
\section{Problem Formulation}
\label{sec:problem-formulation}

% Motivation for this section — why accuracy fails under class imbalance, why
% TPR@FPR gives direct operating-point control, and the Neyman-Pearson framing
% that underpins ROLL — is developed in \Cref{chapter:related_work}
% (\Cref{sec:imbalanced-tpr-fpr,sec:neyman-pearson}).  This section establishes
% only the notation and objective used throughout the chapter.
Given a dataset $\mathcal{D} = \{(\mathbf{x}_i, y_i)\}_{i=1}^{n}$ with inputs
$\mathbf{x}_i \in \mathcal{X}$ and binary labels $y_i \in \{0, 1\}$, we learn a
scoring function $f_\theta : \mathcal{X} \to \mathbb{R}$ by fitting.
The positive and negative subsets of $\mathcal{D}$ are:
\[
  \mathcal{D}_1 = \{(\mathbf{x}_i, y_i) \in \mathcal{D} \mid y_i = 1\},
  \qquad
  \mathcal{D}_0 = \{(\mathbf{x}_i, y_i) \in \mathcal{D} \mid y_i = 0\}.
\]
Applying a threshold $\tau$ produces predictions $\hat{y} = \mathbf{1}[f_\theta(\mathbf{x}) > \tau]$.
The \emph{true positive rate} (TPR) and \emph{false positive rate} (FPR) at $\tau$ are:
\begin{align}
  \operatorname{TPR}(\alpha) &= P\!\bigl(f_\theta(\mathbf{x}) > \tau \mid y = 1\bigr), \\
  \operatorname{FPR}(\alpha) &= P\!\bigl(f_\theta(\mathbf{x}) > \tau \mid y = 0\bigr).
\end{align}
In our scenario, we either wish to learn $f_{\theta}$ in order to maximize $\operatorname{TPR}$
subject to a fixed $\operatorname{FPR}$, or vice versa. Our fixed $\operatorname{TPR}$ or $\operatorname{FPR}$
will be fixed to a value $\alpha \in (0, 1)$.

Therefor, when maximizing our $\operatorname{TPR}$ subject to a fixed $\operatorname{FPR}$, we are solving for:
\begin{equation}
  \max_\theta \; \operatorname{TPR}\!\bigl(\tau^*(\theta)\bigr)
  \quad \text{s.t.} \quad \operatorname{FPR}\!\bigl(\tau^*(\theta)\bigr) = \alpha,
  \label{eq:objective}
\end{equation}

where $\tau^*(\theta)$ is the threshold that achieves exactly $\operatorname{FPR} = \alpha$
for the current model $f_\theta$.
Conversley, when we are minimizing $\operatorname{FPR}$ subject to a fixed $\operatorname{TPR}$ we are solving for:

\begin{equation}
  \min_\theta \; \operatorname{FPR}\!\bigl(\tau^*(\theta)\bigr)
  \quad \text{s.t.} \quad \operatorname{TPR}\!\bigl(\tau^*(\theta)\bigr) = \alpha,
  \label{eq:objective}
\end{equation}

The ROLL framework, introduced in
\Cref{sec:roll-framework}, provides a method of differentiable surrogate for these objectives.

%------------------------------------------------

\section{The ROLL Framework}
\label{sec:roll-framework}

In order to allow gradient-based learning methods to optimize TPR or FPR directly, we introduce
ROLL - Rate Optimized Likelyhood Loss.

The core idea of ROLL is as follows: when training, the user selects that they would like to optimize
TPR given some fixed FPR. Then, during training, on every training step, the actual TPR is calculated for
the given FPR, and the fitting algorithm tries to maximize the TPR directly. This way, while training, we gain direct control over the performance at the threshold that we care about - avoiding needing to pick out a threshold later.

In Neyman-Pearson terms~\cite{tong2016neyman}, our algorithm wishes to directly minimize the Type-1 error given a fixed Type-2 error, or vice versa.

In practice, however, things are not so simple.

\subsection{The Differentiability Problem}
\label{sec:differentiability-problem}

% TPR and FPR are defined via threshold comparisons on model scores, making them
% piecewise-constant functions of the model parameters with zero gradient almost
% everywhere — standard backpropagation cannot move through them. Show this
% concretely with the step-function definition of TPR/FPR and explain why
% common surrogates (AUC proxies, hinge-based losses) only indirectly target
% the operating point a practitioner cares about.

Directly optimizing TPR for FPR, or vice versa, is an indefferentiable problem.

For this sub-section, we will reffer to an example of maximizing TPR given a fixed FPR,
though this can be reversed.

For any loss $\mathcal{L}(\theta)$, in order
to continue fitting our model, we must derive some gradient

$\frac{\partial \mathcal{L}(\theta)}{\partial f_\theta(\mathbf{x}_i)}$ where $f_\theta(\mathbf{x}_i)$ is the score
given to data point $\mathbf{x}_i$ by our model $f_\theta$.
There are two inherent steps built-in to our methods:

\begin{itemize}
    \item Based on fixed rate, calculate threshold that gives said rate.
    For example, when fixing FPR = $\alpha$, find some threshold $\tau$ s.t. $\mathbb{E}[1_{f_{\theta}\mathbf{x}_i > \tau | y_i = 0}] = \alpha$
    \item Given the calculated threshold, calculate the rate we would like to maximize
    (in our example, maximize TPR)
\end{itemize}

TPR or FPR calculation, given a threshold, is non-differentiable.
The reason for that is that the TPR and FPR as a function of $f_\theta(\mathbf{x}_i)$ are
non-continuous. If $f_\theta(\mathbf{x}_i) \neq \tau$, that is, the score is not exactly on the threshold, then changing it
by some $\varepsilon$ will not affect either TPR or FPR at all, and the derivative of the function w.r.t.\ $f_\theta(\mathbf{x}_i)$ will be zero. Likewise,
if $f_\theta(\mathbf{x}_i) = \tau$, the derivative of TPR/FPR w.r.t.\ $f_\theta(\mathbf{x}_i)$ becomes infinite. This derivative cannot be used
to continuously fit our model.

