608 lines
32 KiB
TeX
608 lines
32 KiB
TeX
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\section{Problem Formulation}
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\label{sec:problem-formulation}
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% Motivation for this section — why accuracy fails under class imbalance, why
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% AUC is the natural objective under label noise and imbalance, and the
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% Neyman-Pearson framing for single-threshold optimization — is developed in
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% \Cref{chapter:related_work} (\Cref{sec:imbalanced-tpr-fpr,sec:neyman-pearson}).
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% This section establishes the notation and primary objective used throughout the chapter.
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Given a dataset $\mathcal{D} = \{(\mathbf{x}_i, y_i)\}_{i=1}^{n}$ with inputs
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$\mathbf{x}_i \in \mathcal{X}$ and binary labels $y_i \in \{0, 1\}$, we learn a
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scoring function $f_\theta : \mathcal{X} \to \mathbb{R}$ by minimising a surrogate loss on $\mathcal{D}$.
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The positive and negative subsets of $\mathcal{D}$ are:
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\[
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\mathcal{D}_1 = \{(\mathbf{x}_i, y_i) \in \mathcal{D} \mid y_i = 1\},
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\qquad
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\mathcal{D}_0 = \{(\mathbf{x}_i, y_i) \in \mathcal{D} \mid y_i = 0\}.
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\]
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Applying a threshold $\tau$ produces predictions $\hat{y} = \mathbf{1}[f_\theta(\mathbf{x}) > \tau]$.
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The \emph{true positive rate} (TPR) and \emph{false positive rate} (FPR) at threshold $\tau$ are:
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\begin{align}
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\operatorname{TPR}(\tau) &= P\!\bigl(f_\theta(\mathbf{x}) > \tau \mid y = 1\bigr), \\
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\operatorname{FPR}(\tau) &= P\!\bigl(f_\theta(\mathbf{x}) > \tau \mid y = 0\bigr).
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\end{align}
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For $\alpha \in (0,1)$, let $\tau^*(\alpha)$ denote the threshold achieving
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$\operatorname{FPR}(\tau^*(\alpha)) = \alpha$, and write
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$\operatorname{TPR}(\alpha) = \operatorname{TPR}(\tau^*(\alpha))$ for the true positive rate
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at that operating point. The curve $\alpha \mapsto \operatorname{TPR}(\alpha)$ is the ROC curve.
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The primary learning objective is to maximize its area:
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\begin{equation}
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\max_\theta\;\operatorname{AUC}(f_\theta) = \int_0^1 \operatorname{TPR}(\alpha)\,d\alpha.
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\label{eq:objective-auc}
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\end{equation}
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Maximizing AUC is the natural objective under class imbalance and label noise, where no
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single operating threshold is fixed in advance.
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As a special case, one may instead optimize a single operating point: maximize
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$\operatorname{TPR}$ at a fixed $\operatorname{FPR} = \alpha_0$, or minimize
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$\operatorname{FPR}$ at a fixed $\operatorname{TPR} = \alpha_0$.
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The ROLL framework, introduced in \Cref{sec:roll-framework}, provides differentiable
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surrogates for both the AUC objective \eqref{eq:objective-auc} and these point objectives.
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%------------------------------------------------
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\section{The ROLL Framework}
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\label{sec:roll-framework}
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In order to allow gradient-based learning methods to optimize AUC directly, we introduce
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ROLL, Rate Optimized Likelihood Loss.
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The core idea of ROLL is as follows: at each training step, differentiable distributions
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are fitted to the positive- and negative-class score outputs of $f_\theta$. AUC is then
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approximated by evaluating the estimated $\operatorname{TPR}$ at 99 evenly-spaced
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$\operatorname{FPR}$ values across $(0,1)$ and averaging, a numerical integration of
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the ROC curve. Because each per-threshold evaluation is differentiable through the fitted
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distributions, this AUC approximation can be maximized directly by gradient descent, with
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no threshold selection required at inference time.
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ROLL also supports single-threshold optimization: selecting a target
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$\operatorname{FPR} = \alpha_0$ and maximizing $\operatorname{TPR}$ at exactly that point.
