Related workk

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
@@ -345,4 +455,4 @@ To be continued.
 \end{tabular}
 \caption[Table A -  Example for table]{Some example}
 \label{table:plain_rl_results}
-\end{table}
+\end{table}