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z`BsmQi&R3l)33OoAQj>Qc2$0B)$3HdcD;~1uBthGT|ZnzS~?F7GzEKlUFhz&oZ@;| z+-z2{9Y=o497IW$VdmkXU!bSbz5eWWUENtr;W+aXPrO#X#I3BXeDt|gLr99l-J``q_mgl;CZl~+< zW%b)$y?Rxtdp)pMfunob=omro}rxoTtU>Y@{3Rp*FJ^@8`3=cJgiMOBjPq#RmoLXc~1=>;^Sg zTvo>2>tabNNzcgGkhNdtH`60-VX9Ic*%IfJR&HE!N|z z6w~AA8M;-&c-J}}6UL{TZGW(f$k=8a!|{6LJ+hyZxO7Vf|EEu)6mM{{`I}B*KKUAX z=EqIeU3%Nfc9xdqg?`6*eFcpd`q~A0RNv>ZhBxJG!uJK(Xz0n_$a3$K=>GMA=3#H% zB{%+K<*sw_zTVvvDar6;+N?jBwE3d3@d|UenyTuSk1Y_ShSi+KcRwYvW9_75;~BHa zhhK*_;LO5}pj$Ui=c4Z=*9L0z@k_pF#JyWNFv&@`uH(`Dk?%WH>`G}Gn0dLgSMdhD zNl#-HTmc!`83(Mf??yQ~nj%c^>+5ghEh4@(m;~P5MhK-592cYCU;8DW?IpDk6Zkwg zcg65>e7f&PQ&-m#n`d%4Iz3S7dd12*sJ{+%{bhqU0^5?JXDcc^f8;*y%4}|FIUyyLdx85YT5yjhbX;|F z6MI{VHk7s5xw*Np$)cj7a1K%lN$oZV&5r&OenpR;i>C73qfiX2FAv9uP~dv#Suqv7 z#e{_)%cMTC5Pqh|_hF^I#BSXAq@9C > ${x}_{j}^{\left( i \right)}$: 代表特征矩阵中第 $i$ 行的第 $j$ 个特征,也就是第 $i$ 个训练实例的第 $j$ 个特征。 -参照上图,则记号的举例有,${x}^{(2)}\text{=}\begin{bmatrix} 1416\\\ 3\\\ 2\\\ 40 \end{bmatrix}, {x}^{(2)}_{1} = 1416$ +参照上图,则有 ${x}^{(2)}\text{=}\begin{bmatrix} 1416\\\ 3\\\ 2\\\ 40 \end{bmatrix}, {x}^{(2)}_{1} = 1416$ 多变量假设函数 $h$ 表示为:$h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}}+...+{\theta_{n}}{x_{n}}$ @@ -30,22 +30,80 @@ $h_\theta\left(x\right)=\begin{bmatrix}\theta_0\; \theta_1\; ... \;\theta_n \end > > $x_0$: 为了计算方便我们会假设 $x_0^{(i)} = 1$ -## 4.2 Gradient Descent for Multiple Variables +## 4.2 多变量梯度下降(Gradient Descent for Multiple Variables) +多变量损失函数类似于单变量损失函数, +即 $J\left( {\theta_{0}},{\theta_{1}}...{\theta_{n}} \right)=\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( h_{\theta} \left({x}^{\left( i \right)} \right)-{y}^{\left( i \right)} \right)}^{2}}}$ ,其中 $h_\theta\left(x\right)= \theta^T x$。 -## 4.3 Gradient Descent in Practice I - Feature Scaling +前文提到梯度下降对于最小化损失函数的通用性,则多变量梯度下降公式即 -## 4.4 Gradient Descent in Practice II - Learning Rate +$\begin{align*} & \text{repeat until convergence:} \; \lbrace \newline \; &{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( {\theta_{0}},{\theta_{1}}...