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In Neyman-Pearson terms~\cite{tong2016neyman}, this corresponds to minimizing Type-I error
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at a fixed Type-II error bound.
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In practice, however, making these evaluations differentiable is not straightforward.
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\subsection{The Differentiability Problem}
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\label{sec:differentiability-problem}
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% TPR and FPR are defined via threshold comparisons on model scores, making them
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% piecewise-constant functions of the model parameters with zero gradient almost
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% everywhere — standard backpropagation cannot move through them. Show this
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% concretely with the step-function definition of TPR/FPR and explain why
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% common surrogates (AUC proxies, hinge-based losses) only indirectly target
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% the operating point a practitioner cares about.
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Directly optimizing TPR for FPR, or vice versa, is a non-differentiable problem.
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For this sub-section, we will refer to an example of maximizing TPR given a fixed FPR,
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though this can be reversed.
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For any loss $\mathcal{L}(\theta)$, in order
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to continue fitting our model, we must derive some gradient
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$\frac{\partial \mathcal{L}(\theta)}{\partial f_\theta(\mathbf{x}_i)}$ where $f_\theta(\mathbf{x}_i)$ is the score
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given to data point $\mathbf{x}_i$ by our model $f_\theta$.
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There are two inherent steps built-in to our methods:
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\begin{itemize}
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\item Based on fixed rate, calculate threshold that gives said rate.
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For example, when fixing FPR = $\alpha$, find some threshold $\tau$ such that $\mathbb{E}[\mathbf{1}[f_\theta(\mathbf{x}_i) > \tau \mid y_i = 0]] = \alpha$
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\item Given the calculated threshold, calculate the rate we would like to maximize
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(in our example, maximize TPR)
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\end{itemize}
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TPR or FPR calculation, given a threshold, is non-differentiable.
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The reason for that is that the TPR and FPR as a function of $f_\theta(\mathbf{x}_i)$ are
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non-continuous. If $f_\theta(\mathbf{x}_i) \neq \tau$, that is, the score is not exactly on the threshold, then changing it
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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,
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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
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to continuously fit our model.
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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.
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\subsection{Score Distribution Fitting as a Solution}
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\label{sec:score-distribution-fitting}
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% The key insight: instead of thresholding scores directly, fit a differentiable
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% probability distribution to the positive-class and negative-class score outputs.
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% TPR and FPR then become integrals of that distribution above/below a threshold,
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% which are smooth and differentiable in the distribution parameters — and through
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% them, in the model weights. Explain intuitively why this preserves the semantics
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% of TPR/FPR while making the objective amenable to gradient descent.
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The way to overcome this problem in differentiability is not to calculate the TPR and FPR
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directly, but to estimate them by fitting probability distributions to the score output of the model at
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every step. Continuous probability distributions are continuous with respect to their defining parameters, and a derivation of the parameters with respect to the distributions themselves can be made.
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Additionally, we can calculate the TPR, FPR given the probability distributions CDF and ICDF functions. If we can guarantee these functions to be differentiable, which proved to be the main challenge of this work, we can guarantee the entire process to be differentiable. Once we do so, a derivative of the scores themselves with respect to the loss can be achieved.
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\subsection{Per-Threshold ROLL Loss}
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\label{sec:roll-formulation}
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% Present the abstract ROLL loss in terms of a generic distribution $\mathcal{F}$
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% fitted to model scores, without yet committing to a specific family. Derive the
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% expression for TPR@FPR as a function of $\mathcal{F}$'s parameters. This
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% per-threshold loss is the building block that ROLL-AOC averages over 99 FPR values
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% to obtain the primary AUC objective.
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Using the dataset $\mathcal{D} = \{(\mathbf{x}_i, y_i)\}_{i=1}^{n}$ as introduced in
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\Cref{sec:problem-formulation}, let $\mathcal{B} \subseteq \mathcal{D}$ denote a training batch
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over which we compute the loss.
