Focal LossΒΆ
Focal Loss for Dense Object Detection
address this class imbalance by reshaping the standard cross entropy loss such that it down-weights the loss assigned to well-classified examples
For notational convenience:
\[\begin{split}p_t=
\begin{cases}
\hat{y} , & \text{if } y=1\\
1-\hat{y} , & \text{otherwise}
\end{cases}\end{split}\]
focal loss
\[FL(y) = -(1-p_t)^\gamma log(p_t)\]
\(\alpha\)-balanced variant
\[\begin{split}FL(y) &= -\alpha_t(1-p_t)^\gamma log(p_t)\\
&=-\alpha_t
\begin{cases}
(1-\hat{y})^\gamma log(\hat{y}) , & \text{if } y=1 \\
\hat{y}^\gamma log(1-\hat{y}) , & \text{otherwise}
\end{cases}
\\
&= -\alpha_t(y(1-\hat{y})^\gamma log(\hat{y})
+(1-y)\hat{y}^\gamma log(1-\hat{y}))\end{split}\]