It's done first draft wahoo

content/related_work/related_work.tex

@@ -30,11 +30,8 @@
 
 \subsection{AUC and ROC definitions}
 
-% TODO add formalizations
-% Formalize ROC and integration into AUC
-
 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
+The classifier in question is evaluated on binary-labeled data. The labeled data is ordered based on 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.
@@ -66,16 +63,16 @@ The AUC is a direct representation of the performance of a classifier, as oppose
 
 An example of this is a result for which all true samples receive a score of $0.51$ and all false samples receive 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 separates the samples, so further training is not necessary.
 
-Direct AUC optimization can be ideal 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.
+Direct AUC optimization can be ideal 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.
 
 The noise-robustness properties of ranking-based objectives are discussed in \Cref{sec:noisy-labels}.
 
 \subsection{AUC optimization in literature}
 
-While AUC as a metric for model performance has existed for a while, methods to optimize for it directly are relatively recent. At it's core, it is a non-trivial problem to solve. Existing efforts to directly optimize AUC have been based on the Wilcoxon-Mann-Whitney interpretation of AUC. However, directly calculating the aforementioned probability of a random positive score being higher than a negative one introduces two complications. The first is that one must iterate over $n^2$ pairs to cover every possible pair. Additionally, the pairwise function one must evaluate ($\mathbb{P}(f(x_1) > f(x_0))$) is a discontinuous stepwise function, and is therefore non-differentiable.
+While AUC as a metric for model performance has existed for a while, methods to optimize for it directly are relatively recent. At its core, it is a non-trivial problem to solve. Existing efforts to directly optimize AUC have been based on the Wilcoxon-Mann-Whitney interpretation of AUC. However, directly calculating the aforementioned probability of a random positive score being higher than a negative one introduces two complications. The first is that one must iterate over $n^2$ pairs to cover every possible pair. Additionally, the pairwise function one must evaluate ($\mathbb{P}(f(x_1) > f(x_0))$) is a discontinuous stepwise function, and is therefore non-differentiable.
 
 The discontinuous stepwise nature of the evaluated pairwise function is generally addressed by using a surrogate, either sigmoid-like or hinge functions are applied to the score difference~\cite{cortes2003auc}.
-\citet{yuan2021auc} proposed a compositional stochastic gradient framework that reformulates AUC maximization as a min-max saddle-point problem, enabling mini-batch training without maintaining all pairwise scores in memory. Their PESG optimizer and the LibAUC library are the primary practical tools for deep AUC maximization and serve as the main AUC baseline in this thesis.
+\citet{yuan2021auc} proposed a formulation of the loss that, using specific pairwise surrogate losses, allows training without separately evaluating every pair, but rather, evaluating all given samples once. Their PESG optimizer and the LibAUC library are the primary practical tools for deep AUC maximization and serve as the main AUC baseline in this thesis.
 
 % ============================================================
 % PLACEMENT: Section 2.2, after ROC/AUC.  Core differentiating
@@ -92,7 +89,7 @@ Many data collection methods lead to imperfect labeling, resulting in a muddy gr
 This could be due to a number of factors. The labels might be based on human labeling, which is error prone.
 The labeling might be true, yet the signal might be noisy or wrong, creating a mismatch between signal and label.
 The labeling could be a resource-intensive process, and so it might only be done for a portion of the data.
-And in some cases, the labeling is done based off a different learned model, which too could be faulty.
+And in some cases, the labeling is done based on a different learned model, which too could be faulty.
 
 \subsection{Label noise formalization}
 
@@ -319,7 +316,7 @@ which maximizes the \emph{likelihood ratio test}:
 \]
 Where $\phi^*(\mathbf{x})$ is our theoretical classifier,
 and $\eta \geq 0$ is chosen so that the FPR equals exactly $\alpha$.
-Given these assumptions, and the target FPR that gives $\eta$, the classifier$\phi^*$ is proven by the lemma to be optimal.
+Given these assumptions, and the target FPR that gives $\eta$, the classifier~$\phi^*$ is proven by the lemma to be optimal.
 
 \subsection{The practical gap}