This necessitates a different approach, to make these calculations differentiable w.r.t the scores themselves, in order to fit this type of optimization into a learning platform.

\subsection{Score Distribution Fitting as a Solution}
\label{sec:score-distribution-fitting}

% The key insight: instead of thresholding scores directly, fit a differentiable
% probability distribution to the positive-class and negative-class score outputs.
% TPR and FPR then become integrals of that distribution above/below a threshold,
% which are smooth and differentiable in the distribution parameters — and through
% them, in the model weights. Explain intuitively why this preserves the semantics
% of TPR/FPR while making the objective amenable to gradient descent.

The way to overcome this problem in differentiability is not to calculate the TPR and FPR
directly, but to estimate them by fitting probability distributions to the score output of the model at
every step. Contiguous probability distributions are contiguous w.r.t their defining parameters, and a derivation of the parameters w.r.t the distributions themselves can be made.
Additionally, we can calculate the TPR, FPR given the probability distributions CDF and ICDF functions. If we can gaurantee these functions to be differentiable, which proved to be the main challenge of this work, we can gaurantee the entire process to be differentiable. Once we do so, a derivative of the scores themselves w.r.t. the loss can be acheived.


\subsection{General ROLL Formulation and Derivation}
\label{sec:roll-formulation}

% Present the abstract ROLL loss in terms of a generic distribution $\mathcal{F}$
% fitted to model scores, without yet committing to a specific family. Derive the
% expression for TPR@FPR$_0$ as a function of $\mathcal{F}$'s parameters, and show
% that minimizing the loss is equivalent to maximizing TPR subject to the FPR
% constraint. Include the full derivation here; this is the core theoretical
% contribution of the chapter.

Using the dataset $\mathcal{D} = \{(\mathbf{x}_i, y_i)\}_{i=1}^{n}$ as introduced in
\Cref{sec:problem-formulation}, let $\mathcal{B} \subseteq \mathcal{D}$ denote a training batch
over which we compute the loss.

We then subdivide the batch into positive and negative samples:
%
\begin{align*}
  \mathcal{B}_1 &= \{(\mathbf{x}_i, y_i) \in \mathcal{B} \mid y_i = 1\} \\
  \mathcal{B}_0 &= \{(\mathbf{x}_i, y_i) \in \mathcal{B} \mid y_i = 0\}
\end{align*}
%
and extract the corresponding model score vectors:
%
\begin{equation*}
  \mathbf{s}^{(1)} = \bigl(f_\theta(\mathbf{x}_i)\bigr)_{(\mathbf{x}_i,\, y_i)\,\in\,\mathcal{B}_1},
  \qquad
  \mathbf{s}^{(0)} = \bigl(f_\theta(\mathbf{x}_i)\bigr)_{(\mathbf{x}_i,\, y_i)\,\in\,\mathcal{B}_0}
\end{equation*}

We then fit a distribution family $\mathcal{F} = \{P_\phi : \phi \in \Phi\}$ separately to each
score vector in order to obtain a differentiable model of each class's score distribution:
%
\begin{equation*}
  \hat{\phi}_1 = \operatorname{Fit}_{\mathcal{F}}\!\left(\mathbf{s}^{(1)}\right),
  \qquad
  \hat{\phi}_0 = \operatorname{Fit}_{\mathcal{F}}\!\left(\mathbf{s}^{(0)}\right)
\end{equation*}
%
Let $\hat{F}_1$ and $\hat{F}_0$ denote the CDFs of $P_{\hat{\phi}_1}$ and $P_{\hat{\phi}_0}$,
respectively. Specific choices of $\mathcal{F}$ (Gaussian, Beta, KDE) are discussed in
\Cref{sec:roll-instantiations}.

Using this notation, at a threshold $\tau$:
\begin{align}
  \operatorname{TPR}(\tau) &= 1 - \hat{F}_1(\tau), \label{eq:tpr-cdf} \\
  \operatorname{FPR}(\tau) &= 1 - \hat{F}_0(\tau). \label{eq:fpr-cdf}
\end{align}t

Let $\hat{F}_1^{-1}$ and $\hat{F}_0^{-1}$ denote the quantile functions of $P_{\hat{\phi}_1}$ and
$P_{\hat{\phi}_0}$, respectively. The threshold achieving a target TPR of $\alpha$ follows directly
from \eqref{eq:tpr-cdf}:
\[
  \operatorname{TPR}(\tau) = \alpha \;\Longrightarrow\; 1 - \hat{F}_1(\tau) = \alpha
  \;\Longrightarrow\; \tau = \hat{F}_1^{-1}(1 - \alpha)
\]
and similarly for a target FPR of $\alpha$, from \eqref{eq:fpr-cdf}:
\[
  \operatorname{FPR}(\tau) = \alpha \;\Longrightarrow\; 1 - \hat{F}_0(\tau) = \alpha
  \;\Longrightarrow\; \tau = \hat{F}_0^{-1}(1 - \alpha)
\]

This yields two versions of our objective. Substituting the ICDF threshold into the rate
expressions, our estimated TPR at a fixed FPR of $\alpha$ is $1 - \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))$.
Therefore, to maximise TPR we minimise:
\begin{equation}
  \mathcal{L}_{\text{ROLL-TPR@FPR}}(f_{\theta}(\mathcal{B}) ; \alpha) = \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))
  \label{eq:roll-tpr-at-fpr}
\end{equation}

