• Coursera - Machine Learning Specialization
• Discourse - DeepLearning.AI

C1 Supervised Machine Learning: Regression and Classification

C1_W1 Introduction to Machine Learning

C1_W1_M1 Supervised vs Unsuperverside Machine Learning

C1_W1_M1_1 What is machine learning?

Field of study that gives computers the ability learn without being explicitly pogrammed -- Arthur Samuel

Supervised vs Unsupervised

C1_W1_M1_2 Supervised Learning: Regression Algorithms

learn x to y or input to output mappings

Input (x)	Output (Y)	Application
email	spam? (0/1)	spam filtering
audio	text transcripts	speech recognition
Enlish	Spanish	machine translation
ad, user	click? (0/1)	online advertising
image, radar	position of other cars	self-driving car
image of phone	defect? (0/1)	visual inspection

• Use different algorithms to predict price of house based on data

C1_W1_M1_3 Supervised Learning: Classification

Regression attempts to predict ininite possible results Classification predicts categories ie from limited possible results

• e.g. Breast cancer detection

• we can have multiple outputs

• we can draw a boundary line to separate our output

Supervised Learning

	Regression	Classification
Predicts	numbers	categories
Outputs	infinite	limited

C1_W1_M1_4 Unsupervised Learning

• With unsupervised we don't have predetermined expected output
• we're trying to find structure in the pattern
• in this example it's clustering
• e.g. Google News will "cluster" news related to pandas given a specific article about panda/birth/twin/zoo

• e.g. given set of individual genes, "cluster" similar genes

• e.g. how deeplearning.ai categorizes their learners

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

Question: Of the following examples, which would you address using an unsupervised learning algorithm?

(Check all that apply.)

• [ ] Given a set of news articles found on the web, group them into sets of articles about the same stories.
• [ ] Given email labeled as spam/not spam, learn a spam filter.
• [ ] Given a database of customer data, automatically discover market segments and group customers into different market segments.
• [ ] Given a dataset of patients diagnosed as either having diabetes or not, learn to classify new patients as having diabetes or not.

Lab 01: Python and Jupyter Notebooks

• Coursera Jupyter
• Local: Jupyter Notebook

Quiz: Supervised vs Unsupervised Learning

Which are the two common types of supervised learning (choose two)

• [ ] Classificaiton
• [ ] Regression
• [ ] Clustering

Which of these is a type of unsupervised learning?

• [ ] Clustering
• [ ] Regression
• [ ] Classification

C1_W1_M2 Regression Model

C1_W1_M2_1 Linear regression model part 1

Linear Regression Model => a Supervised Learning Model that simply puts a line through a dataset

• most commonly used model

• e.g. Finding the right house price based on dataset of houses by sq ft.

Terminology
Training Set	data used to train the model
x	input variable or feature
y	output variable or target
m	number of training examples
(x,y)	single training example
(xⁱ,yⁱ)	i-th training example

C1_W1_M2_2 Linear regression model part 2

f is a linear function with one variable

• $ f_{w,b}(x) = wx + b $ is equivalent to
• $ f(x) = wx + b $

• Univariate linear regression => one variable

Lab 02: Model representation

• Coursera Jupyter: Model representation
• Local: Model Representation

Here is a summary of some of the notation you will encounter.

General Notation	Python (if applicable)	Description
$ a $		scalar, non bold
$ \mathbf{a} $		vector, bold
Regression
$ \mathbf{x} $	x_train	Training Example feature values (in this lab - Size (1000 sqft))
$ \mathbf{y} $	y_train	Training Example targets (in this lab Price (1000s of dollars))
$ x^{(i)}$, $y^{(i)} $	x_i, y_i	$ i_{th} $ Training Example
m	m	Number of training examples
$ w $	w	parameter: weight
$ b $	b	parameter: bias
$ f_{w,b}(x^{(i)}) $	f_wb	The result of the model evaluation at $ x^{(i)} $ parameterized by $ w,b $: $ f_{w,b}(x^{(i)}) = wx^{(i)}+b $

Code

• NumPy, a popular library for scientific computing
• Matplotlib, a popular library for plotting data
  • scatter() to plot on a graph
    • marker for symbol to use
    • c for color

