+% Compare ROLL-AoC against a hard-threshold variant (no KDE; threshold estimated
+% by sorting scores and selecting the rank corresponding to FPR $= \alpha$).
+% Expected finding: without smooth KDE the gradient signal collapses and training
+% is unstable, confirming that differentiability is not just a theoretical convenience.
+
+\subsection{Effect of Kernel Choice}\label{sec:experiments:ablation:kernel}
+
+% Compare Gaussian vs.\ sigmoid kernel in ROLL-AoC on representative datasets.
+% The sigmoid kernel admits closed-form gradients (see \Cref{appendix:kde-sigmoid});
+% this subsection assesses whether kernel choice materially affects accuracy or
+% whether performance is robust to kernel selection.
+
+\subsection{ROLL-AoC vs.\ ROLL-TPR}\label{sec:experiments:ablation:variants}
+
+% Systematically compare the two ROLL variants across all benchmarks.  Discuss when
+% targeting a single operating point (ROLL-TPR) helps (e.g.\ Credit Card Fraud, where
+% a strict FPR budget is natural) vs.\ when whole-curve optimization (ROLL-AoC) is
+% preferable (heterogeneous or unconstrained evaluation).
+
+
+\section{Sensitivity Analysis}\label{sec:experiments:sensitivity}
+\addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{ניתוח רגישות}}
+
+% Characterize ROLL's robustness to its key hyperparameters so practitioners can
+% apply it without exhaustive tuning.
+
+\subsection{Bandwidth Sensitivity}\label{sec:experiments:sensitivity:bandwidth}
+
+% Plot AUC vs.\ log-bandwidth $h$ for ROLL-AoC across a representative subset of
+% datasets (at least one from KEEL, CIFAR-10N, and large-scale).  Report the range
+% of $h$ over which performance stays within 1\% of its peak --- this defines a
+% practical safe tuning zone.  Show that Silverman's rule falls within this zone,
+% justifying it as a default.
+
+\subsection{Operating-Point Sensitivity (ROLL-TPR)}\label{sec:experiments:sensitivity:alpha}
+
+% For ROLL-TPR, sweep the target FPR budget $\alpha$ and measure TPR achieved at
+% each setting.  Validate that the trained model's operating point tracks the
+% specified $\alpha$, demonstrating that ROLL-TPR reliably targets the operating
+% point it was trained for and generalises to nearby values.
+
+
+\section{Summary}\label{sec:experiments:summary}
+\addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{סיכום}}
+
+% Synthesise the empirical findings in 2--3 paragraphs.
+% (1) Where ROLL-AoC wins clearly: CIFAR-10N noise robustness, Credit Card Fraud,
+%     Home Credit --- relate to the distributional objective and KDE smoothing.
+% (2) Where it underperforms: Bank Marketing --- flag as an open question; discuss
+%     possible causes (feature scale, label proxy quality, optimizer interaction).
+% (3) What the ablation reveals: which components are load-bearing vs.\ incidental.
+% Close with a sentence bridging to \Cref{chapter:discussion_and_conclusions}.
+
+TODO write this section.

content/related_work/related_work.tex

@@ -83,6 +83,116 @@ TODO finish this section
 % surrogate that only loosely correlates with it.
 