{\theta_{n}} \right) \newline \rbrace \end{align*}​$ -## 4.5 Features and Polynomial Regression +对其求导: + +$\begin{align*}& \text{repeat until convergence:} \; \lbrace \newline \; & \theta_j := \theta_j - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0,1...n}\newline \rbrace\end{align*}$ + +可展开为: + +$\begin{align*} & \text{repeat until convergence:} \; \lbrace \newline \; & \theta_0 := \theta_0 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_0^{(i)}\newline \; & \theta_1 := \theta_1 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_1^{(i)} \newline \; & \theta_2 := \theta_2 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_2^{(i)} \newline & \vdots \newline \; & \theta_n := \theta_n - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_n^{(i)} &\newline \rbrace \end{align*}$ + +当然,同单变量梯度下降一样,计算时需要**同时更新**所有参数。 + +## 4.3 梯度下降实践1-特征值缩放(Gradient Descent in Practice I - Feature Scaling) + +在应用梯度下降算法实践时,由于各特征值的范围不一,可能会导致损失函数收敛过慢。 + +以房价预测问题为例,这里选取房屋面积大小和房间数量这两个特征。 + +下图中,左图是以原始数据绘制的损失函数轮廓图,右图为采用特征缩放(都除以最大值)后图像。左图中呈现的图像较扁,相对于使用特征缩放方法的右图,梯度下降算法需要更多次的迭代。 + +![](image/20180108_100751.png) + + + +为了优化梯度下降的收敛速度,采用特征缩放方法,使各特征值的**范围尽量一致**。 + +除了以上图人工选择并除以一个参数的方式,**均值归一化(Mean normalization)**方法更为便捷,可采用它来对所有特征值统一缩放: + + $x_i=\frac{x_i-average(x)}{maximum(x)-minimum(x)}, 使得 $ $x_i \in (-1,1)$$ + +对于特征的范围,并不一定需要使得 $-1 < x < 1$,类似于 $1\leqslant x \leqslant 3$ 也是可取的。 + +另外注意,一旦采用特征缩放,我们就需对所有的输入采用特征缩放,包括训练集、测试集、预测输入等。 + +## 4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate) + +对于梯度下降,一般给定迭代次数来得出最小化损失函数的参数值,当然也可以设定比如 $J\left(\theta\right) < {10}^{-3}$ 等下限值来得到参数值,不过我们更常用的是**给定迭代次数**来得到参数值。 + +通过绘制损失函数关于迭代次数的图像,我们可以可视化梯度下降的执行过程,直观的图形可以帮助我们发现损失函数趋向于多少时能趋于收敛,参考图像来调整诸如学习速率的取值,迭代次数的选定等问题。 + +![](image/20180108_103357.png) + +对于学习速率 $\alpha$,一般上图为合适,下图中左图可能表明 **$\alpha$ 过大**,损失函数**无法收敛**,右图可能表明 **$\alpha$ 过小**,损失函数**收敛的太慢**。当然,$\alpha$ 足够小时,损失函数在每轮迭代后一定会减少。$\alpha$ 可选取如 $\dots\;0,001,\;0.003,\;0.01,\;0.03,\;0.1,\;\dots$ + +![](image/20180108_104701.png) + +另外,迭代次数刚开始可尽量高些,后面再根据需要同 $\alpha$ 进行调整。 + +## 4.5 特征和多项式回归(Features and Polynomial Regression) + +线性回归只能以直线来对数据进行拟合,有时候需要使用曲线来对数据进行拟合,即多项式回归(Polynomial Regression)。 + +比如一个二次方模型:$h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}$ + +或者三次方模型:$h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}+{\theta_{3}}{x_{3}^3}$ + +或者平方根模型: $h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}+{\theta_{3}}{\sqrt{x_3}}$ + +![](image/20180108_113132.png) + +在使用多项式回归时,要记住非常有必要进行特征缩放。 ## 4.6 Normal Equation ## 4.7 Normal Equation Noninvertibility -## 4.8 Working on and Submitting Programming Assignments - # 5 Octave Matlab Tutorial 复习时可直接倍速回顾视频,笔记整理暂留。