吴恩达机器学习笔记

Error analysis

Methods to solve over fitting

• more training examples
• try smaller sets of features
• try increasing $\lambda$

Methods to solve under fitting

• getting additional features
• try adding polynomial features
• try decreasing $\lambda$

Recommend approach

• start with a simple algorithm that you can implement quickly. Implement it and test it on your cross-validation data
• plot learning curves to decide if more data, more features, etc. are likely to help.
• Error analysis: See if you spot any systematic trend in what type of examples it is making errors on

Error metrics for skewed classes

	1	0
1	True Positive	False Positive
0	False Negative	True Negative

$precision=\frac{TP}{TP+FP}$ 准确率

$recall=\frac{TP}{TP+FN}$ 召回率

$F_{1}score=2\frac{PR}{P+R}$ F值

Data for machine learning

In the following conditions, more data makes sense.

• Assume feature $x\in R^{n+1}$ has sufficient information to predict y accurately.
• Use a learning algorithm with many parameters such as logistic regression or linear regression with many features or neural network with many hidden units.

Support Vector Machine

Logistic regression Cost function
$$mi{n}_{\theta }\frac{1}{m}[\sum _{i=1}^m{y}^{(i)}(-\log h_{\theta }(x^{(i)}))+(1-y^{(i)})(-\log (1-h_{\theta }(x^{(i)})))]+\frac{\lambda }{2m}\sum _{j=1}^n{\theta }_j^2$$
SVM hypothesis
$$mi{n}_{\theta }C\sum _{i=1}^m[y^{(i)}cos{t}_1({\theta }^T{x}^{(i)})+(1-y^{(i)})cos{t}_0({\theta }^T{x}^{(i)})]+\frac{1}{2}\sum _{i=1}^n{\theta }_j^2$$

$$h_{\theta }(x)=\{\begin{array}{l}0\\ 1\end{array}$$

SVM parameters

$C=\frac{1}{\lambda }$

• large C low bias, high variance
• small C higher bias, low variance

${\sigma }^2$

• large ${\sigma }^2$. feature $f_i$ vary more smoothly. higher bias, lower variance
• small ${\sigma }^2$. feature $f_i$ vary less smoothly. lower bias, higher variance

Kernel function

• no kernel

• 高斯kernel
  $$f=e^{-\frac{||x_1-x_2||}{2{\sigma }^2}}$$

• 多项式kernel $k(x,l)=(x^{T}l+constant{)}^{degree}$

Muti-class classification

train K SVMs, one to distinguish $y=i$ from the rest

Logistic regression vs SVMs

n=number of features $x\in R^{n+1}$，m=number of training examples

• n is large relative to m: use logistic regression or SVM without kernel
• n is small , m is intermediate: use SVM with Gaussian kernel
• n is small and m is large: create or add more features and then use logistic regression or svm without kernel

K-means

Input

• K (number of clusters)
• training set { x ( 1 ) , x ( 2 ) , ⋯   , x ( m ) } \{x^{(1)},x^{(2)},\cdots,x^{(m)}\} {x(1),x(2),⋯,x(m)}

K-means algorithm

1. randomly initialize K cluster centroids ${\mu }_1,{\mu }_2,\cdots ,{\mu }_K\in R^n$

2. repeat{
for i in m:
    C^{(i)}=index(from 1 to K) of cluster centroid closest to x^{(i)}
for k=1 to K:
    u_k=mean of points assigned to cluster k
}

K-means optimization objective

• $c^{(i)}= index\ (1,2,\dots,K) $ to which example $x^{(i)}$ is currently assigned

• ${\mu }_k$ cluster centroid k $({\mu }_k\in R^n)$

• ${\mu }_{c^{(i)}}$ cluster centroid of cluster to which example $x^{(i)}$ has been assigned

• $$mi{n}_{c^{(1)},c^{(2)},\cdots ,c^{(m)},{\mu }_1,{\mu }_2,\cdots ,{\mu }_K}\frac{1}{m}\sum _{i=1}^m||x^{(i)}-{\mu }_{c^{(i)}}|{|}^2$$

Random initialization

for i in range(100):
    randomly initialize K-means
    run k-means. get c^{(i)},c^{(m)},...
    compute cost function J
pick clustering that gave lowest cost J(get c^{(i)},c^{(m)},....)

Usually, set k=2-10, kmeans can work well. And random initialization can avoid local optimal.

Choosing the number of clusters

Elbow method

choose the number for the purpose

Principle Component Analysis

Data preprocessing

Training set: $x^{(1)},x^{(2)},\cdots ,x^{(m)}$

preprocessing ${\mu }_j=\frac{1}{m}{\sum }_{i=1}^m{x}_j^{(i)}$, replace each $x_j^{(i)}$ with $x_j-{\mu }_j$, if different features on different scales, scale features to have comparable range of values.

