一、Introduction
略
二、Linear Regression with One Variable
0 Model
本节课的问题是房价预测问题:
1 model and cost function
Andrew Ng在cost function Intuition I中对hypothesis和cost function做了对比
\(\theta_0\)已经假设等于0,于是只剩下一个参数\(\theta_1\)。- hypothesis \(h_{\theta}(x)\):是x的函数(对于一个固定的\(\theta_1\))
cost function \(J(\theta_1)\):是参数\(\theta_1\)的函数
2 Gradient Descent
(1)针对这个单变量线性回归问题,如下图,有个要点:
\(\theta_1\)和\(\theta_2\)要同时更新,不然就会出错
(2)梯度下降算法公式:
\[\theta_j := \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta_0, \theta_1)\]
无论\(\frac{\partial}{\partial \theta_j} J(\theta_0, \theta_1)\)的符号是什么,\(\theta_1\)都会收敛到使得cost function取得最小值的点,符号是正时,\(\theta_1\)减小,符号是负时,\(\theta_1\)增大。
(3)$\alpha的值要合理
- 此外
(4)\(\theta_j := \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta_0, \theta_1)\)的推导过程
\[ \begin{aligned} \frac{\partial}{\partial \theta_{j}} J(\theta) &=\frac{\partial}{\partial \theta_{j}} \frac{1}{2}\left(h_{\theta}(x)-y\right)^{2} \\ &=2 \cdot \frac{1}{2}\left(h_{\theta}(x)-y\right) \cdot \frac{\partial}{\partial \theta_{j}}\left(h_{\theta}(x)-y\right) \\ &=\left(h_{\theta}(x)-y\right) \cdot \frac{\partial}{\partial \theta_{j}}\left(\sum_{i=0}^{n} \theta_{i} x_{i}-y\right) \\ &=\left(h_{\theta}(x)-y\right) x_{j} \end{aligned} \]
(5)一个梯度下降的例子
梯度下降的轨迹,初始值为(48,30)