Our estimated FPR at a fixed TPR of $\alpha$ is $1 - \hat{F}_0(\hat{F}_1^{-1}(1 - \alpha))$.
Therefore, to minimise FPR:
\begin{equation}
  \mathcal{L}_{\text{ROLL-FPR@TPR}}(f_{\theta}(\mathbf{X}) ; \alpha) = 1 - \hat{F}_0(\hat{F}_1^{-1}(1 - \alpha))
  \label{eq:roll-fpr-at-tpr}
\end{equation}

Observe that \eqref{eq:roll-tpr-at-fpr} and \eqref{eq:roll-fpr-at-tpr} share the same
inner structure: each evaluates one empirical CDF at the $(1-\alpha)$-quantile of the other.
The two objectives are interchangeable at the implementation level: replacing
$f_\theta(\mathbf{x})$ with $-f_\theta(\mathbf{x})$ and $y$ with $1 - y$ swaps the roles
of the two populations, and passing target rate $1-\alpha$ to the resulting
TPR@FPR objective recovers the FPR@TPR objective at rate $\alpha$. Because
the two objectives are therefore interchangeable at the implementation level, all derivations
and implementation details below are given for \eqref{eq:roll-tpr-at-fpr} only;
\eqref{eq:roll-fpr-at-tpr} follows by the same transformation applied to the inputs, with
the target rate complemented to $1-\alpha$.
\Cref{fig:roll-principle} provides a geometric view of \eqref{eq:roll-tpr-at-fpr}.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.85\textwidth]{content/method/figures/roll_principle.png}
  \caption{Geometric picture of the ROLL TPR@FPR objective. \textit{Upper}: score PDFs
    for the negative (blue, $y=0$) and positive (orange, $y=1$) classes. The operating
    threshold $\tau$ (dashed) is placed at the $(1-\alpha)$-quantile of the negative
    class, so that exactly $\alpha$ fraction of negative scores exceed it ($\operatorname{FPR}=\alpha$).
    The shaded area under the positive-class PDF to the left of $\tau$ is
    $\hat{F}_1(\tau)$ --- the ROLL loss we minimize. Tick marks along the bottom represent
    observed positive-class scores. \textit{Lower}: corresponding CDFs $\hat{F}_0$ (blue)
    and $\hat{F}_1$ (orange). Reading horizontally at height $1-\alpha$ on $\hat{F}_0$
    gives $\tau$; reading $\hat{F}_1(\tau)$ off $\hat{F}_1$ at the same $\tau$ gives the
    loss; the distance from $\hat{F}_1(\tau)$ to $1$ is the resulting TPR.}
  \label{fig:roll-principle}
\end{figure}

We now derive the gradient of \eqref{eq:roll-tpr-at-fpr} with respect to $f_\theta(\mathbf{x}_i)$.

If $y_i = 1$, then $(\mathbf{x}_i, y_i) \in \mathcal{B}_1$ and $f_\theta(\mathbf{x}_i)$ has no
effect on $\hat{F}_0^{-1}(1-\alpha)$. Therefore:
\begin{equation}
  \left.\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}(f_\theta(\mathcal{B}) ; \alpha)}
             {\partial f_\theta(\mathbf{x}_i)}\right|_{y_i = 1}
  = \frac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial f_\theta(\mathbf{x}_i)}
\end{equation}

If $y_i = 0$, then $(\mathbf{x}_i, y_i) \in \mathcal{B}_0$ and $f_\theta(\mathbf{x}_i)$ affects
$\hat{F}_0^{-1}(1-\alpha)$ instead. Applying the chain rule:
\begin{equation}
  \left.\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}(f_\theta(\mathbf{X}) ; \alpha)}
             {\partial f_\theta(\mathbf{x}_i)}\right|_{y_i = 0}
  = \frac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial \hat{F}_0^{-1}(1-\alpha)}
    \cdot \frac{\partial \hat{F}_0^{-1}(1-\alpha)}{\partial f_\theta(\mathbf{x}_i)}
\end{equation}

Unifying both cases, the general gradient applicable to all ROLL instantiations is:
\begin{equation}
  \frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}(f_\theta(\mathbf{X}) ; \alpha)}
       {\partial f_\theta(\mathbf{x}_i)} =
  \begin{cases}
    \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial f_\theta(\mathbf{x}_i)}
      & \text{if } y_i = 1 \\[10pt]
    \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial \hat{F}_0^{-1}(1-\alpha)}
    \cdot \dfrac{\partial \hat{F}_0^{-1}(1-\alpha)}{\partial f_\theta(\mathbf{x}_i)}
      & \text{if } y_i = 0
  \end{cases}
  \label{eq:roll-gradient}
\end{equation}
The concrete form of each partial derivative depends on the choice of distribution family
$\mathcal{F}$, and is derived for each instantiation in \Cref{sec:roll-instantiations}.

\subsection{ROLL variant - direct AUC optimisation}

In addition to optimising for a specific rate, ROLL provides us an opportunity to
optimize the AUC of the ROC directly.

The AUC is defined as the area under the curve of the ROC. The ROC is a function that
for every FPR gives us the performing TPR of the model we have fitted.

Formally, for the function $TPR(\alpha)$ as defined above, we may define AUC as:

\begin{equation}
  \operatorname{AUC} = \int_{0}^{1}\operatorname{TPR}(\alpha) d \alpha
\end{equation}

The AUC is something we would like to maximize. In order to do so, we would like to
minimize the area over the curve, which is $1-AUC$.

