Fixes fixes fixes

content/method/method.tex

@@ -286,20 +286,32 @@ Direct integration is not tractable, but the integral is well-approximated by av
   \mathcal{L}_{\text{ROLL-AOC}} = \frac{1}{99}\sum_{i=1}^{99}\mathcal{L}_{\text{ROLL-TPR@FPR}}\!\left(\frac{i}{100}\right)
   \label{eq:roll-aoc}
 \end{equation}
+Since $\operatorname{TPR}(\alpha)$ is monotone in $\alpha$ along the ROC curve,
+$g(\alpha) := 1-\operatorname{TPR}(\alpha)$ is monotone and bounded in $[0,1]$, so
+\eqref{eq:roll-aoc} approximates $\int_0^1 g(\alpha)\,d\alpha$ to within
+$1/99 \approx 0.01$ by a standard Darboux-sum argument, assuming each per-threshold
+term is evaluated exactly. This bound covers only the quadrature step: it does not
+extend to the gradient \eqref{eq:roll-aoc-gradient}, nor does it account for the gap
+between the KDE-fitted rates and the raw empirical scores (\Cref{sec:roll-instantiations})
+or any residual numerical error in the per-threshold bisection for $\tau_k$.
+
 The gradient follows by linearity of the sum:
 \begin{equation}
   \frac{\partial \mathcal{L}_{\text{ROLL-AOC}}(f_\theta(\mathbf{X}))}
-       {\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)}
+       {\partial f_\theta(\mathbf{x}_i)} = \frac{1}{99}\sum_{r=1}^{99}\frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}\!\left(f_\theta(\mathbf{X}) ; \frac{r}{100}\right)}
        {\partial f_\theta(\mathbf{x}_i)}
      \label{eq:roll-aoc-gradient}
 \end{equation}
+(the summation index $r$ ranges over the $99$ thresholds; $i$, fixed throughout the sum,
+indexes the sample being differentiated — distinct roles that \eqref{eq:roll-aoc} does
+not need to distinguish, since it has no outer per-sample index to collide with.)
 The concrete per-threshold gradient for the KDE instantiation is derived in
 \Cref{sec:roll-instantiations}; gradients for the parametric cases are in
 \Cref{appendix:gauss-grad,appendix:beta-roll}.
 
 \Cref{fig:roll-aoc-roc} gives a geometric view of this objective.
 
-\begin{figure}[H]
+\begin{figure}[H] 
   \centering
   \includegraphics[width=0.62\textwidth]{content/method/figures/roll_aoc_roc.pdf}
   \caption{Geometric interpretation of ROLL-AOC. The ROC curve (orange) maps
@@ -312,7 +324,8 @@ The concrete per-threshold gradient for the KDE instantiation is derived in
     segments show the per-threshold loss contribution
     $1 - \operatorname{TPR}(\alpha_i) = \mathcal{L}_{\text{ROLL-TPR@FPR}}(\alpha_i)$
     at each point. Averaging these contributions (99 in practice, 10 shown
-    for clarity) gives the value of \eqref{eq:roll-aoc}.}
+    for clarity) approximates the continuous integral, giving the value of
+    \eqref{eq:roll-aoc}.}
   \label{fig:roll-aoc-roc}
 \end{figure}
 
@@ -335,15 +348,16 @@ Parametric families, however, impose distributional assumptions on the score out
 no trained model can be guaranteed to satisfy.  The primary instantiation in this thesis
 is therefore non-parametric: Kernel Density Estimation (KDE) places no assumption on the
 shape of the score distribution and adapts to whatever the model produces at each training
-step.  The cost is a numerical CDF inversion and a custom backward pass; the gain is a
-faithful surrogate for the ROC curve regardless of how scores evolve during training.
+step.  The cost is a numerical CDF inversion and a custom backward pass. The gain is a
+surrogate that adapts to the score distribution's shape as it evolves during training,
+without committing to a parametric assumption that may not hold.
 
 \subsection{KDE ROLL}
 \label{sec:roll-kde}
 
 KDE estimates the score distribution directly from the observed batch scores, placing no
 assumption on their shape.  It is the primary instantiation of ROLL used throughout this
-thesis: its non-parametric flexibility makes it the most faithful surrogate for the ROC
+thesis: its non-parametric flexibility makes it the most adaptable surrogate for the ROC
 curve when score distributions are unknown, at the cost of a numerical CDF inversion and
 a custom backward pass evaluated 99 times per training step under ROLL-AOC.
 
