New figure! New content!

content/method/method.tex

@@ -763,6 +763,64 @@ additive shift of the raw scores then produces a non-uniform shift of the transf
 changing the shape of the fitted distribution rather than merely translating it, and the
 balance property no longer holds.
 
+\paragraph{Gradient locality.}
+\label{para:gradient-locality}
+
+A second structural consequence of the ROLL formulation is that gradient mass is
+\emph{concentrated near the operating threshold $\tau$}. Samples whose scores lie
+close to $\tau$ receive disproportionately large 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 comfortably classified
+on the correct side contribute negligible signal, not because they are deemed unimportant,
+but because their contribution to the CDF estimate at $\tau$ is already saturated.
+
+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. Under heavy class imbalance this can cause the dominant class to
+``push'' the threshold away from the intended operating point, a problem that gradient
+locality avoids by construction.
+
+\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}
+
 
 \subsection{Gradient Computation and the Custom Backward Pass}
 \label{sec:roll-backward}