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We then subdivide the batch into positive and negative samples:
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%
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\begin{align*}
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\mathcal{B}_1 &= \{(\mathbf{x}_i, y_i) \in \mathcal{B} \mid y_i = 1\} \\
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\mathcal{B}_0 &= \{(\mathbf{x}_i, y_i) \in \mathcal{B} \mid y_i = 0\}
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\end{align*}
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%
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and extract the corresponding model score vectors:
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%
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\begin{equation*}
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\mathbf{s}^{(1)} = \bigl(f_\theta(\mathbf{x}_i)\bigr)_{(\mathbf{x}_i,\, y_i)\,\in\,\mathcal{B}_1},
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\qquad
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\mathbf{s}^{(0)} = \bigl(f_\theta(\mathbf{x}_i)\bigr)_{(\mathbf{x}_i,\, y_i)\,\in\,\mathcal{B}_0}
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\end{equation*}
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We then fit a distribution family $\mathcal{F} = \{P_\phi : \phi \in \Phi\}$ separately to each
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score vector in order to obtain a differentiable model of each class's score distribution:
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%
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\begin{equation*}
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\hat{\phi}_1 = \operatorname{Fit}_{\mathcal{F}}\!\left(\mathbf{s}^{(1)}\right),
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\qquad
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\hat{\phi}_0 = \operatorname{Fit}_{\mathcal{F}}\!\left(\mathbf{s}^{(0)}\right)
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\end{equation*}
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%
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Let $\hat{F}_1$ and $\hat{F}_0$ denote the CDFs of $P_{\hat{\phi}_1}$ and $P_{\hat{\phi}_0}$,
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respectively. The KDE instantiation is developed in \Cref{sec:roll-instantiations};
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the Gaussian and Beta instantiations, including full derivations, are in
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\Cref{appendix:gauss-grad,appendix:beta-roll}.
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Using this notation, at a threshold $\tau$:
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\begin{align}
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\operatorname{TPR}(\tau) &= 1 - \hat{F}_1(\tau), \label{eq:tpr-cdf} \\
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\operatorname{FPR}(\tau) &= 1 - \hat{F}_0(\tau). \label{eq:fpr-cdf}
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\end{align}
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Let $\hat{F}_1^{-1}$ and $\hat{F}_0^{-1}$ denote the quantile functions of $P_{\hat{\phi}_1}$ and
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$P_{\hat{\phi}_0}$, respectively. The threshold achieving a target TPR of $\alpha$ follows directly
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from \eqref{eq:tpr-cdf}:
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\[
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\operatorname{TPR}(\tau) = \alpha \;\Longrightarrow\; 1 - \hat{F}_1(\tau) = \alpha
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\;\Longrightarrow\; \tau = \hat{F}_1^{-1}(1 - \alpha)
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\]
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and similarly for a target FPR of $\alpha$, from \eqref{eq:fpr-cdf}:
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\[
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\operatorname{FPR}(\tau) = \alpha \;\Longrightarrow\; 1 - \hat{F}_0(\tau) = \alpha
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\;\Longrightarrow\; \tau = \hat{F}_0^{-1}(1 - \alpha)
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\]
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This yields two versions of our objective. Substituting the ICDF threshold into the rate
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expressions, our estimated TPR at a fixed FPR of $\alpha$ is $1 - \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))$.
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Therefore, to maximise TPR we minimise:
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\begin{equation}
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\mathcal{L}_{\text{ROLL-TPR@FPR}}(f_{\theta}(\mathcal{B}) ; \alpha) = \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))
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\label{eq:roll-tpr-at-fpr}
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\end{equation}
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Our estimated FPR at a fixed TPR of $\alpha$ is $1 - \hat{F}_0(\hat{F}_1^{-1}(1 - \alpha))$.
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Therefore, to minimise FPR:
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\begin{equation}
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\mathcal{L}_{\text{ROLL-FPR@TPR}}(f_{\theta}(\mathbf{X}) ; \alpha) = 1 - \hat{F}_0(\hat{F}_1^{-1}(1 - \alpha))
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\label{eq:roll-fpr-at-tpr}
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\end{equation}
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Observe that \eqref{eq:roll-tpr-at-fpr} and \eqref{eq:roll-fpr-at-tpr} share the same
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inner structure: each evaluates one empirical CDF at the $(1-\alpha)$-quantile of the other.