@@ -437,13 +451,11 @@ Appendix~\ref{appendix:bandwidth-freeze} shows by counterexample.
 % 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.
+A structural consequence of the ROLL formulation is that the total signed
+gradient contribution from each class cancels exactly, 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.
 
 \begin{proposition}[Gradient balance]
 \label{prop:gradient-balance}
@@ -471,9 +483,15 @@ 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.
+is equivalent to an equal shift in $\tau$. The condition is sufficient, not necessary: it is
+what the proof uses to guarantee a rigid translation of $\hat{F}_k$, but the balance identity
+can hold without it. For example, $\hat{F}_k(t;\mathbf{s}) = \Phi(t - 2\bar{s})$ (with
+$\bar{s}$ the class mean) does not satisfy $\hat{F}_k(t;\mathbf{s}+\delta) = \hat{F}_k(t-\delta;\mathbf{s})$
+— shifting the scores by $\delta$ shifts $\hat{F}_k$ by $2\delta$, not $\delta$ — yet applying
+this same construction to both classes still yields $\sum_{\mathcal{B}_1}\partial L/\partial f_i
+= -\sum_{\mathcal{B}_0}\partial L/\partial f_i$ exactly, since the extra factor of $2$ enters
+both classes' gradients identically and cancels in the same shift argument. Translation
+invariance is a convenient sufficient condition, not the weakest one that would do.
 
 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
@@ -486,6 +504,19 @@ average of 99 per-threshold objectives each satisfying \eqref{eq:grad-balance},
 is preserved by linearity of summation.
 \end{corollary}
 
+The identity \eqref{eq:grad-balance} constrains only the signed sum per class. It
+does not by itself say that each class contributes equal gradient mass,
+$\sum_i |\partial\mathcal{L}/\partial f_\theta(\mathbf{x}_i)|$. Equal gradient-mass is satisfied
+when every sample within a class shares the same gradient sign, so that the signed and
+absolute sums are equal in magnitude. For KDE-ROLL,
+this extra property does hold (\Cref{obs:gradient-sign-correctness} below), so the
+signed identity does translate into balanced gradient mass for the method used
+throughout this thesis. However, notably, this does not hold true for the Gaussian formulation.
+
+Additionally, \Cref{prop:gradient-balance} is a statement about the per-sample loss
+gradients $\partial\mathcal{L}/\partial f_\theta(\mathbf{x}_i)$ at the model's output. It
+does not extend to the resulting parameter gradients $\partial\mathcal{L}/\partial\theta$,
+as this depends on the model's structure.
 \paragraph{Gradient sign correctness.}
 \label{para:gradient-sign-correctness}
 
@@ -518,28 +549,42 @@ the fully adaptive gradient violates this sign guarantee.
 \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.
+single operating threshold, is that gradient magnitude concentrates near $\tau$, but
+the mechanism differs between the two classes, and so does how literally the claim
+holds.
 
-This follows directly from \Cref{eq:kde-grad-combined}. For the positive class ($y_i = 1$),
-the gradient magnitude is
+This follows from \Cref{eq:kde-grad-combined}. For the positive class ($y_i = 1$), the
+gradient magnitude is an absolute, unnormalised function of the sample's own
+distance to $\tau$:
 \[
   \left|\frac{\partial \mathcal{L}}{\partial f_\theta(\mathbf{x}_i)}\right|
   = \frac{1}{|\mathcal{B}_1|}\,\sigma_1'(f_\theta(\mathbf{x}_i) - \tau),
 \]
-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 $\tau$.
+a bell-shaped kernel PDF that peaks when $f_\theta(\mathbf{x}_i) = \tau$ and decays to
+exactly $0$ as $|f_\theta(\mathbf{x}_i)-\tau|\to\infty$, regardless of where any other
+sample lies. For this class, "far from $\tau$ implies negligible gradient" holds without
+qualification.
 
-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.
+For the negative class ($y_i = 0$), the gradient is instead a \emph{normalised share} of
+the class's total kernel mass, $\sigma_0'(\tau-f_\theta(\mathbf{x}_i)) \big/
+\sum_j \sigma_0'(\tau-f_\theta(\mathbf{x}_j))$, and this ratio doesn't vanish just
+because $f_\theta(\mathbf{x}_i)$ is far from $\tau$: what determines
+sample $i$'s share of the gradient mass is its position relative to the rest of $\mathcal{B}_0$,
+not its absolute distance to $\tau$. If the whole negative class drifts far from $\tau$
+together, every individual $\sigma_0'$ value shrinks, but so does their sum, and the
+ratio can converge to a nonzero constant no matter how far the class has drifted.
+Concretely, for two negative samples at $\tau+M$ and $\tau+M+1$ (a fixed unit
+separation), the closer sample's normalised weight converges, as $M\to\infty$, to
+$1/(1+e^{-v_0})$, $\approx 0.731$ at $v_0=1$, not to $0$.
+However, if a single score drifts far from $\tau$ while the rest of it's class scores,
+stay close, its individual score will converge to zero.
+
+The effect, for both classes, is that each gradient step focuses the model's attention
+on the samples whose re-ranking relative to $\tau$ would most change the operating-point
+performance. For the positive class this is an absolute statement about each sample's
+own distance to $\tau$, and for the negative class, it is a statement about each sample's
+distance relative to its classmates. Typically, but not necessarily, this correlates
+with its absolute distance to $\tau$.
 
