Wrote stuff?

content/related_work/related_work.tex

@@ -74,7 +74,7 @@ The noise-robustness properties of ranking-based objectives are discussed in \Cr
 
 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 afformentioned probability of a random positive score behing higher than a negative one introduces two complications. The first is that one must iterate over $n^2$ pairs to cover every posible pair. Additionally, the pairwise function one must evaluate ( $\mathbb{P}(f(x_1) > f(x_0))$ ) is a non-contiguous stepwise function, and is therefor on-differentiable.
 
-The non-contiguous stepwise nature of the evaluated pairwise function is generally fixed by using a surrogate — either sigmoid-like or hinge functions are applied to the score difference~\cite{cortes2003auc}.
+The non-contiguous stepwise nature of the evaluated pairwise function is generally fixed 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.
 
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@@ -272,7 +272,8 @@ on the rare class during training, as it contributes less to the loss metric, an
 generalize on the rare class poorly.
 
 Dealing with this depends on the loss used. For most losses, for which the loss for each sample is calculated seperately,
-is to simply weight the losses differently, by multiplying the loss for the rare class
+the solution is to simply weight the losses for each class differently,
+by multiplying the loss for the rare class
 so that it contributes equally. Other approaches using pairwise losses, such as those implemented in LibAUC \cite{yuan2021auc}, naturally weight positive and negative contributions to learning through their implementation.
 
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