一、Introduction
- 略
二、Linear Regression with One Variable
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0 Model
本节课的问题是房价预测问题:
![图片名称](https://img2018.cnblogs.com/blog/1140624/202002/1140624-20200229163736905-258914032.png)
![图片名称](https://img2018.cnblogs.com/blog/1140624/202002/1140624-20200229163838211-1392269789.png)
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hypothesis (h_{ heta}(x)):是x的函数(对于一个固定的( heta_1))
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cost function (J( heta_1)):是参数( heta_1)的函数
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2 Gradient Descent
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(1)针对这个单变量线性回归问题,如下图,有个要点:
![图片名称](https://img2018.cnblogs.com/blog/1140624/202002/1140624-20200229163907360-1387868711.png)
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(2)梯度下降算法公式:
[ heta_j := heta_j - alpha frac{partial}{partial heta_j} J( heta_0, heta_1)
]
无论(frac{partial}{partial heta_j} J( heta_0, heta_1))的符号是什么,( heta_1)都会收敛到使得cost function取得最小值的点,符号是正时,( heta_1)减小,符号是负时,( heta_1)增大。
![图片名称](https://img2018.cnblogs.com/blog/1140624/202002/1140624-20200229164321712-1009781011.png)
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(3)$alpha的值要合理
![图片名称](https://img2018.cnblogs.com/blog/1140624/202002/1140624-20200229164343486-196270054.png)
- 此外
![图片名称](https://img2018.cnblogs.com/blog/1140624/202002/1140624-20200229164546601-1933954210.png)
[egin{aligned}
frac{partial}{partial heta_{j}} J( heta) &=frac{partial}{partial heta_{j}} frac{1}{2}left(h_{ heta}(x)-y
ight)^{2} \
&=2 cdot frac{1}{2}left(h_{ heta}(x)-y
ight) cdot frac{partial}{partial heta_{j}}left(h_{ heta}(x)-y
ight) \
&=left(h_{ heta}(x)-y
ight) cdot frac{partial}{partial heta_{j}}left(sum_{i=0}^{n} heta_{i} x_{i}-y
ight) \
&=left(h_{ heta}(x)-y
ight) x_{j}
end{aligned}
]
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(5)一个梯度下降的例子
梯度下降的轨迹,初始值为(48,30)
![图片名称](https://img2018.cnblogs.com/blog/1140624/202002/1140624-20200229164624739-1797995554.png)