We note that $1-\operatorname{TPR}(\alpha) = \mathcal{L}(TPR@FPR)$.

From here, we would like to minimize:

\begin{equation}
  \mathcal{L}_{AOC} = \int_0^1\mathcal{L}_{\text{ROLL-TPR@FPR}}(\alpha) d \alpha
\end{equation}

A direct integration is not part of this work. However, integration can be approximated. Instead
of integrating directly, it is possible to evaluate the loss for every FPR between 0 and 1, sum them, and re-normalize.

Therefor, subdividing the interval $(0, 1)$ to 99 steps (every alpha starting from $0.01$ and ending with $0.99$, since a value of $0$ or $1$ would be illegal), our loss for minimizing AOC becomes:

\begin{equation}
  \mathcal{L}_{ROLL-AOC} = \frac{1}{99}\sum_{i=1}^{99}\mathcal{L}_{ROLL-TPR@FPR}\left(\frac{i}{100}\right)
  \label{eq:roll-aoc}
\end{equation}

The gradient per score then simply becomes the average gradient over every threshold.

\begin{equation}
  \frac{\partial \mathcal{L}_{ROLL-AOC}(f_\theta(\mathbf{X}))}
       {\partial f_\theta(\mathbf{x}_i)} = \frac{1}{99}\sum_{i=1}^{99}\left(\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}\left(f_\theta(\mathbf{X}) ; \frac{i}{100}\right)}
       {\partial f_\theta(\mathbf{x}_i)} \right)
     \label{eq:roll-aoc-gradient}
\end{equation}


%------------------------------------------------

\section{Distribution Instantiations}
\label{sec:roll-instantiations}

% Each subsection below instantiates the general ROLL framework with a specific
% distributional assumption. Gaussian and Beta are included as parametric
% special cases; KDE is the primary contribution and receives the fullest treatment.

\subsection{Gaussian ROLL}
\label{sec:roll-gaussian}

% Assume positive- and negative-class scores are Gaussian distributed. The
% TPR/FPR integrals reduce to closed-form expressions in terms of the normal
% CDF, giving a simple and fast-to-compute instantiation of ROLL. Note the
% limitations: the Gaussian assumption can be violated in practice, motivating
% the more flexible instantiations below.

A simple method of implementing the ROLL loss is to approximate the distribution
of the score distributions as gaussian. This method is simple, as there exist
simple closed-form estimations for the parameters of gaussian distributions, and
they are differentiable.

While the distributions of scores cannot be gauranteed to follow any form, there are
some instances where the score distributions might be similar or exactly gaussian.
For example, when finding linear separability between gaussian distributions, the resulting
scores will be distributed normally, and this method will fit exactly.

\subsubsection{Forward}
\label{sec:roll-gaussian-forward}

% Instantiate \eqref{eq:roll-tpr-at-fpr} under Gaussian score distributions.
% Each class's scores are modelled as $\mathcal{N}(\mu_k, \sigma_k^2)$, fitted
% by MLE. The CDF is the standard normal CDF $\Phi$, giving a closed-form loss.

Firstly, we must estimate the parameters to fit the gaussian distributions
to the true and false populations of the score output.

\begin{equation}
  \mu_0 = \frac{1}{|\mathcal{B}_0|}\sum_{\mathbf{x}_j \in \mathcal{B}_0}{f_{\theta}(\mathbf{x}_j)}
  \qquad
  \mu_1 = \frac{1}{|\mathcal{B}_1|}\sum_{\mathbf{x}_j \in \mathcal{B}_1}{f_{\theta}(\mathbf{x}_j)}
\end{equation}

\begin{equation}
  \sigma_0^2 = \frac{1}{|\mathcal{B}_0|}\sum_{\mathbf{x}_j \in \mathcal{B}_0}(f_\theta(\mathbf{x}_j) - \mu_0)^2
  \qquad
  \sigma_1^2 = \frac{1}{|\mathcal{B}_1|}\sum_{\mathbf{x}_j \in \mathcal{B}_1}(f_\theta(\mathbf{x}_j) - \mu_1)^2
\end{equation}

Once these parameters are estimated, we can then define our ROLL objective as required.

The CDF of the normal distribution with mean $\mu$ and standard deviation $\sigma$ is:
\[
  F(x) = \frac{1}{2}\left[1 + \text{erf}\!\left(\frac{x - \mu}{\sigma\sqrt{2}}\right)\right]
\]
and its inverse (quantile function) is:
\[
  F^{-1}(p) = \mu + \sigma\sqrt{2}\,\text{ierf}(2p - 1)
\]
where $\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$ is the Gauss
error function and $\operatorname{ierf}$ denotes its functional inverse, satisfying
$\operatorname{ierf}(\operatorname{erf}(x)) = x$. Neither has a closed form, but both are
available in standard numerical libraries (e.g.\ \texttt{torch.erf} and
\texttt{torch.erfinv}). Plugging into the general ROLL framework, the threshold
$\tau$ achieving FPR $= \alpha$ and the resulting loss are:
\begin{align}
  \tau &= \mu_0 + \sigma_0\sqrt{2}\,\text{ierf}(1 - 2\alpha) \nonumber \\
  \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{GAUSSIAN}}(f_{\theta}(\mathcal{B}) ; \alpha)
    &= \frac{1}{2}\left[1 + \text{erf}\!\left(\frac{\tau - \mu_1}{\sigma_1\sqrt{2}}\right)\right]
  \label{eq:roll-tpr-at-fpr-gaussian}
\end{align}