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The two objectives are interchangeable at the implementation level: replacing
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$f_\theta(\mathbf{x})$ with $-f_\theta(\mathbf{x})$ and $y$ with $1 - y$ swaps the roles
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of the two populations, and passing target rate $1-\alpha$ to the resulting
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TPR@FPR objective recovers the FPR@TPR objective at rate $\alpha$. All derivations
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and implementation details below are therefore given for \eqref{eq:roll-tpr-at-fpr} only;
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\eqref{eq:roll-fpr-at-tpr} follows by the same transformation applied to the inputs, with
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the target rate complemented to $1-\alpha$.
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\Cref{fig:roll-principle} provides a geometric view of \eqref{eq:roll-tpr-at-fpr}.
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.85\textwidth]{content/method/figures/roll_principle.png}
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\caption{Geometric picture of the ROLL TPR@FPR objective. \textit{Upper}: score PDFs
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for the negative (blue, $y=0$) and positive (orange, $y=1$) classes. The operating
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threshold $\tau$ (dashed) is placed at the $(1-\alpha)$-quantile of the negative
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class, so that exactly $\alpha$ fraction of negative scores exceed it ($\operatorname{FPR}=\alpha$).
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The shaded area under the positive-class PDF to the left of $\tau$ is
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$\hat{F}_1(\tau)$, the ROLL loss we minimize. Tick marks along the bottom represent
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observed positive-class scores. \textit{Lower}: corresponding CDFs $\hat{F}_0$ (blue)
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and $\hat{F}_1$ (orange). Reading horizontally at height $1-\alpha$ on $\hat{F}_0$
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gives $\tau$; reading $\hat{F}_1(\tau)$ off $\hat{F}_1$ at the same $\tau$ gives the
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loss; the distance from $\hat{F}_1(\tau)$ to $1$ is the resulting TPR.}
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\label{fig:roll-principle}
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\end{figure}
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We now derive the gradient of \eqref{eq:roll-tpr-at-fpr} with respect to $f_\theta(\mathbf{x}_i)$.
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If $y_i = 1$, then $(\mathbf{x}_i, y_i) \in \mathcal{B}_1$ and $f_\theta(\mathbf{x}_i)$ has no
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effect on $\hat{F}_0^{-1}(1-\alpha)$. Therefore:
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\begin{equation}
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\left.\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}(f_\theta(\mathcal{B}) ; \alpha)}
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{\partial f_\theta(\mathbf{x}_i)}\right|_{y_i = 1}
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= \frac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial f_\theta(\mathbf{x}_i)}
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\end{equation}
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If $y_i = 0$, then $(\mathbf{x}_i, y_i) \in \mathcal{B}_0$ and $f_\theta(\mathbf{x}_i)$ affects
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$\hat{F}_0^{-1}(1-\alpha)$ instead. Applying the chain rule:
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\begin{equation}
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\left.\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}(f_\theta(\mathbf{X}) ; \alpha)}
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{\partial f_\theta(\mathbf{x}_i)}\right|_{y_i = 0}
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= \frac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial \hat{F}_0^{-1}(1-\alpha)}
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\cdot \frac{\partial \hat{F}_0^{-1}(1-\alpha)}{\partial f_\theta(\mathbf{x}_i)}
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\end{equation}
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Unifying both cases, the general gradient applicable to all ROLL instantiations is:
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\begin{equation}
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\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}(f_\theta(\mathbf{X}) ; \alpha)}
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{\partial f_\theta(\mathbf{x}_i)} =
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\begin{cases}
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\dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial f_\theta(\mathbf{x}_i)}
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& \text{if } y_i = 1 \\[10pt]
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\dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial \hat{F}_0^{-1}(1-\alpha)}
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\cdot \dfrac{\partial \hat{F}_0^{-1}(1-\alpha)}{\partial f_\theta(\mathbf{x}_i)}
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& \text{if } y_i = 0
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\end{cases}
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\label{eq:roll-gradient}
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\end{equation}
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The concrete form of each partial derivative depends on the choice of distribution family
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$\mathcal{F}$; the KDE case is derived in \Cref{sec:roll-instantiations}, and the
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parametric cases are in \Cref{appendix:gauss-grad,appendix:beta-roll}.
