Machine Learning Notes

# Machine Learning

标签（空格分隔）： ML

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Introduction

Supervised Learning

Supervised Learning: “right answers” given
Classification: Discrete valued output (0 or 1)
Regression: Predict continuous valued output

Unsupervised Learning

Model and Cost Function

Linear regression with one variable
Univariate linear regression
Hypothesis:

$$h_{\theta }(x)=\theta _{0}+\theta _{1}(x)$$
Parameters:
$$\theta _{0},\theta _{1}$$
Idea: Choose $\theta _{0},\theta _{1}$so that $h_{\theta }(x)$is close to $y$ for our training examples$(x,y)$
Cose Function:
$$\min J(\theta _{0},\theta _{1}) = \frac{1}{2m}\sum_{i=1}^{m}(h_{\theta }(x^{i})-y^{i})^{2}$$
We called that square error function.
$h_{\theta }(x)$ (for fixed $\theta _{0},\theta _{1}$, this is a function of $x$)

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Matrix

Dimension of matrix: number of rows $\times$ number of columns. $\mathbb{R}^{m \times n}$

Gradient descent algorithm

repeat until convergence

$$\theta _{j}:=\theta _{j}-\alpha \frac{\partial J(\theta _{0},\theta _{1})}{\partial \theta _{j}}\; \;\; for(j=0 \, and\, j=1)$$
:= is assignment, and $\alpha$ is learning rate.
the subtlety of how you implement gradient descent

Correct: simultaneous update:
$temp0:=\theta _{0}-\alpha \frac{\partial J(\theta _{0},\theta _{1})}{\partial \theta _{0}}$
$temp1:=\theta _{1}-\alpha \frac{\partial J(\theta _{0},\theta _{1})}{\partial \theta _{1}}$
$\theta _{0}=temp0$
$\theta _{1}=temp1$
Gradient descent can converge to a local minimum, even with the learning rate in a fixed.
As we approach a local minimum, gradient descent will automatically take smaller steps. So, no need to decrease &\alpha$ over time.

WEEK 2

Linear Regression with multiple variables: Multiple Features

$x^{(i)}$=input (features) of $i^{(th)}$ training example.
$x_{j}^{i}$=value if feature $j$ in $i^{th}$ training example.

Hypothesis

$h_{\theta}(x)=\theta_{0}+\theta_{1}x_{1}+\theta_{2}x_{2}+\cdots +\theta_{n}x_{n}$
For convenience of notation, define $x_{0}=1$

$$x=\begin{bmatrix} x_{0}\\ x_{1}\\ x_{2}\\ \dots\\ x_{n} \end{bmatrix} \: \: \: \: \: \: \: \: \: \: \: \: \theta =\begin{bmatrix} \theta _{0}\\ \theta _{1}\\ \theta _{2}\\ \dots\\ \theta _{n} \end{bmatrix}$$
$$h_{\theta}(x)=\theta^{T}x$$

Gradient Descent for Multiple Variables

Hypothesis:$h_{\theta}(x)=\theta^{T}=\theta_{0}x_{0}+\theta_{1}x_{1}+\theta_{2}x_{2}+\cdots +\theta_{n}x_{n}$
Parameters:$\theta_{0},\theta_{1},\dots,\theta_{n}$
Cost function:

$$J(\theta _{0},\theta _{1},\dots,\theta_{n}) =J(\theta)= \frac{1}{2m}\sum_{i=1}^{m}(h_{\theta }(x^{i})-y^{i})^{2}$$
Gradient Descent
repeat until convergence
$$\theta _{j}:=\theta _{j}-\alpha \frac{\partial J(\theta _{0},\theta _{1})}{\partial \theta _{j}}\; \;\; for(j=0 ,1,\dots,n)$$

Feature Scaling

Mean Normalization

Learning rate

• “Debugging”: How to make sure gradient descent is working correctly.
• How to choose learning rate $\alpha$

Housing prices prediction

$$h_{\theta}(x)=\theta_{0}+\theta_{1}\times frontage+\theta_{2}\times depth$$
define new features
$$h_{\theta}(x)=\theta_{0}+\theta_{1}\times Area$$

Polynomial regression

Choice of features

$$h_{\theta}(x)=\theta_{0}+\theta_{1}\times(size)+\theta_{2}\times (size)^{2}$$
$$h_{\theta}(x)=\theta_{0}+\theta_{1}\times(size)+\theta_{2}\times \sqrt {size}$$

Computing Parameters Analytically

Normal Equation

Method to solve for $\theta$ analytically.

$$\theta = (X^{T}X)^{-1}X^{T}y$$
+ No need to chhose $\alpha$
+ Don’t need to iterate.
+ Need to compute $(X^{T}X)^{-1}$
+ Slow if $n$ is very large.