\subsubsection{Gradient Derivation}
\label{sec:roll-gaussian-backward}

The full derivation is given in Appendix~\ref{appendix:gauss-grad}.
Letting $\tau = \hat{F}_0^{-1}(1-\alpha) = \mu_0 + \sigma_0\sqrt{2}\,\operatorname{ierf}(1-2\alpha)$, the gradient is:

\begin{equation}
  \frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{GAUSSIAN}}}{\partial f_\theta(\mathbf{x}_i)} =
  \begin{cases}
    \dfrac{-1}{|\mathcal{B}_1|\,\sigma_1\sqrt{2\pi}}
    \exp\!\left(-\dfrac{(\tau-\mu_1)^2}{2\sigma_1^2}\right)
    \!\left(1 + \dfrac{(\tau-\mu_1)(f_\theta(\mathbf{x}_i)-\mu_1)}{\sigma_1^2}\right)
      & \text{if } y_i = 1 \\[18pt]
    \dfrac{1}{|\mathcal{B}_0|\,\sigma_1\sqrt{2\pi}}
    \exp\!\left(-\dfrac{(\tau-\mu_1)^2}{2\sigma_1^2}\right)
    \!\left(1 + \dfrac{(\tau-\mu_0)(f_\theta(\mathbf{x}_i)-\mu_0)}{\sigma_0^2}\right)
      & \text{if } y_i = 0
  \end{cases}
  \label{eq:gauss-grad-combined}
\end{equation}

\subsection{Beta ROLL}
\label{sec:roll-beta}

Beta ROLL applies when model scores are constrained to the open interval $(0,1)$,
as is the case when $f_\theta$ uses a sigmoid output activation.  The Beta distribution
is supported on $(0,1)$ and its two shape parameters allow it to capture a wide range
of unimodal and skewed score distributions that a Gaussian cannot represent.

\subsubsection{Forward}
\label{sec:roll-beta-forward}

Given score vectors $\mathbf{s}^{(k)} \in (0,1)^{|\mathcal{B}_k|}$ for class $k$, let
$\mathbf{t}^{(k)} = \mathbf{1} - \mathbf{s}^{(k)}$ denote the complementary scores.
For any vector $\mathbf{v}$, write $\bar{v} = \frac{1}{|\mathbf{v}|}\sum_i v_i$.
We estimate the shape parameters using the closed-form log-moment estimator
of~\cite{chen2022beta}: define for each class $k$,

\begin{equation}
  D_k = \overline{s^{(k)}\!\ln s^{(k)}} - \bar{s}^{(k)}\cdot\overline{\ln s^{(k)}}
      + \overline{t^{(k)}\!\ln t^{(k)}} - \bar{t}^{(k)}\cdot\overline{\ln t^{(k)}}
  \label{eq:beta-denom}
\end{equation}

\begin{equation}
  \hat{a}_k = \frac{\bar{s}^{(k)}}{D_k}, \qquad \hat{b}_k = \frac{\bar{t}^{(k)}}{D_k}
  \label{eq:beta-params}
\end{equation}

These estimators follow from two simultaneous moment conditions:
$\mathbb{E}[X] = a/(a+b)$ and
$\operatorname{Cov}(X,\ln X) + \operatorname{Cov}(Y,\ln Y) = 1/(a+b)$,
where $Y=1-X\sim\operatorname{Beta}(b,a)$.
Replacing expectations by sample means and solving in closed form yields~\eqref{eq:beta-params}.

The CDF of $\operatorname{Beta}(\hat{a}_k,\hat{b}_k)$ at $x\in(0,1)$ is the
regularised incomplete beta function:
\begin{equation}
  \hat{F}_k(x) = I_x(\hat{a}_k,\hat{b}_k)
               = \frac{B(x;\,\hat{a}_k,\hat{b}_k)}{B(\hat{a}_k,\hat{b}_k)},
  \qquad B(x;\,a,b) = \int_0^x u^{a-1}(1-u)^{b-1}\,\mathrm{d}u
  \label{eq:beta-cdf}
\end{equation}

Substituting into the general ROLL framework \eqref{eq:roll-tpr-at-fpr}:
\begin{equation}
  \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{BETA}}(f_\theta(\mathcal{B});\,\alpha)
  = I_{\hat\tau}(\hat{a}_1,\hat{b}_1),
  \qquad
  \hat\tau = I^{-1}_{1-\alpha}(\hat{a}_0,\hat{b}_0)
  \label{eq:roll-tpr-at-fpr-beta}
\end{equation}
where $I^{-1}_{1-\alpha}(a,b)$ denotes the $(1-\alpha)$-quantile of $\operatorname{Beta}(a,b)$.

\subsubsection{Gradient Derivation}
\label{sec:roll-beta-backward}

Unlike the Gaussian case, we do not derive closed-form gradients for Beta ROLL.
The regularised incomplete beta function $I_x(a,b)$ and its inverse are smooth
functions of all their arguments and are available as differentiable operations in
PyTorch (\texttt{torch.special.betainc} and \texttt{torch.special.betaincinv}).
Gradients of \eqref{eq:roll-tpr-at-fpr-beta} with respect to $f_\theta(\mathbf{x}_i)$
are therefore propagated via automatic differentiation through the estimator
\eqref{eq:beta-params}, the CDF evaluation, and the quantile inversion.


\subsection{KDE ROLL}
\label{sec:roll-kde}

% KDE background: see \Cref{sec:kde-background} in \Cref{chapter:related_work}.
% The primary instantiation: non-parametric, no assumption on score shape.
% Discuss the trade-off between flexibility and computational cost, and how
% the bandwidth (see \Cref{sec:kde-bandwidth}) controls bias-variance.

