From 342efd836d88f1d2e41521b46a78eca5c01c8732 Mon Sep 17 00:00:00 2001 From: Aner Zakobar Date: Thu, 2 Jul 2026 15:31:32 +0300 Subject: [PATCH] A lot written. --- content/method/method.tex | 151 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 151 insertions(+) diff --git a/content/method/method.tex b/content/method/method.tex index a5a3e5f..81991d9 100644 --- a/content/method/method.tex +++ b/content/method/method.tex @@ -379,6 +379,50 @@ $\tau$ achieving FPR $= \alpha$ and the resulting loss are: % functions of the scores), and through $\Phi$ and $\Phi^{-1}$, which have % simple closed-form derivatives. +In order to compute the gradient derivation for the gaussian estimates we must first compute the gradient to each of the parameters. +Firstly, for $y_i = 1$. + +\[ + \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \mu_1} = + \frac{\partial \frac{1}{2}\left[1 + \text{erf}\!\left(\frac{\hat{F}_0^{-1}(\alpha) - \mu_1}{\sigma_1\sqrt{2}}\right)\right]}{\partial \mu_1} +\] + +\[ + = \frac{1}{\sqrt{2}\sigma_1}\exp\left(\frac{\hat{F}_0^{-1}(\alpha) - \mu_1}{\sqrt{2}\sigma_1}\right) +\] + +\[ + \frac{\partial \mu_1}{\partial f_{\theta}(\mathbf{x}_i)} = \frac{1}{|\mathcal{B}_1|} +\] + +\[ + \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \sigma_1} = + \frac{\partial \frac{1}{2}\left[1 + \text{erf}\!\left(\frac{\hat{F}_0^{-1}(\alpha) - \mu_1}{\sigma_1\sqrt{2}}\right)\right]}{\partial \sigma_1} +\] + +\[ + = \frac{\hat{F}_0^{-1}(\alpha)-\mu_1}{\sqrt{2\pi}\sigma_1^2} \exp\left( - {\left( \frac{\hat{F}_0^{-1}(\alpha) - \mu_1}{\sqrt{2}\sigma_1} \right)}^2 \right) +\] +\[ + \frac{\partial \sigma_1}{\partial f_{\theta}(\mathbf{x}_i)} = \frac{\mathbf{x}_i - \mu_1}{|\mathcal{B}_1| \sigma_1} +\] + +The final derivation, given $y_i = 1$, is given as: + +\[ + \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial f_{\theta}(\mathbf{x}_i)} = \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \mu_1} \cdot \frac{1}{|\mathcal{B}_1|} + + \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \sigma_1} \cdot \frac{\partial \sigma_1}{\partial f_{\theta}(\mathbf{x}_i)} = \frac{\mathbf{x}_i - \mu_1}{|\mathcal{B}_1| \sigma_1} +\] + +TODO - above equation is based off of mathexchange derivation. Link and reference properly! + +Given $y_i = 0$ we first must derive: + +\[ + \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \hat{F}_0^{-1}(\alpha)} = \frac{1}{\sqrt{2 \pi \sigma_1^2}}\exp\left( -{\left( \frac{\hat{F}_0^{-1}(\alpha) - \mu_1}{2\sigma_1^2}\right)}^2\right) +\] + +TODO finish somehow - later! \subsection{Beta ROLL} \label{sec:roll-beta} @@ -410,6 +454,10 @@ $\tau$ achieving FPR $= \alpha$ and the resulting loss are: % 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} @@ -417,6 +465,55 @@ $\tau$ achieving FPR $= \alpha$ and the resulting loss are: % 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: + + +\[ + \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{KDE}}(f_{\theta}(\mathcal{B}) ; \alpha) = \hat{F}_1(\hat{F}_0^{-1}(\alpha)) + \label{eq:roll-tpr-at-fpr} +\] + +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 nice formula, the inverse CDF has no known closed formula. It instead must be calculated numerically. Thus, the