Choosing the number of principal components

Averaged squared projection error $\frac{1}{m}{\sum }_{i=1}^m\Vert x^{(i)}-x_{approx}^{(i)}{\Vert }^2$

Total variation in the data $\frac{1}{m}{\sum }_{i=1}^m\Vert x^{(i)}{\Vert }^2$

Typically, choosing k to the smallest value so that $\frac{\frac{1}{m}{\sum }_{i=1}^m\Vert x^{(i)}-x_{approx}^{(i)}{\Vert }^2}{\frac{1}{m}{\sum }_{i=1}^m\Vert x^{(i)}{\Vert }^2}<=0.01(1\%)$(99% of variance is retained)

[U,S,V]=svd(Sigma)

pick the smallest value of k for which $\frac{{\sum }_{i=1}^k{S}_{ii}}{{\sum }_{i=1}^m{S}_{ii}}>=0.99$

Application of PCA

• Compression

  • reduce memory or disk needed to store data
  • speed up learning algorithm

• Visualization

  • plot 2 or 3 dimensional data

• A bad way to prevent over fitting

  • Fewer features might work ok, but isn’t a good way to address over fitting. Use regularization instead.

• A bad use to design ML systems

Anomaly detection

Examples

• Fraud detection
• Manufacturing
• Monitoring computers in a data center

Anomaly detection algorithm

1. Choose features $x_i$ that you think might be indicative anomalous examples.

2. fit parameters ${\mu }_1,\cdots ,{\mu }_n,{\sigma }_1^2,\cdots ,{\sigma }_n^2$
   $$\begin{gathered} {\mu }_j=\sum _{i=1}^m{x}_j^{(i)} \\ {\sigma }_j^2=\frac{1}{m}\sum _{i=1}^m(x_j^{(i)}-{\mu }_j^2) \end{gathered}$$

3. given new example x, compute $p(x)$
   $$p(x)=\prod _{j=1}^{n}p(x_j;{\mu }_j,{\sigma }_j^2)=\prod _{j=1}^n\frac{1}{\sqrt{2\pi {\sigma }_j}}{e}^{-\frac{(x_j-{\mu }_j{)}^2}{2{\sigma }_j^2}}$$

4. Anomaly if $p(x)<\epsilon $

Algorithm evaluation

• fit model $p(x)$ on training set { x ( 1 ) , ⋯   , x ( m ) } \{x^{(1)},\cdots,x^{(m)}\} {x(1),⋯,x(m)}

• on a cross validation or test example $x$, predict
  $$y=\{\begin{array}{l}1ifp(x)<ϵanomaly\\ 0ifp(x)>=ϵnormal\end{array}$$

• possible evaluation

  • precision or recall
  • $F_{1}score$
  • can choose suitable $ϵ$ through maximize the $F_1$ score

Anomaly detection vs supervised learning

• Use Anomaly detection when

  • very small number of positive examples (0-20) and large number of negative examples
  • many different types of anomalies. Hard for any algorithm to learn from positive examples what the anomalies look like
  • future anomalies may look nothing like any of the anomalous examples we’ve seen so far
  • examples like
    • fraud detection
    • manufacturing
    • monitoring machines in a data center

• Use Supervised learning when

  • large number of positive and negative examples
  • enough positive examples for algorithm to get a sense of what positive examples are like, future examples likely to be similar to ones in training set
  • examples like
    • email spam classification
    • weather prediction
    • cancer classification

Choosing what features to use

make non-gaussian features gaussian

• $x_i\leftarrow \log (x_i)$
• $x_2\leftarrow \log (x_2+c)$
• $x_3=\sqrt{x_3}$

Multivariate Gaussian distribution

$x\in R^n$. Don’t model $p(x_1),p(x_2),\cdots$ separately, model $ p(x)$ all in one goal. Parameters $\mu \in R^n$, $\Sigma \in R^{n\times n}$
$$p(x;\mu ,\Sigma )=\frac{1}{\sqrt{2\pi |\Sigma |}}ex{p}^{-\frac{1}{2}(x-\mu {)}^T{\Sigma }^{-1}(x-\mu )}$$
Use original model $p(x)={\prod }_{i=1}^{n}p(x_i;{\mu }_i,{\sigma }_i^2)$

• Manually create features to capture anomalies where $x_1,x_2$ take unusual combinations of values
• Computationally cheaper
• ok even if m (training set size) is small

Use Multivariate Gaussian

• Automatically captures correlations between features
• Computationally more expensive
• Must have $m>n$ or else $\Sigma$ is non-invertible

Recommender Systems

$n_u=number$ of users

$n_m=number$ of movies

$m_j=numberofmoviesratedbyuseri$

$r(i,j)=1$ if user j has rated movie i

$y^{(i,j)}=rating$ given by user j to movie i(defined only if $r(i,j)=1 $)

Content-based recommendations

For each user j, learn a parameter ${\theta }^{(j)}\in R^3$. Predict user j as rating movie i with $({\theta }^{(j)}{)}^T{x}^{(i)}$ stars.