Octave

Moving Data Around

    load('featuresX.dat')
    load featuresX.dat
    who %veriables in the current scope
    whos
    clear
    save hello.mat v
    save hello.txt v %save as text(ASCII)
    A(2,:)  %':' means every elements along that row
    A([1 3],:)
    A = [A,[100;200;300]];
    A[:] %put all elements of A into a single vector

Computing on Data

    A.*B
    A*B
    A.^2
    v=[1;2;3]
    1 ./ v
    log(v)
    exp(v)
    abs(v)
    -v
    v + ones(length(v),1)
    a = [1 2 3 4]
    [val,ind]=max(a)
    a < 1
    find(a<3)
    A = magic(3)
    [r,c] = find(A>=7)
    sum(a)
    prod(a)
    floor(a)
    ceil(a)
    max(rand(3），rand(3))
    max(A,[],1)
    max(A,[],2)
    max(max(A))
    A.*eye(3)
    pinv(A)

Plotting Data

    plot

Vectorization

Week 3

Classification and Represstation

Classification

$y=0$ or $1$
$h_\theta(x)$ can be $>1$or
Logistic Regression: $0 \leq h_\theta(x) \leq 1$

Hypothesis Representation

Logistic Regression Model
want $0 \leq h_\theta(x) \leq 1$

$$h_\theta(x)=g(\theta^{T}x)$$
$$g(z)=\frac{1}{1+e^{-z}}$$
Sigmoid function or logistic function
Interpretation of Hypothesis Output
$h_\theta(x)=$ estimated probability that $y=1$ on input $x$

Cost Function

convex
Logistic regression cost function

$$J(\theta)=\frac{1}{m}\sum_{i=1}^{m}Cost(h_{\theta}(x^{i}),y^{i})$$
$$Cost(h_\theta(x),y)=\left\{\begin{matrix} -log(h_\theta(x))\: \: if y=1\\ -log(1-h_\theta(x))\: \: if y=0 \end{matrix}\right.$$
Note: $y=0$ or $1$ always
Simplified Cost Function
$$Cost(h_{\theta}(x^{i}),y^{i})=-y^{(i)}log(h_{\theta}(x^{(i)}))-(1-y^{(i)})log(1-h_{\theta}(x^{(i)}))$$
$$J(\theta)=-\frac{1}{m} \sum_{i=1}^{m}\left [y^{(i)}log(h_{\theta}(x^{(i)}))+(1-y^{(i)})log(1-h_{\theta}(x^{(i)}))\right ]$$
Want $min_{\theta}J(\theta):$
Repeat{
$$\theta_{j}:=\theta_{j}-\alpha \sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})x_{j}^{(i)}$$
(simultaneously update all $\theta_{j}$)
}

Algorithm looks identical to linear regression!

Regularization

Overfitting

Cost Function

$$J(\theta)= \frac{1}{2m}\left [ \sum_{i=1}^{m}(h_{\theta }(x^{(i)})-y^{(i)})^{2}+\lambda\sum_{j=1}^{n}\theta_{j}^{2} \right ]$$

Gadient descent

Repeat{

$$\theta _{0}:=\theta _{0}-\alpha \frac{1}{m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})x_{0}^{i}$$
$$\theta _{j}:=\theta _{j}-\alpha \left [\frac{1}{m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})x_{j}^{i} +\frac{\lambda}{m}\theta _{j} \right ]$$
}
$$\theta _{j}:=\theta _{j}(1-\alpha\frac{\lambda}{m})-\alpha \frac{1}{m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})x_{j}^{i}$$

Non-invertibility (optional/advanced)

Suppose $m \leq n,$ (#examples) (#features)

$$\theta=(X^{T}X)^{-1}X^{T}y$$
if$\;ambda>0,$
$$\theta =\left ( X^{T}X+\lambda\begin{bmatrix} 0 & & & & \\ & 1& & & \\ & & 1& & \\ & & & \ddots & \\ & & & & 1 \end{bmatrix} \right )^{-1}X^{T}y$$

Week 4

$a_i^{(j)}=$”activation” of unit i in layer j
$\Theta^{(j)}=$matrix of weights controlling function mapping from layer j to layer j+1

$$a_{1}^{(2)}=g(\Theta_{10}^{(1)}x_{0}+\Theta_{11}^{(1)}x_{1}+\Theta_{12}^{(1)}x_{2}+\Theta_{13}^{(1)}x_{3})$$
$$a_{2}^{(2)}=g(\Theta_{20}^{(1)}x_{0}+\Theta_{21}^{(1)}x_{1}+\Theta_{22}^{(1)}x_{2}+\Theta_{23}^{(1)}x_{3})$$
$$a_{3}^{(2)}=g(\Theta_{30}^{(1)}x_{0}+\Theta_{31}^{(1)}x_{1}+\Theta_{32}^{(1)}x_{2}+\Theta_{33}^{(1)}x_{3})$$
$$h_{\Theta}(x)=a_{1}^{(3)}=g(\Theta_{10}^{(2)}a_{0}+\Theta_{11}^{(2)}a_{1}+\Theta_{12}^{(2)}a_{2}+\Theta_{13}^{(2)}a_{3})$$
If network has $s_j$ units in layer $j$, $s_{j+1}$ units in layer $j+1$, then $\Theta^{(j)}$ will be of demension $s_{j+1}\times (s_{j}+1)$.
$$z_{1}^{2}=\Theta_{10}^{(1)}x_{0}+\Theta_{11}^{(1)}x_{1}+\Theta_{12}^{(1)}x_{2}+\Theta_{13}^{(1)}x_{3}$$

Forward propagation: Vectorized implementation

$$x=\begin{bmatrix} x_{0}\\ x_{1}\\ x_{2}\\ x_{3} \end{bmatrix} \: \: \: \: \: \: \: \: z^{(2)}=\begin{bmatrix} z_{1}^{(2)}\\ z_{2}^{(2)}\\ z_{3}^{(2)} \end{bmatrix}$$
$$z^{(2)}=\Theta^{(1)}x$$
$$a^{2}=g(z^{(2)})$$
$$Add \: \: a_{0}^{(2)}=1$$
$$z^{(3)}=\Theta^{(2)}a^{(2)}$$