Unfortunately, the score output of trained models remains unpredictable. A solution for this is to use a Kernel Density Estimation (KDE) for probability
estimation. This benefits from being far more adaptable to real-world distributions of outputs of our trained models. The tradeoff of course is computational
complexity - computing the gradient derivation in such a case is non-trivial.

\subsubsection{Forward}
\label{sec:roll-kde-forward}

% Instantiate \eqref{eq:roll-tpr-at-fpr} under KDE. The CDF becomes a sum of
% kernel CDFs; the ICDF is computed numerically. Derive the loss expression
% explicitly and discuss computational cost relative to the parametric cases.

As before, to calculate the loss, we calculate:


\[
  \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{KDE}}(f_{\theta}(\mathcal{B}) ; \alpha) = \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))
\]

To calculate this, we first define the kernel function as $\sigma '$, and the CDF of the kernel function as $\sigma$. The approach used here is to define $\sigma$ as a sigmoid function,.

\begin{equation}
  \sigma(x ; v) = \left( 1 + \exp(-vx)\right)^{-1} \qquad \sigma'(x ; v) = \frac{v \cdot \exp(-v \cdot x)}{(1 + \exp(-vx))^2}
\end{equation}

Where $v$ is the bandwidth parameter. This affects the calculation. We calculate $v_0$ as the bandwidth for the false population and $v_1$ for the true population. Bandwidth calculation
will be discussed later.

We then denote $\sigma_1(x) = \sigma(x ; v_1), \sigma_0(x) = \sigma(x ; v_0)$ to differentiate between the kernel functions for the true and false population, which only differentiate by bandwidth.

The CDF of the KDE function then becomes:

\[
  \hat{F}_{KDE}(\tau ; \mathbf{X}) = \frac{1}{|X|}\sum_i \sigma(\tau - x_i)
\]

While the CDF has a closed form, its inverse has none and must be found numerically.
We use bisection, which requires no derivative evaluation and is guaranteed to converge
for any strictly monotone CDF; the bracket construction and iteration are detailed in
Appendix~\ref{appendix:kde-bisection}.

We can now relatively easily calculate $\hat{F}_1(\tau ; \mathbf{X}_1)$ using the formulat for $\hat{F}_{KDE}(\tau ; \mathbf{X})$ above.

Importantly, we must save the calculation of $\tau$ for the backward derivation, as well as the calculated bandwidths $v_1, v_0$.


\subsubsection{Gradient Derivation}
\label{sec:roll-kde-backward}

% The primary technical contribution. Standard autograd cannot differentiate
% through the numerical ICDF inversion; describe the custom backward pass.
% Gradients flow through the kernel evaluations back to the model scores.
% This is the section that connects to \Cref{sec:roll-backward} motivation.

The derivation is given in Appendix~\ref{appendix:kde-grad}; letting $\tau = \hat{F}_0^{-1}(1-\alpha ; \mathcal{B}_0)$, the gradient is:

\begin{equation}
  \frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}}{\partial f_\theta(\mathbf{x}_i)} =
  \begin{cases}
    -\dfrac{1}{|\mathcal{B}_1|}\,\sigma_1'(\tau - f_{\theta}(\mathbf{x}_i))
      & \text{if } y_i = 1 \\[14pt]
    +\dfrac{1}{|\mathcal{B}_1|}\!\displaystyle\sum_{\mathbf{x}_j \in \mathcal{B}_1}\!\sigma_1'(\tau - f_{\theta}(\mathbf{x}_j))
    \;\cdot\;
    \dfrac{\sigma_0'(\tau - f_{\theta}(\mathbf{x}_i))}{\displaystyle\sum_{\mathbf{x}_j \in \mathcal{B}_0}\sigma_0'(\tau - f_{\theta}(\mathbf{x}_j))}
      & \text{if } y_i = 0
  \end{cases}
  \label{eq:kde-grad-combined}
\end{equation}

where $\tau = \hat{F}_0^{-1}(1-\alpha ; \mathcal{B}_0)$.

The explicit sigmoid-kernel form of \eqref{eq:kde-grad-combined} — forward pass CDF, efficient gradient, and implementation notes — is given in Appendix~\ref{appendix:kde-sigmoid}.
Positive-class samples are weighted by their individual kernel
PDF at $\tau$; negative-class samples are weighted by the mean positive-class kernel
PDF scaled by each sample's share of the negative-class kernel mass.

%------------------------------------------------

\subsection{Properties of the ROLL Formulation}
\label{sec:roll-properties}

% State and prove (or sketch) key properties: consistency of the estimator as
% the number of samples grows, monotonicity of the loss with respect to the
% TPR/FPR trade-off, and behaviour at the constraint boundary. Discuss what
% guarantees ROLL provides and under what assumptions they hold.
% NOTE - these properties are auto-generated, consider suggestions.

A structural consequence of the ROLL formulation is that gradient mass is automatically
balanced between the two classes, regardless of their relative sizes in the batch.
This stands in contrast to standard losses such as cross-entropy, where the total gradient
contribution from each class is proportional to its count, making heavy class imbalance
a direct source of biased updates. Therefor, instead of needing to compensate for this
with loss-weighting per-class, fitting updates are made based equally off the results
of both classes.

\begin{proposition}[Gradient balance]
\label{prop:gradient-balance}
Let $\mathcal{F}$ be any distribution family whose CDF estimator is translation-invariant,
meaning $\hat{F}_k(t\,;\,\mathbf{s}+\delta) = \hat{F}_k(t-\delta\,;\,\mathbf{s})$
for all $\delta \in \mathbb{R}$. Then for any batch $\mathcal{B} = \mathcal{B}_0 \cup \mathcal{B}_1$:
\begin{equation}
  \sum_{\mathbf{x}_i \in \mathcal{B}_1}
    \frac{\partial \mathcal{L}_{\emph{\text{ROLL-TPR@FPR}}}}{\partial f_\theta(\mathbf{x}_i)}
  \;=\;
  -\!\sum_{\mathbf{x}_i \in \mathcal{B}_0}
    \frac{\partial \mathcal{L}_{\emph{\text{ROLL-TPR@FPR}}}}{\partial f_\theta(\mathbf{x}_i)}
  \label{eq:grad-balance}
\end{equation}
with both sides equal in magnitude to $\dfrac{\partial \hat{F}_1(\tau)}{\partial \tau}$,
independently of $|\mathcal{B}_0|$ and $|\mathcal{B}_1|$.
\end{proposition}

\begin{proof}
We treat each class separately via a uniform-shift argument.