optimization objective

to learn ${\theta }^{(j)}$, parameter for user j, you should
$$mi{n}_{\theta (j)}\frac{1}{2}\sum _{i:r(i,j)=1}(({\theta }^{(j)}{)}^T{x}^{(i)}-y^{(i,j)}{)}^2+\frac{\lambda }{2}\sum _{k=1}^n({\theta }_k^{(j)}{)}^2$$
to learn ${\theta }^{(1)},{\theta }^{(2)},\cdots ,{\theta }_{n_u}$
$$mi{n}_{{\theta }^{(1)},{\theta }^{(2)},\cdots ,{\theta }_{n_u}}\frac{1}{2}\sum _{j=1}^{n_u}\sum _{i:r(i,j)=1}(({\theta }^{(j)}{)}^T{x}^{(i)}-y^{(i,j)}{)}^2+\frac{\lambda }{2}\sum _{j=1}^{n_u}\sum _{k=1}^n({\theta }_k^{(j)}{)}^2$$

Collaborative filtering

Given $x^{(1)},\cdots ,x^{(n_m)}$ and movie ratings can estimate ${\theta }^{(1)},\cdots ,{\theta }^{(n_u)}$
$$mi{n}_{{\theta }^{(1)},{\theta }^{(2)},\cdots ,{\theta }_{n_u}}\frac{1}{2}\sum _{j=1}^{n_u}\sum _{i:r(i,j)=1}(({\theta }^{(j)}{)}^T{x}^{(i)}-y^{(i,j)}{)}^2+\frac{\lambda }{2}\sum _{j=1}^{n_u}\sum _{k=1}^n({\theta }_k^{(j)}{)}^2$$
Given ${\theta }^{(1)},\cdots ,{\theta }^{n_u}$ can estimate $x^{(1)},\cdots ,x^{(n_m)}$
$$mi{n}_{x^{(1)},x^{(2)},\cdots ,x_{n_u}}\frac{1}{2}\sum _{j=1}^{n_u}\sum _{j:r(i,j)=1}(({\theta }^{(j)}{)}^T{x}^{(i)}-y^{(i,j)}{)}^2+\frac{\lambda }{2}\sum _{i=1}^{n_u}\sum _{k=1}^n(x_k^{(i)}{)}^2$$
We can iterate $x\to \theta \to x\to \theta \cdots$
$$\begin{gathered} mi{n}_{{\theta }^{(1)},{\theta }^{(2)},\cdots ,{\theta }_{n_u},x^{(1)},x^{(2)},\cdots ,x_{n_u}}\frac{1}{2}\sum _{(i,j):r(i,j)=1}(({\theta }^{(j)}{)}^T{x}^{(i)}-y^{(i,j)}{)}^2 \\ +\frac{\lambda }{2}\sum _{i=1}^{n_u}\sum _{k=1}^n(x_k^{(i)}{)}^2+\frac{\lambda }{2}\sum _{j=1}^{n_u}\sum _{k=1}^n({\theta }_k^{(j)}{)}^2 \end{gathered}$$
predicted ratings

低秩矩阵分解
$$\left[\begin{array}{cccc}{\theta }^{(1)}{x}^{(1)}& {\theta }^{(2)}{x}^{(1)}& \cdots & {\theta }^{(n_u)}{x}^{(1)}\\ \cdots & \cdots & \cdots & \cdots \\ {\theta }^{(1)}{x}^{(n_m)}& {\theta }^{(2)}{x}^{(n_m)}& \cdots & {\theta }^{(n_u)}{x}^{(n_m)}\end{array}\right]=X{O}^T$$

Learning with large datasets

${\theta }_j:={\theta }_j-\alpha \frac{1}{m}{\sum }_{i=1}^m(h_{\theta (x^{(i)})}-y^{(i)})x_j^{(i)}$

如果m特别大，那么全部遍历就不太容易。

Stochastic gradient descent

1. Randomly shuffle training examples

2. # 随机梯度下降，每次只输入一个实例，就直接更新所有参数
   repeat{
       for i in range(m):
       	for j in range(n):
       		update $\theta$
   }
   

Mini-batch gradient descent

• Batch gradient descent: use all m examples in each iteration
• Stochastic gradient descent: use 1 example in each iteration
• Mini-batch gradient: use b examples in each iteration

b=10,m=10000
repeat{
    for i in range(1,m,10):
    	for j in range(n):
    		update $\theta$
}

Online learning

repeat forever{
    get (x,y) corresponding to user
    update \theta using (x,y)
}

可以适应客户变化和环境变化。

Application example Photo OCR

Photo OCR pipeline 机器学习流水线
$$Image\to Textdetection\to Charactersegmentation\to Characterrecognition$$
Synthesizing data by introducing distortions

• Distortion introduced should be representation of the type of noise or distortions in the test set
• Usually does not help to add purely random or meaningless noise to your data. 高斯噪声

Ceiling analysis

What part of the pipeline should you spend the most time to work on next?

天花板分析，在一个组件前提供100%正确的内容，查看某一组件如果是perfect的，会增加多少accuracy，找到增加accuracy最多的组件并首先花费大量时间来完善该组件。