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\subsection{ROLL-AOC: AUC Optimization}
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\label{sec:roll-aoc}
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The primary objective of ROLL is to maximize AUC directly, using the per-threshold
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loss \eqref{eq:roll-tpr-at-fpr} as a building block. Recall from
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\Cref{sec:problem-formulation} that AUC is the integral of $\operatorname{TPR}(\alpha)$
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over all FPR values:
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\begin{equation}
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\operatorname{AUC} = \int_{0}^{1}\operatorname{TPR}(\alpha)\,d\alpha.
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\end{equation}
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Maximizing AUC is equivalent to minimizing the \emph{Area Over the Curve} (AOC),
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$1 - \operatorname{AUC}$. Since $1 - \operatorname{TPR}(\alpha) =
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\mathcal{L}_{\text{ROLL-TPR@FPR}}(\alpha)$ by definition \eqref{eq:roll-tpr-at-fpr}:
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\begin{equation}
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\mathcal{L}_{\text{AOC}} = \int_0^1\mathcal{L}_{\text{ROLL-TPR@FPR}}(\alpha)\,d\alpha.
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\end{equation}
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Direct integration is not tractable, but the integral is well-approximated by averaging out sampled points. Evaluating at 99 evenly-spaced FPR values $\alpha = i/100$ for $i = 1, \ldots, 99$
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(excluding the degenerate endpoints 0 and 1), the ROLL-AOC loss is:
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\begin{equation}
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\mathcal{L}_{\text{ROLL-AOC}} = \frac{1}{99}\sum_{i=1}^{99}\mathcal{L}_{\text{ROLL-TPR@FPR}}\!\left(\frac{i}{100}\right)
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\label{eq:roll-aoc}
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\end{equation}
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The gradient follows by linearity of the sum:
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\begin{equation}
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\frac{\partial \mathcal{L}_{\text{ROLL-AOC}}(f_\theta(\mathbf{X}))}
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{\partial f_\theta(\mathbf{x}_i)} = \frac{1}{99}\sum_{i=1}^{99}\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}\!\left(f_\theta(\mathbf{X}) ; \frac{i}{100}\right)}
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{\partial f_\theta(\mathbf{x}_i)}
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\label{eq:roll-aoc-gradient}
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\end{equation}
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The concrete per-threshold gradient for the KDE instantiation is derived in
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\Cref{sec:roll-instantiations}; gradients for the parametric cases are in
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\Cref{appendix:gauss-grad,appendix:beta-roll}.
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\Cref{fig:roll-aoc-roc} gives a geometric view of this objective.
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.62\textwidth]{content/method/figures/roll_aoc_roc.pdf}
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\caption{Geometric interpretation of ROLL-AOC. The ROC curve (orange) maps
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each FPR value $\alpha$ to $\operatorname{TPR}(\alpha)$.
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\textbf{Green}: area under the curve (AUC), the quantity we maximize.
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\textbf{Red}: area over the curve
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($\mathrm{AOC} = 1 - \mathrm{AUC} = \mathcal{L}_{\text{ROLL-AOC}}$),
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the loss we minimize.
