1. Introduction to machine learning

Supervised learning

input x → output y (learns from being right answers)

• Examples
  • Spam email filtering
  • Audio text transcripts (speech recognition)
  • Language translations
  • Online advertisement
  • Self-driving car
  • Visual inspection

Regression

Regression predicts a number infinitely many possible outputs

Classification

Classification predicts categories, small number of possible outputs

Two or more inputs

Unsupervised Learning

Find something interesting in unlabeled data

• example

  • Google news

    

  • DNA microarray

    

Unsupervised learning

Data only comes with inputs x, but not output labels y: Algorithm has to find structure in the data

• clustering - group similar data points together
• Anomaly detection - find unusual data points
• Dimensionality reduction - compress data using fewer numbers

Linear Regression Model

Terminology

• Training set - data used to train the model
• x - input variable feature
• y - output variable, target variable
• m - number of training examples
• (x,y) - single training example
• $(x^{(i)},y^{(i)})$ - ith training example
  • ex) $(x^{(1)}, y^{(1)}) = (2104,400)$

Cost Function

What do w, b do?

Find w,b:

$\hat{y}^{(i)}$ is close to $y^{(i)}$ for all $(x^{(i)},y^{(i)})$

Cost function

cost function is to see how well the model is doing

$$J(w,b) = \frac{1}{2m}\sum_{i=1}^m (f_w,_b(x^{(i)})-y^{(i)})^2$$

• error : $\hat{y}^{(i)}-y^{(i)}$ (prediction_i - realvalue_i)
  • different people use different cost function
  • squared error cost function is the most commonly used one

Cost function intuition

• model
  • $f_w,_b(x) = wx + b$
• parameter
  • $w, b$
• cost function
  • $J(w,b) = \frac{1}{2m}\sum_{i=1}^m (f_w,_b(x^{(i)})-y^{(i)})^2$
• goal
  • $minimize_w,_bJ(w,b)$

Simplified cost function

• model

  • $f_w(x) = wx$

• parameter

  • $w$

• cost function

  • $J(w) = \frac{1}{2m}\sum_{i=1}^m (f_w(x^{(i)})-y^{(i)})^2$

• goal

  • $minimize_wJ(w)$

• examples

  

  

  

Visualizing the cost function

• examples

  

  

  

Gradient Descent

Have some function $J(w,b)$ for linear regression or any function

Want $min_w,_bJ(w,b)$

Outline:

• Start with some w, b (set w=0,b=0)
• keep changing w,b to reduce J(w,b)
• until we settle at or near a minimum

  • may have > 1 minimum

    

Implementing gradient descent

Gradient descent algorithm

simultaneously update w and b

$$w = w - a\frac{d}{dw}J(w,b)$$ $$b = b - a\frac{d}{db}J(w,b)$$

$a$: learning rate

$\frac{d}{dw}J(w,b)$: Derivative

Gradient descent intuition

Learning rate

if a is too small, gradient descent may be slow

if a is too large, gradient descent may:

• overshoot, never reach minimum
• fail to converge, diverge

Gradient descent for linear regression

Linear regression model

$$f_w,_b(x) = wx + b$$

Cost function

$$J(w,b) = \frac{1}{2m}\sum_{i=1}^m (f_w,_b(x^{(i)})-y^{(i)})^2$$

Gradient descent algorithm

repeat until convergence

$$w = w - a\frac{d}{dw}J(w,b)$$

• $w = w - a\frac{d}{dw}J(w,b)$
  • $\frac{1}{m}\sum_{i=1}^m(f_w,_b(x^{(i)})-y^{(i)})x^{(i)}$

$$b = b - a\frac{d}{db}J(w,b)$$

• $b = b - a\frac{d}{db}J(w,b)$
  • $\frac{1}{m}\sum_{i=1}^m(f_w,_b(x^{(i)})-y^{(i)})$

• squared error cost will never have local minimum
• gradient descent with convex function will always converge with global minimum

Mathematics

$\frac{1}{m}\sum_{i=1}^{m}(f_w,_b(x^{(i)})-y^{(i)})x^{(i)}$

1. $\frac{d}{dw}J(w,b)$
2. $\frac{d}{dw}\frac{1}{2m}\sum_{i=1}^{m}(f_w,_b(x^{(i)})-y^{(i)})^2$
3. $\frac{d}{dw}\frac{1}{2m}\sum_{i=1}^{m}(wx^{(i)}+b-y^{(i)})^2$
4. $\frac{1}{2m}\sum_{i=1}^{m}(wx^{(i)}+b-y^{(i)})2x^{(i)}$

$\frac{1}{m}\sum_{i=1}^{m}(f_w,_b(x^{(i)})-y^{(i)})$

1. $\frac{d}{db}J(w,b)$
2. $\frac{d}{db}\frac{1}{2m}\sum_{i=1}^{m}(f_w,_b(x^{(i)})-y^{(i)})^2$
3. $\frac{d}{db}\frac{1}{2m}\sum_{i=1}^{m}(wx^{(i)}+b-y^{(i)})^2$
4. $\frac{1}{2m}\sum_{i=1}^{m}(wx^{(i)}+b-y^{(i)})2$

Running gradient descent

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Source

Machine Learning