线性分类-感知机

样本集：$\lbrace (x_i, y_i) \rbrace_{i=1}^N$
模型:
\begin{aligned}
&f(x)=\operatorname{sign}\left(w^{\top} x\right), \quad x_i \in \mathbb{R}^{p}, w \in \mathbb{R}^{p}\\
&\operatorname{sign}(a)=\begin{cases} +1,\quad a>0 \\{-1, \quad a<0 }
\end{cases}
\end{aligned}

策略：构造损失函数
\begin{array}{c}
L(w)=\sum_{i=1}^{N} I\lbrace y_{i} w^{\top} x_{i}<0\rbrace \\
{ L(w)=\sum_{x_{i} \in D}-y_{i} w^{\top} x_{i}} \\
{\nabla_{w} L=-y_{i} x_{i}}
\end{array}

算法：随机梯度SGD
\begin{aligned}
W^{(t+1)} & \leftarrow w^{(t)}-\lambda \nabla_{w} L \\
& w^{(t)}+\lambda y_{i} x_{i}
\end{aligned}