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Dots mark 10 evenly-spaced FPR sample points $\alpha_i$; vertical
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segments show the per-threshold loss contribution
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$1 - \operatorname{TPR}(\alpha_i) = \mathcal{L}_{\text{ROLL-TPR@FPR}}(\alpha_i)$
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at each point. Averaging these contributions (99 in practice, 10 shown
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for clarity) gives the value of \eqref{eq:roll-aoc}.}
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\label{fig:roll-aoc-roc}
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\end{figure}
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%------------------------------------------------
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\section{Distribution Instantiations}
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\label{sec:roll-instantiations}
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The abstract ROLL framework (\Cref{sec:roll-formulation}) accommodates any distribution
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family $\mathcal{F}$ whose CDF and quantile function are differentiable with respect to
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the model scores. As part of this work, we instantiated the framework for two parametric
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families: a Gaussian model, fitted by maximum likelihood, and a Beta model, fitted by a
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closed-form log-moment estimator~\cite{chen2022beta}. Both admit fast parameter estimation
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and differentiable CDFs; for the Gaussian case we derive analytic gradients in full, and
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for the Beta case gradients propagate via automatic differentiation. Complete derivations
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are given in \Cref{appendix:gauss-grad,appendix:beta-roll}.
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Parametric families, however, impose distributional assumptions on the score output that
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no trained model can be guaranteed to satisfy. The primary instantiation in this thesis
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is therefore non-parametric: Kernel Density Estimation (KDE) places no assumption on the
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shape of the score distribution and adapts to whatever the model produces at each training
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step. The cost is a numerical CDF inversion and a custom backward pass; the gain is a
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faithful surrogate for the ROC curve regardless of how scores evolve during training.
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\subsection{KDE ROLL}
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\label{sec:roll-kde}
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KDE estimates the score distribution directly from the observed batch scores, placing no
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assumption on their shape. It is the primary instantiation of ROLL used throughout this
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thesis: its non-parametric flexibility makes it the most faithful surrogate for the ROC
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curve when score distributions are unknown, at the cost of a numerical CDF inversion and
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a custom backward pass evaluated 99 times per training step under ROLL-AOC.
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\subsubsection{Forward}
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\label{sec:roll-kde-forward}
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% Instantiate \eqref{eq:roll-tpr-at-fpr} under KDE. The CDF becomes a sum of
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% kernel CDFs; the ICDF is computed numerically. Derive the loss expression
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% explicitly and discuss computational cost relative to the parametric cases.
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As before, to calculate the loss, we calculate:
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\[
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\mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{KDE}}(f_{\theta}(\mathcal{B}) ; \alpha) = \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))
|
|
\]
|
|
|
|
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 trivially compute $\hat{F}_1(\tau ; \mathbf{X}_1)$ using the formula for $\hat{F}_{KDE}(\tau ; \mathbf{X})$ above.
|
|
|
|
Note that $\tau$ is cached for use in the backward pass, along with the computed 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. Therefore, 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}
|
|
See \Cref{appendix: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.
|
|
|
|
\begin{corollary}[Gradient balance under ROLL-AOC]
|
|
\label{cor:grad-balance-aoc}
|
|
$\mathcal{L}_{\emph{\text{ROLL-AOC}}}$ also satisfies gradient balance: since it is a uniform
|
|
average of 99 per-threshold objectives each satisfying \eqref{eq:grad-balance}, the property
|
|
is preserved by linearity of summation.
|
|
\end{corollary}
|
|
|
|
\paragraph{Gradient locality.}
|
|
\label{para:gradient-locality}
|
|
|
|
A second structural consequence of the ROLL-KDE formulation, when optimizing at a
|
|
single operating threshold, is that gradient mass is \emph{concentrated near that
|
|
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, single-threshold]
|
|
\label{obs:gradient-locality}
|
|
For KDE-ROLL at a single operating threshold, 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 $\tau$. 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}
|
|
|
|
\begin{remark}[Gradient coverage under ROLL-AOC]
|
|
\label{rem:aoc-coverage}
|
|
\Cref{obs:gradient-locality} describes gradient concentration at a single threshold.
|
|
Under ROLL-AOC, 99 distinct thresholds spanning $\operatorname{FPR} \in \{0.01,\ldots,0.99\}$
|
|
are active simultaneously, so the gradient is a superposition of 99 bell-shaped kernels
|
|
spread across the full score range. This broadens gradient coverage relative to
|
|
single-threshold optimization, making ROLL-AOC more suitable when no specific operating
|
|
point is fixed in advance.
|
|
\end{remark}
|
|
|
|
%------------------------------------------------
|
|
|
|
\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.
|
|
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.
|
|
|