 
+% ============================================================
+% PLACEMENT: Section 2.2, immediately after the imbalanced
+% classification section (sec:imbalanced-tpr-fpr).  Establishes
+% AUC as the standard measure and surveys how the literature
+% optimizes it, before narrowing to partial AUC / single-point
+% objectives in the next section.
+% ============================================================
+\section{ROC Curves and AUC Optimization}
+\label{sec:roc-auc}
+\addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{עקומות ROC ואופטימיזציה של AUC}}
+
+% \subsection{The ROC Curve}
+% Define the ROC curve as the plot of TPR vs.\ FPR as the threshold sweeps its full range.
+% Show that AUC equals $P(f(x^+) > f(x^-))$ for a random positive/negative pair (the
+% Wilcoxon--Mann--Whitney statistic): AUC is threshold-independent and measures ranking
+% quality, not calibration.  Cite Hanley \& McNeil (1982).
+
+% \subsection{Surrogate AUC Losses}
+% Explain why directly maximizing the empirical AUC is hard: the WMW indicator is pairwise
+% and discontinuous ($O(n^2)$ pairs).  Introduce smooth surrogates --- sigmoid approximation
+% to the pairwise indicator, squared loss on score differences --- and their consistency
+% guarantees.  Cite Cortes \& Mohri (2003/2004).
+
+% \subsection{Stochastic AUC Maximization}
+% Describe mini-batch and compositional stochastic methods that make AUC optimization
+% practical at scale.  Cover PESG (Yuan et al., NeurIPS 2020) and LibAUC (Yuan et al.,
+% ICML 2021), which is used as a baseline in \Cref{chapter:experiments}.
+% Key limitation shared by all: they optimize the \emph{whole-curve} AUC, not a specific
+% operating point on the ROC --- motivating the next section.
+
+A Receiver Operating Characteristic (ROC) is a graph that represents the performance of a classifier.
+The classifier in question is evaluated on binary-labeled data. The labeled data is ordered based off of the
+scores given by the classifier. Given any numerical threshold, it is then easy to see which labeled data points fall above and below.
+
+For each threshold, the true positive rate and the false positive rate is given. The results for each threshold, and so for each possible TPR and FPR, are put on a graph. The resulting curve is the ROC curve.
+
+The AUC is a number in the range $[0, 1]$ representing the total area under the curve. It is calculated by integrating over all TPRs and taking the resulting average FPRs. A result of $1$ represents a perfect classifier,
+while a result of $0$ represents a classifier with a perfect opposite result. A result of $0.5$ represents a classifier that found no separation whatsoever.
+
+The AUC is a direct representation of the performance of a classifier, as opposed to a loss. There are instances where the loss may be minimized, yet seperation might be minimal. These two metrics might correlate, but do not necessarily correspond.
+
+An example of this is a result for which all true samples receive a score of $0.51$ and all false samples receie a score of $0.49$. Using many losses, including BCE, this would result in a relatively high loss value. On this score however, the AUC is $1$. The model perfectly seperates the samples, so further training is not necessary.
+
+Direct AUC optimization can be ieal with optimizing for imbalanced datasets. With most losses such as BCE or focal, the loss is calculated per-sample and then summed (or averaged). In order to properly train for both true and false populations, the scores are weighted, so that equal weight falls on both the true and the false populations. With direct AUC optimization, the imbalance does not affect the calculation, and the reweighting can be forgone.
+
+Additionally, direct AUC methods are more resilient to label noise, as opposed to classical methods. Due to inseperability, correctly classifying a sample might require a model to incorrectly classify similar or even identical samples that were mislabeled. This is only worth it for the model in training if the total loss of the correctly labeled samples outweighs the total loss of the incorrectly labeled samples. For most methods, the weight per class is given by the class imbalance in the dataset, but this doesn't take into account the label noise in any way. Optimizing for AUC specifically would eliminate this risk.
+
+% ============================================================
+% PLACEMENT: Section 2.3, after ROC/AUC section above and
+% before the KDE background section.  Narrows from whole-curve
+% AUC to the single operating-point objective that ROLL targets.
+% ============================================================
+\section{Partial AUC and TPR at Fixed FPR}
+\label{sec:partial-auc}
+\addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{AUC חלקי ו-TPR בנקודת FPR קבועה}}
+
+% \subsection{Partial AUC}
+% Define partial AUC as the area under the ROC curve restricted to $\mathrm{FPR} \in [0, \alpha]$.
+% Motivate: in high-stakes settings (fraud, medical triage) only the low-FPR region matters.
+% Survey recent pAUC optimizers that extend the PESG/compositional framework
+% (Lin et al., 2022; Yang \& Ying, 2022).
+
+% \subsection{TPR at a Fixed FPR: The Single-Point Objective}
+% Distinguish pAUC (integral over an interval) from the exact single-point objective:
+% maximize $\mathrm{TPR}(f, \alpha)$ for a fixed budget $\alpha$.
+% Argue that post-hoc threshold search and pAUC integration both fail to provide a
+% differentiable end-to-end training signal for this precise target --- this is the gap
+% that ROLL fills.
+
+% \subsection{Neyman-Pearson Classification (Brief)}
+% Introduce NP classification (Cannon et al., 2002; Tong, 2013) as the statistical
+% framing: minimize type-II error subject to $\mathrm{FPR} \le \alpha$.
+% Note that classical NP methods are not gradient-based and cannot be applied to DNNs.
+% Full treatment deferred to \Cref{sec:neyman-pearson} at the end of this chapter.
+
+TODO write this section.
+
+
+% ============================================================
+% PLACEMENT: Section 2.4, after partial AUC and before the
+% noisy labels section.  Provides the technical KDE background
+% that Ch.\ 3 (sec:roll-kde) builds on directly.  End with a
+% forward pointer to sec:roll-kde.
+% ============================================================
+\section{Kernel Density Estimation}
+\label{sec:kde-background}
+\addcontentsline{tocheb}{section}{\protect\numberline{\secnumforhebrewtoc}{אמידת צפיפות גרעין (KDE)}}
+
+% \subsection{The Parzen Window Estimator}
+% Define KDE: $\hat{p}(x) = \frac{1}{nh}\sum_i K\!\bigl(\frac{x-x_i}{h}\bigr)$,
+% where $K$ is a kernel and $h > 0$ is the bandwidth.
+% Present common kernels: Gaussian, Epanechnikov, sigmoid.  Discuss how kernel
+% smoothness affects gradient properties.  Cite Parzen (1962), Silverman (1986).
+
+% \subsection{Bandwidth Selection}
+% Describe Silverman's rule-of-thumb ($h = 1.06\,\hat\sigma\,n^{-1/5}$) as the default.
+% Mention cross-validation as the principled alternative, expensive at training time.
+% In ROLL, bandwidth is fixed (Silverman) or treated as a learnable parameter;
+% details deferred to Ch.\ 3.
+
+% \subsection{Differentiability and Use as a Loss Ingredient}
+% Explain why KDE with a smooth kernel is differentiable w.r.t.\ the support points $x_i$:
+% the density at any query is a smooth weighted sum over all support points, so gradients
+% flow back to network parameters.  This property is why KDE is suitable as a building
+% block inside a loss function.  Forward pointer: \Cref{sec:roll-kde} exploits this to
+% construct ROLL's differentiable threshold estimate from the estimated score distributions.
+
+TODO write this section.
+
+
 % ============================================================
 % PLACEMENT: Section 2.5, AFTER the KDE background section
 % (sec:kde-background) and BEFORE the Neyman-Pearson bridge

main.tex

@@ -124,6 +124,7 @@
 
 \usepackage{graphicx}
 \graphicspath{ {content/images/} }
+\usepackage{booktabs}
 
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 \usepackage{titling}