Focal Loss 的前向与后向公式推导

时间 2019-12-13

标签 focal loss 公式推导繁體版

原文原文链接

把Focal Loss的前向和后向进行数学化描述。本文的公式可能数学公式比较多。本文尽可能采用分解的方式一步一步的推倒。达到能易懂的目的。spa

Focal Loss 前向计算

$Loss(x, class) = -\alpha_{class}(1-\frac {e^{x[class]}} {\sum_j e^{x[j]}} )^\gamma \log{(\frac {e^{x[class]}} {\sum_j e^{x[j]}} )} （1）$

其中是输入的数据是输入的标签。数学

$= \alpha_{class}(1-\frac {e^{x[class]}} {\sum_j e^{x[j]}} )^\gamma \cdot (-x[class] + \log{\sum_j e^{x[j]}}) （2）$

$= -\alpha_{class}(1- softmax(x)[class] )^\gamma \cdot \log\big(softmax(x)[class]\big) （3）$

其中 $softmax(x) = \frac {e^{x[class]}} {\sum_j e^{x[j]}} = p_{class}$ io

Focal Loss 后向梯度计算

为了计算前向公式（3）的梯度咱们，首先计算单元 $\log p_t$ 的导数。class

$\begin {aligned} \frac{\partial}{\partial x_i} \log p_t & =\frac{1}{p_t}\cdot\frac{\partial p_t}{\partial x_i} \\\ &=\frac{1}{p_t}\cdot\frac{\partial}{\partial x_i} \frac{e^{x_t}} {\sum_j{e^{x_j}}} \\\ & = \begin {cases} \frac{1}{p_t}\cdot(p_t-p_t^2) = 1-p_t, & i=t \\\ \frac{1}{p_t}\cdot(-p_i \cdot p_t) = -p_i, & i \neq t \end {cases} \end {aligned} （4）$

计算计算导数：im

$\begin {aligned} \frac{\partial}{\partial x_i}p_t & = \frac{\partial}{\partial x_i} \frac{e^{x_t}}{\sum_j{e^{x_j}}} \\\ & = \begin {cases} \frac{e^{x_t}\cdot \sum_j{e^{x_j}} - e^{x_t}\cdot e^{x_t}}{\sum_j{e^{x_j}} \cdot \sum_j{e^{x_j}}} = p_t -p_t^2 & i=t \\\ \frac{-e^{x_t}\cdot e^{x_i}}{\sum_j{e^{x_j}} \cdot \sum_j{e^{x_j}}} = -p_i \cdot p_t, &i \neq t \end {cases} \end {aligned}（5）$

有了（4）和（5）咱们就来对（3）进行推倒。数据

$\because \begin {aligned} FL(x, t) &= -(1-p_t)^{\gamma}\log{p_t} \end {aligned}$

$\begin {aligned} \therefore \frac{\partial{FL(x, t)}}{\partial x_i} &= -\gamma(1-p_t)^{\gamma-1} \cdot \frac {\partial (-p_t)} {\partial x_i}\cdot \log p_t-(1-p_t)^\gamma \cdot \frac {\partial \log p_t}{\partial x_i} \\\ & = \gamma(1-p_t)^{\gamma-1} \cdot \log p_t \cdot \frac {\partial (p_t)} {\partial x_i}-(1-p_t)^\gamma \cdot \frac {\partial \log p_t}{\partial x_i} \\\ &= \begin {cases} \gamma(1-p_t)^{\gamma-1} \cdot \log p_t \cdot (p_t-p_t^2)-(1-p_t)^\gamma \cdot (1-p_t), & i=t \\\ \gamma(1-p_t)^{\gamma-1} \cdot \log p_t \cdot (-p_i\cdot p_t)-(1-p_t)^\gamma \cdot (-p_i), & i \neq t \end {cases} \end {aligned} （6）$

在（6）中把（4）（5）带入并合并整理就获得(7)img

$\therefore \frac{\partial{FL(x, t)}}{\partial x_i} = \begin {cases} -(1-p_t)^{\gamma}\cdot(1-p_t-\gamma p_t\log p_t),&i = t \\\ p_i\cdot(1-p_t)^{\gamma - 1 } \cdot(1-p_t-\gamma p_t \log p_t),&i\neq t \end {cases} （7）$

(7)就是Focal loss的后向的最后结果。要是在TF, Pytorch等中实现Focal Loss 便可采用（7）实现backward。标签