\medskip\noindent\textit{Sum over $\mathcal{B}_0$.}
From \Cref{eq:roll-gradient}, for $y_i = 0$:
\[
  \frac{\partial \mathcal{L}}{\partial f_\theta(\mathbf{x}_i)}
  = \frac{\partial \hat{F}_1(\tau)}{\partial \tau}
    \cdot \frac{\partial \tau}{\partial f_\theta(\mathbf{x}_i)}
\]
We claim $\sum_{\mathbf{x}_i \in \mathcal{B}_0}\frac{\partial \tau}{\partial f_\theta(\mathbf{x}_i)} = 1$.
Shift all negative scores uniformly: $f_\theta(\mathbf{x}_i) \to f_\theta(\mathbf{x}_i) + \delta$ for
$\mathbf{x}_i \in \mathcal{B}_0$. By translation invariance, $\hat{F}_0$ simply translates, so
$\tau = \hat{F}_0^{-1}(1-\alpha)$ shifts by exactly $\delta$. Differentiating with respect to $\delta$ at
$\delta = 0$:
\[
  1 = \frac{\mathrm{d}\tau}{\mathrm{d}\delta}\bigg|_{\delta=0}
    = \sum_{\mathbf{x}_i \in \mathcal{B}_0} \frac{\partial \tau}{\partial f_\theta(\mathbf{x}_i)}
\]
Summing the $y_i = 0$ gradients therefore gives:
\begin{equation}
  \sum_{\mathbf{x}_i \in \mathcal{B}_0}
    \frac{\partial \mathcal{L}}{\partial f_\theta(\mathbf{x}_i)}
  = \frac{\partial \hat{F}_1(\tau)}{\partial \tau} \cdot 1
  = \frac{\partial \hat{F}_1(\tau)}{\partial \tau}
  \label{eq:grad-balance-b0}
\end{equation}

\medskip\noindent\textit{Sum over $\mathcal{B}_1$.}
For $y_i = 1$, $\frac{\partial \mathcal{L}}{\partial f_\theta(\mathbf{x}_i)} = \frac{\partial \hat{F}_1(\tau)}{\partial f_\theta(\mathbf{x}_i)}$.
Shifting all positive scores by $\delta$ and applying translation invariance of $\hat{F}_1$:
$\hat{F}_1(\tau\,;\,\mathbf{s}+\delta\mathbf{1}) = \hat{F}_1(\tau-\delta\,;\,\mathbf{s})$.
Differentiating at $\delta = 0$:
\[
  \sum_{\mathbf{x}_i \in \mathcal{B}_1}
    \frac{\partial \hat{F}_1(\tau)}{\partial f_\theta(\mathbf{x}_i)}
  = -\frac{\partial \hat{F}_1(\tau)}{\partial \tau}
\]
Combining with \eqref{eq:grad-balance-b0} gives \eqref{eq:grad-balance}.
\end{proof}

The translation-invariance assumption holds for both the Gaussian distribution,
as well as the KDE-based distribution, but not for the Beta distribution.
For Gaussian MLE, a uniform shift in scores moves $\mu_k$ by the same amount while
leaving $\sigma_k$ unchanged, which translates $\hat{F}_k$; for KDE, each kernel term is
evaluated at $\tau - f_\theta(\mathbf{x}_j)$, so a uniform shift in the $f_\theta(\mathbf{x}_j)$
is equivalent to an equal shift in $\tau$. The condition is necessary: without it, a uniform
score shift need not produce a rigid translation of $\hat{F}_k$, so the shift argument in the
proof breaks down and the gradient sums need not be equal.

The Beta distribution only accepts values in the interval $\[0, 1\]$. Since it
is mapped only over an interval, the translation-invariance assumption does not hold
for this distribution.

\paragraph{Gradient locality.}
\label{para:gradient-locality}

A second structural consequence of the ROLL-KDE formulation is that gradient mass is
\emph{concentrated near the operating threshold $\tau$}. Samples whose scores lie
close to $\tau$ receive larger gradient updates, while samples
already far on the correct side contribute very little.

This follows directly from \Cref{eq:kde-grad-combined}. For the positive class ($y_i = 1$),
the gradient magnitude is
\[
  \left|\frac{\partial \mathcal{L}}{\partial f_\theta(\mathbf{x}_i)}\right|
  = \frac{1}{|\mathcal{B}_1|}\,\sigma_1'(f_\theta(\mathbf{x}_i) - \tau),
\]
which is a bell-shaped kernel PDF that peaks when $f_\theta(\mathbf{x}_i) = \tau$ and
decays to zero as the score moves away from the threshold in either direction.
For the negative class ($y_i = 0$), the gradient is proportional to
$\sigma_0'(\tau - f_\theta(\mathbf{x}_i))$ (normalized over $\mathcal{B}_0$), which is
again bell-shaped and centred on $\tau$.

