1 Evaluating the Hypothesis

1.1 Training / Testing Procedure

Training sets	Testing sets
70%	30%
$(x^{(1)},y^{(1)})···(x^{(m)},y^{(m)})$	$(x_{test}^{(1)},y_{test}^{(1)})···(x_{test}^{(m_{test})},y_{test}^{(m_{test})})$

datas are all randomly ordered

1.1.1 for Linear Regression

1. Learn parameter $\theta$ from training data
2. Compute test set error：
   $J_{test}(\theta)=\frac{1}{2m_{test}}\sum_{i=1}^{m_{test}}{(h_\theta(x_{test}^{(i)})-y_{test}^{(i)})}^2$

1.1.2 for Logistic Regression

1. Learn parameter $\theta$ from training data
2. Compute test set error：

way(1)

$J_{test}(\theta)=-\frac{1}{m_{test}}\sum_{i=1}^{m_{test}}\left(y_{test}^{(i)}\text{log}h_\theta(x_{test}^{(i)})+(1-y_{test}^{(i)})\text{log}h_\theta(x_{test}^{(i)})\right)$

way(2)

0/1 misclassification error：
$err(h_\theta(x),y)=\begin{cases} 1,&\text{if $h(x)≥0.5$ and $y=0$, or $h(x)＜0.5$ and $y=1$}\\ 0,&\text{otherwise} \end{cases}$
$Test\ \ error=\frac{1}{m_{test}}\sum_{i=1}^{m_{test}}err(h_\theta(x_{test}^{(i)}),y_{test}^{(i)})$

2 Model Selection and Training / Validation / Test sets（交叉验证集）

Training set	Cross validation set	Test set
60%	20%	20%
$(x^{(1)},y^{(1)})···(x^{(m)},y^{(m)})$	$(x_{cv}^{(1)},y_{cv}^{(1)})···(x_{cv}^{(m_{cv})},y_{cv}^{(m_{cv})})$	$(x_{test}^{(1)},y_{test}^{(1)})···(x_{test}^{(m_{test})},y_{test}^{(m_{test})})$

• Training error：
  $J_{train}(\theta)=\frac{1}{2m}\sum_{i=1}^m{\left(h_\theta(x^{(i)})-y^{(i)}\right)}^2$
  Cross Validation error：
  $J_{cv}(\theta)=\frac{1}{2m_{cv}}\sum_{i=1}^{m_{cv}}{\left(h_\theta(x_{cv}^{(i)})-y_{cv}^{(i)}\right)}^2$
  Test error：
  $J_{test}(\theta)=\frac{1}{2m_{test}}\sum_{i=1}^{m_{test}}{\left(h_\theta(x_{test}^{(i)})-y_{test}^{(i)}\right)}^2$
• Model selection：
  1° 使用训练集训练出$n$个模型
  2° 用$n$个模型分别对交叉验证集计算得出交叉验证误差（代价函数的值）
  3° 选取代价函数值最小的模型
  4° 用步骤3°中选出的模型对测试集计算得出推广误差（代价函数的值）

3 Diagnosing Bias（偏差，欠拟合） vs. Variance（方差，过拟合）

Bias（Underfit）	Variance（overfit）
$J_{train}(\theta)$ will be high	$J_{train}(\theta)$ will be low
$J_{cv}(\theta)≈J_{train}(\theta)$	$J_{cv}(\theta)>>J_{train}(\theta)$

3.1 Regularization and Bias / Variance

3.2 Learning Curves

• 将训练集误差和交叉验证集误差作为训练集实例数量$m$的函数绘制的图表

3.2.1 High Bias

3.2.2 High Variance

3.2.3 Solutions

to solve high bias	to solve high variance
Try getting additional features	Get more training examples
Try adding polynomial features	Try smaller sets of features
Try decreasing $\lambda$	Try increasing $\lambda$

4 Nerual networks and overfitting

“small” neural network	“large” neural network
fewer parameters	more parameters
more prone to underfitting	more prone to overfitting
computationally cheaper	computationally more expensive
	use regularization（$\lambda$）to address overfitting

5 Reference

吴恩达 机器学习 coursera machine learning
黄海广 机器学习笔记