 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
@@ -547,27 +592,36 @@ 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.
+For KDE-ROLL at a single operating threshold: (a) a positive-class sample's gradient
+magnitude is the kernel PDF evaluated at its own distance to $\tau$, and vanishes as
+that distance grows, independent of the rest of the batch. (b) a negative-class
+sample's gradient magnitude is its kernel-PDF value \emph{relative to the sum over
+$\mathcal{B}_0$} — samples near $\tau$ receive a larger share than those far from it
+\emph{within the same batch}, but this relative weighting does not by itself vanish
+for an individual sample merely because the whole class is far from $\tau$.
 \end{observation}
 
-\Cref{fig:threshold-locality} illustrates this for a simulated two-class setting. The
+\Cref{fig:threshold-locality} illustrates this for a simulated two-class setting in
+which both classes are broadly spread around $\tau$ — the typical case, and the one in
+which the negative class's relative weighting (\Cref{obs:gradient-locality}b) and the
+positive class's absolute decay (\Cref{obs:gradient-locality}a) happen to agree. 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.
+shows the gradient magnitude of each sample, with a clear bell-shaped concentration
+around $\tau$ for both classes in this regime.
 
 \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
+  \caption{Gradient locality in KDE-ROLL, for a batch where both classes are spread
+    broadly around $\tau$. \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.}
+    show a bell-shaped concentration of gradient mass near $\tau$ here; for the negative
+    class this reflects relative weighting within $\mathcal{B}_0$
+    (\Cref{obs:gradient-locality}b), which need not track absolute distance to $\tau$ if
+    the whole class is displaced together. Smooth curves are the kernel PDFs scaled for
+    reference.}
   \label{fig:threshold-locality}
 \end{figure}
 
@@ -596,19 +650,39 @@ deviation:
   \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.
+This keeps the values given to the sigmoid kernel at a consistent scale. Why that
+matters depends entirely on which bandwidth estimator is in play.
 
-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.
+\begin{proposition}[Normalisation invariance under scale-equivariant bandwidth]
+\label{prop:normalization-invariance}
+Let $\hat{h}$ be any bandwidth estimator satisfying $\hat{h}(a\cdot x) = a\cdot\hat{h}(x)$
+for all $a>0$ (e.g.\ Silverman's rule, $h \propto \hat\sigma \cdot n^{-1/5}$). Under the
+frozen-bandwidth convention of \Cref{sec:roll-kde-backward} and a frozen normalisation
+constant $\hat\sigma_\mathcal{B}$, the loss value and every per-sample gradient
+$\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}/\partial f_\theta(\mathbf{x}_i)$ computed from
+the normalised scores $\tilde{s}$ are exactly equal to those computed directly from the
+raw scores $s$.
+\end{proposition}
+\begin{proof}
+See Appendix~\ref{appendix:score-normalization}.
+\end{proof}
+
+For Silverman's rule, normalisation is therefore a pure floating-point reparametrisation:
+it changes neither the loss nor the gradient used for training, so it cannot be what
+prevents overflow for that estimator — there is nothing left for it to change.
+ISJ, the primary bandwidth estimator (\Cref{sec:kde-bandwidth}), does \emph{not} satisfy
+the proposition's hypothesis (Appendix~\ref{appendix:score-normalization} measures its
+scale-dependence directly). For ISJ, both the loss value and the gradient genuinely
+depend on whether the scores are normalised, so normalisation is doing real work: it
+pins ISJ's input to the one scale regime it behaves consistently in, rather than leaving
+it to drift with the model's raw output scale over training.
+
+As with the bandwidths $v_0,v_1$, the normalisation constant $\hat\sigma_\mathcal{B}$ is
+treated as frozen (stop-gradient) in the backward pass: gradients with respect to the
+normalised scores are simply multiplied by $1/\hat\sigma_\mathcal{B}$. Differentiating
+fully through $\hat\sigma_\mathcal{B}$ instead introduces a further correction term that,
+like the bandwidth correction of \Cref{sec:roll-kde-backward}, can violate
+\Cref{obs:gradient-sign-correctness} (Appendix~\ref{appendix:score-normalization}).
 
 \subsection{Bandwidth Estimation for KDE}
 \label{sec:kde-bandwidth}