The effect is that each gradient step focuses the model's attention on the region that
directly determines the operating-point performance: samples that could plausibly be
re-ranked relative to the threshold at the current step. Samples classified with strong
confidence relative to the threshold, with scores far enough away from said threshold,
receive small gradient updates, since a small change to the score of these samples will not
bring any change to the placement of the threshold itself, or the performance at said threshold.

This stands in contrast to standard losses such as cross-entropy, where the per-sample
gradient is determined by the predicted probability, with no direct dependence on an
operating threshold.

\begin{observation}[Gradient locality]
\label{obs:gradient-locality}
For KDE-ROLL, the gradient magnitude of sample $\mathbf{x}_i$ is proportional to
the kernel PDF evaluated at the distance between $f_\theta(\mathbf{x}_i)$ and the
operating threshold $\tau$. Samples nearest to $\tau$ receive the largest updates;
samples far from $\tau$ in either direction contribute negligible gradient.
\end{observation}

\Cref{fig:threshold-locality} illustrates this for a simulated two-class setting. The
upper panel shows the score distributions and the operating threshold; the lower panel
shows the gradient magnitude of each sample. Both classes exhibit a clear bell-shaped
concentration of gradient mass around $\tau$, with smooth decay on either side.

\begin{figure}[H]
  \centering
  \includegraphics[width=0.82\textwidth]{content/method/figures/threshold_weighting.pdf}
  \caption{Gradient locality in KDE-ROLL. \textit{Upper}: score distributions for the
    negative (blue, $y=0$) and positive (orange, $y=1$) classes; dashed line marks the
    operating threshold $\tau$ at FPR $= \alpha$. \textit{Lower}: gradient magnitude
    $|\partial\mathcal{L}/\partial f_\theta(\mathbf{x})|$ for each sample. Both classes
    show a bell-shaped concentration of gradient mass near $\tau$, with samples far from
    the threshold contributing negligible updates. Smooth curves are the kernel PDFs scaled
    for reference.}
  \label{fig:threshold-locality}
\end{figure}

%------------------------------------------------

\section{Implementation Considerations}
\label{sec:roll-implementation}

\subsection{Combatting Numerical Instability With Score Scaling}
\label{sec:roll-numerical-stability}

Before computing the KDE, all batch scores are divided by their overall standard
deviation:
\[
  \tilde{s}_i = \frac{f_\theta(\mathbf{x}_i)}{\hat\sigma_{\mathcal{B}}},
  \qquad
  \hat\sigma_{\mathcal{B}} = \sqrt{\operatorname{Var}(f_\theta(\mathbf{x}_i) : (\mathbf{x}_i, y_i) \in \mathcal{B})}.
\]
This keeps the values given to the sigmoid kernel ($v(\tau - \tilde{s}_j)$)
near unit scale, preventing floating-point overflow and underflow in the exponential
evaluations.

The scaling has no effect on the loss value itself or derived gradients.
Silverman's rule derives the bandwidth proportionally to the data standard deviation,
$h \propto \hat\sigma \cdot n^{-1/5}$, so dividing all scores by $\hat\sigma_\mathcal{B}$
causes the bandwidth to scale by the same factor.
The product $v \cdot (\tau - s_j)$ — the only quantity entering the sigmoid — is
therefore invariant under the rescaling.
In the backward pass, gradients with respect to the normalised scores are multiplied
by $1/\hat\sigma_\mathcal{B}$ to apply the chain rule, providing gradients mathematically
identical to those that would have been obtained without normalisation.

\subsection{Bandwidth Estimation for KDE}
\label{sec:kde-bandwidth}

% The KDE bandwidth $h$ controls how smoothly the score distribution is estimated:
% too small and the loss becomes noisy; too large and the gradient signal is
% washed out. Describe the bandwidth estimation strategy used in ROLL (e.g.
% Silverman's rule, a learned or scheduled $h$), justify the choice, and discuss
% sensitivity of results to this hyperparameter.
% NOTE - not silverman's rule, we do use scheduling, but we also use different
% estimator - see impl.

When computing KDE, bandwidth estimation strongly affects the result. Large bandwidths
produce an overly smooth CDF that fails to track the true score distribution; small
bandwidths produce a jagged CDF, yielding negligible gradients for all but the
nearest-threshold samples.

We estimate the bandwidth using the Improved Sheather-Jones (ISJ)
method~\cite{botev2010kde}, which minimises the asymptotic mean integrated squared
error via an iterative plug-in procedure.  ISJ tracks the true score distribution more
closely than closed-form rules, as illustrated in \Cref{fig:density-fit-comparison}.
Because ISJ is iterative, it can fail to converge on small or degenerate batches early
in training.  In those cases we fall back to Silverman's rule of
thumb~\cite{silverman1986density},
\[
  h = 1.06\,\hat{\sigma}\,n^{-1/5},
\]
which is closed-form and always produces a valid bandwidth.
The kernel parameter is then set to $v = 1/h$.

\subsubsection{Bandwidth Scaling for KDE}
\label{sec:kde-bandwidth-scaling}

Although we may select a valid bandwidth at each step, it tends to be
too small near the start of training, when scores are not yet
spread out and $\hat{\sigma}$ is small. This causes the KDE to undersmooth the score
distribution, producing very low gradient magnitudes and stalling convergence.

The solution was to use a bandwidth scheduler. Starting with large bandwidths resulted in gradients that spread out more evenly across all scores, and fewer non-convergence runs occurred. The bandwidth scheduler artificially scales up the bandwidth at the beginning to speed up initial training, and then tapers back to restore the intended bandwidth in order to maximise results.


% TODO: algorithm box — full training loop (forward → ROLL loss → custom backward → weight update)
