监督学习中的损失函数及应用研究

邓建国,张素兰,张继福,荀亚玲,刘爱琴

Loss function and application research in supervised learning

Jianguo DENG,Sulan ZHANG,Jifu ZHANG,Yaling XUN,Aiqin LIU

表3 基于绝对损失的演化损失函数

名称 演化形式 平均绝对误差函数 $\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|Y_i-f\left(x_i\right)\right|}$ 平均相对误差函数 $\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{\left|Y_i-f\left(x_i\right)\right|}{Y_i}}$ L1损失函数 ${\displaystyle \sum _{i=1}^n\left|Y_i-f\left(x_i\right)\right|}$ Chebyshev损失函数 $\underset{i=1}{\overset{n}{\max }}\left(Y_i-f\left(x_i\right)\right)$ Minkowski损失函数 ${\left({\displaystyle \sum _{i=1}^n{\left|Y_i-f\left(x_i\right)\right|}^p}\right)}^{\frac{1}{p}}$ smooth L1损失函数 $\{\begin{array}{l}\frac{1}{2}{\left(Y-f\left(x\right)\right)}^2,\left|Y-f\left(x\right)\right|＜1\\ \left|Y-f\left(x\right)\right|-\frac{1}{2}，\left|Y-f\left(x\right)\right|\ge 1\end{array}$ huber损失函数 $\{\begin{array}{l}\frac{1}{2}{\left(Y-f\left(x\right)\right)}^2,\left|Y-f\left(x\right)\right|\le \delta \\ \delta \left|Y-f\left(x\right)\right|-\frac{1}{2}{\delta }^2，\left|Y-f\left(x\right)\right|＞\delta \end{array}$ 分位数损失函数 ${\displaystyle \sum _{i;y_i＜f\left(x_i\right)}\left(1-\gamma \right)}\left|Y_i-f\left(x_i\right)\right|+{\displaystyle \sum _{i;y_i\ge f\left(x_i\right)}\gamma }\left|Y_i-f\left(x_i\right)\right|$