# 梯度下降法的推导过程

感知机的损失函数：

$$
L(w, b) = - \sum\_{x\_i \in M}y\_i (w \cdot x\_i + b)
$$

目标是最小化这个损失函数。

使用梯度下降法求出$$L(w,b)$$的偏导，使w,b向导数的负方向移动。

$$
\begin{cases}
\nabla\_wL(w,b) = - \sum\_{x\_i \in M}y\_ix\_i \\
\nabla\_bL(w,b) = - \sum\_{x\_i \in M}y\_i && {2}
\end{cases}
$$

其中M是错误分类点的集合

由于perceptron使用随机梯度下降法，一次只基于一个点来调整w,b。\
假设当前选择的误分类点是$$(x\_i, y\_i)$$，那就相当集合M中只有$$(x\_i, y\_i)$$这一个点，偏导公式（2）可简化为

$$
\begin{cases}
\nabla\_wL(w,b) = - y\_ix\_i \\
\nabla\_bL(w,b) = - y\_i
\end{cases}
$$

令(w,b)向导数的负方向移动，学习率为$$\eta$$，得到

$$
\begin{cases}
w\_{new} = w\_{old} + \eta y\_ix\_i \\
b\_{new} = b\_{old} + \eta y\_i
\end{cases}
$$


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