Machine Learning 学习笔记（第一周）

机器学习 吴恩达

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1. welcome to machine learning

(quote) machine learning is the science of getting computers to learn, without being explicitly programmed.
Machine Learning at Stanford Visit ml-class.org to enroll
leacture slide:
Machine Learning
- Grew out of work in AI
- New capability for computers
Examples:
- Database mining
Large datasets from growth of automation/web.
E.g., Web click data ,medical records, biology, engineering
- Applications can't program by hand.
E.g., Autonomous helicopter, handwriting recognition, most of Natural Language Processing(NLP), Computer Vision.
- Self-customizing programs
E.g., Amazon, Netflix product recommendations
- Understanding human learning(brain, real AI)

2.Introduction-Supervised Learning

• Housing price prediction
  supervised learning is often called regression problems:predict continous values
• Estimate Breast Cancer probability(malignant or benign):predict discontinous values
  features:clump thickness,uniformity of cell size, uniformity of cell shape
  key words:features, infinity features
  (note)in supervised learning, in every example in our dataset, we are told what is the "correct answer" that we would have quite liked the algorithms have predicted on that example.

3.Unsupervised Learning

no labels or features
there is a dataset, and can you find the structure

• cluster algorithm, e.g.google news gather news in simillar topics

• applications:

  1. organize large computer clusters, and try to figure out which machine should work together
  2. social network analysis
  3. market segmentation
  4. astronomical data analysis

example: cocktail party problem

use Octave to code

4.Linear Regression with One Variable

• Regression Problem
  Predict real-valued output
  training set <--> test set
  $(x,y)$one training sample
  $(x^{（i）},y^{（i)}）$, i th training sample
  Process of learning algorithms:
  Training Set------> Learning Algorithm------>h(hypothesis):input x, output estimated_y
  h is a function that maps from x to y
  How to represent h?
  in this linear regression problem,
  

  $$h_{\theta}(x)=\theta_0+\theta_1x$$
  a linear regression with one variable(univariate linear regression).

• Cost Function
  problem to choose(calculate) parameters $\theta_0, \theta_1$
  idea:choose $\theta_0,\theta_1$ so that $h_\theta (x)$ is close to $y$ for our training samples$(x,y)$
  it's a minimizing problem, let's see:
  

  $$\min_{\theta_0,\theta_1}\{{\frac{1}{2m}\sum_{i=1}^m(h_\theta(x^{(i)})-y^{(i)})^2}\}$$
  $m$ is the size of training set
  above formula is the overall objective function for linear regression
  rewrite:
  $$J(\theta_0,\theta_1)=\{{\frac{1}{2m}\sum_{i=1}^m(h_\theta(x^{(i)})-y^{(i)})^2}\}$$
  $$\min_{\theta_0,\theta_1}J(\theta_0,\theta_1)$$
  and $J(\theta_0,\theta_1)$ is called the cost function, and also called the squared error function

• Cost Function -- intuition I

• Cost Function -- intuition II
  contour plot: two parameters for $J(\theta_0,\theta_1)$

5.Parameter Learning

• Gradient Descent
  gradient descent is going to minimize the cost function $J(\theta_0,\theta_1)$
  Have some function $J(\theta_0,\theta_1)$
  want

  $$\min_{\theta_0,\theta_1}J(\theta_0,\theta_1)$$
  outline :
  start with some $\theta_0,\theta_1$
  keep changing $\theta_0,\theta_1$, to reduce $J(\theta_0,\theta_1)$
  until we hopefully end up at a minimum

• Gradient Descent Algorithm
  

  $$repeat\; until\; convergence{ \theta_j:\theta_j-\alpha\frac{\partial}{\partial\theta_j}J(\theta_0,\theta_1)\quad (for\; j=0 and j=1) }$$
  correct: Simultaneous update
  $$temp0 :=\theta_0-\alpha\frac{\partial}{\partial\theta_0}J(\theta_0,\theta_1)\\ temp1 :=\theta_1-\alpha\frac{\partial}{\partial\theta_1}J(\theta_0,\theta_1)\\ \theta_0:=temp0\\ \theta_1:=temp1$$
  $\alpha$ is called the learning rate

• Gradient Descent -- intuition
  gradient descent can converge to a local minimum, even with the learning rate $\alpha$ fixed
  $\theta_1 :=\theta_1-\alpha\frac{d}{d\theta_1}J(\theta_0,\theta_1)$
  as we approach a local minimum, gradient descent will automatically take smaller steps. So, no need to decrease $\alpha$ over time

• Gradient Descent for linear regression
  for the linear regression model, the partial derivative of $J(\theta_0,\theta_1)$ is:
  

  $$\frac{\partial}{\partial\theta_j}J(\theta_0,\theta_1)=\frac{\partial}{\partial\theta_j}*\frac{1}{2m}\sum_{i=1}^{m}(h_{\theta}(x^{(i)})-y^{(i)})^2$$
  $$for\; j=0:\frac{\partial}{\partial\theta_0}J(\theta_0,\theta_1)=\frac{1}{m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})$$
  $$for\; j=1:\frac{\partial}{\partial\theta_1}J(\theta_0,\theta_1)=\frac{1}{m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})*x^{(i)}$$

so put the slope back into the gradient algorithm:

$$repeat\; until\; convergence\{\\ \theta_0:=\theta_0-\alpha\frac{1}{m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})\\ \theta_1:=\theta_1-\alpha\frac{1}{m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})*x^{(i)}\\ \}$$

it turns out to be that the cost function of linear regression model is often a convex function, which doesn't have any local optimum, except for one global optimum

• "Batch" Gradient Descent
  "Batch":each step of gradient descent uses all trainingexamples

  1. Linear Algebra review(optional)

• Matrices and vectors

• Addition and scalar multiplication
• Matrix Vector multiplication
• Matrix Matrix multiplication

• Matrix multiplication properties

  1. let A and B be matrices. Then in general, $A\times B\ne B\times A$
  2. Identity Matrix $I_{n\times n}$

• Inverse and Transpose
  sigular and degenerate matrices