calculation of the decision threshold $\tau$ must be achevied +using a numerical calculation algorithm. In this work, Newton Raphson is chosen. + +Firstly we must calculate $\hat{F}_0^{-1}(\alpha)$ = \tau. We start with our initial guess, $\tau_0$, and for each step, calculate: + +\[ + \tau_{n+1} = \tau_n - \frac{\hat{F}_0(\tau_n ; \mathcal{X}_0)}{\frac{\partial \hat{F}_0(\tau_n ; \mathcal{X}_0)}{\partial \tau}} +\] + +This step is taken until $\hat{F}_0(\tau_n ; \mathcal{B}_0) \approx \alpha$ within some acceptable error (in our case $1e-3$). + +Our derivative w.r.t. $\tau$ is + +\[ % + \frac{\partial \hat{F}_0(\tau_n ; \mathcal{X}_0)}{\partial \tau} = \frac{1}{\mathcal{X}_0}\sum_i \sigma'(\tau - x_i) +\] + +At this point, we have succesfully calculated $\tau$ for which $\tau = \hat{F}_0^{-1}(\alpha ; \mathbf{X}_0)$. + +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} @@ -425,6 +522,60 @@ $\tau$ achieving FPR $= \alpha$ and the resulting loss are: % Gradients flow through the kernel evaluations back to the model scores. % This is the section that connects to \Cref{sec:roll-backward} motivation. +Firstly, for the true population, we calculate the derivatives. + +\[ + \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial f_{\theta}(x_i)} = + \frac{\partial \sum_{x_j \in \mathcal{B}_1} (\sigma_1(f_{\theta}(x_j) - \tau))}{\partial f_{\theta}(x_i)} +\] + +\[ + = \frac{\partial \left(\sum_{x_j \in \mathcal{B}_1 ; j \neq i} \frac{1}{|\mathcal{B}_1|}(\sigma_1(f_{\theta}(x_j) - \tau)) + \frac{1}{|\mathcal{B}_1|}\sigma_1(f_{\theta}(x_i) - \tau)\right)}{\partial f_{\theta}(x_i)} +\] + +\[ + = \frac{1}{|\mathcal{B}_1|}\sigma_1'(f_{\theta}(x_i) - \tau) +\] + +Now, for the false population. As stated in the roll framework equation above (TODO LINK) we must first compute the derivation w.r.t the threshold $\tau = \hat{F}_{0}^{-1}(\alpha)$: + + +\[ + \frac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \hat{F}_0^{-1}(\alpha)} = \frac{\partial \sum_{x_j \in \mathcal{B}_1} (\sigma_1(f_{\theta}(x_j) - \tau))}{\partial \tau} +\] + +\[ + = -\frac{1}{|\mathcal{B}_1|}\sum_{x_i \in \mathcal{B}_1}\sigma_1'(f_{\theta}(x_i) - \tau) +\] + +At this point, we must calculate the derivative of the threshold w.r.t each score from the false population. + +Unfortunately for us, this involves calculating the derivative of a calculation we computed numerically. + +In order to derive this, we can utilize our earlier computation, along with the inverse derivative rule: + +\[ + f^{-1}'(x) = \frac{1}{f'(f^{-1}(x))} +\] + +We can use this to calculate the derivative we need + +\[ + \frac{\partial \hat{F}_0^{-1}(\alpha)}{\partial f_\theta(\mathbf{x}_i)} +\] +\[ + \frac{\partial \hat{F}_0^{-1}(\alpha ; \mathbf{X}_0)}{\partial f_\theta(\mathbf{x}_i)} = \frac{\partial \tau}{\partial f_{\theta}(x_i)} +\] + +\[ + = \frac{1}{\frac{\partial\hat{F}_0(\tau ; X_0)}{\partial x_i}} +\] + +We must now calculate the derivative $\frac{\partial\hat{F}_0(\tau ; X_0)}{\partial x_i}$. + +\[ + \frac{\partial\hat{F}_0(\tau ; X_0)}{\partial x_i} = \frac{\frac{1}{|\mathcal{B}_0|}\sum_{x_j \in \mathcal{B}_0}\sigma_0(\tau - x_j)}{\partial x_i} +\] %------------------------------------------------