C1_W1_M2_4 Cost function formula

• We can play with w & b to find the best fit line

• 1st step to implement linear function is to define Cost Function
• Given $ f_{w,b}(x) = wx + b $ where w is the slope and b is the y-intercept
• Cost function takes predicted $ \hat{y} $ and compares to y
• ie error = $ \hat{y} - y $
• $ \sum\limits_{i=1}^{m} (\hat{y}^{(i)} - y^{(i)})^{2} $
  • where m is the number of training examples
• Dividing by 2m makes the calculation neater $ \frac{1}{2m} \sum\limits_{i=1}^{m} (\hat{y}^{(i)} - y^{(i)})^{2} $
• Also known as squared error cost function $ J_{(w,b)} = \frac{1}{2m} \sum\limits_{i=1}^{m} (\hat{y}^{(i)} - y^{(i)})^{2} $
• Which can be rewritten as $ J_{(w,b)} = \frac{1}{2m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)}) - y^{(i)})^{2} $
• Remember we want to find values of w,b where $ \hat{y}^{(i)} $ is close to $ y^{(i)} $ for all $ (x^{(i)}, y^{(i)}) $

Question: Which of these parameters of the model that can be adjusted?

• $ w $ and $ b $
• $ f_{w,b} $
• $ w $ only, because we should choose $ b = 0 $
• $ \hat{y} $

C1_W1_M2_5 Cost Function Intuition

To get a sense of how to minimize $ J $ we can use a simplified model

		simplified
model	$ f_{w,b}(x) = wx + b`	$ f_{w}(x) = wx` by setting $ b=0 $
parameters	$ w $, $ b $	$ w $
cost function	$ J_{(w,b)} = \frac{1}{2m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)}) - y^{(i)})^{2} $	$ J_{(w)} = \frac{1}{2m} \sum\limits_{i=1}^{m} (f_{w}(x^{(i)}) - y^{(i)})^{2} $
goal	we want to minimize $ J_{(w,b)} $	we want to minimize $ J_{(w)} $

• we can use simplified function to find the best fit line

• the 2nd graph shows that when $ w = 1 $ then $ J(1) = 0 $

• the 2nd graph shows that when $ w = 0.5 $ then $ J(0.5) ~= 0.58 $

• the 2nd graph shows that when $ w = 0 $ then $ J(0) ~= 2.33 $

• We can do this calculation for various $ w $ even negative numbers
• when $ w = -0.5 $ then $ J(-0.5) ~= 5.25 $

• We can plot various values for w and get a graph (on the right)
• As we can see the cost function with $ w = 1 $ is the best fit line for this data

The goal of linear regression is to find the values of $ w,b $ that allows us to minimize $ J_{(w,b)} $

C1_W1_M2_6 Visualizing the cost function

model	$ f_{w,b}(x) = wx + b $``$
parameters	$ w $, $ b $
cost function	$ J_{(w,b)} = \frac{1}{2m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)}) - y^{(i)})^{2} $
goal	minimize $ J_{(w,b)} $

• When we only have w, then we can plot J vs w in 2-dimensions

• However, when we add b then it's 3-dimensional
• The value of J is the height

• this is easier to visualize as a contour plot

• Same as used to visualize height of mountains

• take a horizontal slice which gives you the same J for given w,b
• the center of the contour is the minimum
• Countour allows us to visualize the 3-D J in 2-D

C1_W1_M2_7 Visualization examples

Here are some examples of J

• We can see this is a pretty bad J

• This is pretty good and close to minimal (but not quite perfect)

In the next lab, you can click on different points on the contour to view the cost function on the graph

Gradient Descent is an algorithm to train linear regression and other complex models

Lab 03: Cost function

• Coursera Jupyter: Cost Function
• Local: Cost Function
  • from lab_utils_uni import plt_intuition, plt_stationary, plt_update_onclick, soup_bowl

Quiz: Regression Model

1. Which of the following are the inputs, or features, that are fed into the model and with which the model is expected to make a prediciton?

• [ ] $ m $
• [ ] $ w $ and $ b $
• [ ] $ (x,y) $
• [ ] $ x $

2. For linear regression, if you find parameters $ w $ and $ b $ so that $ J_{(w,b)} $ is very close to zero, what can you conclude?

• [ ] The selected values of the parameters $ w, b $ cause the algorithm to fit the training set really well
• [ ] This is never possible. There must be a bug in the code
• [ ] The selected values of the parameters $ w, b $ cause the algorithm to fit the training set really poorly

Ans

4, 1

C1_W1_M3 Train the model with gradient descent

C1_W1_M3_1 Gradient descent

Want a systematic way to find values of $ w,b $ that allows us to easily find smallest $ J $

Gradient Descent is an algorithm used for any function, not just in linear regression but also in advanced neural network models

• start with some $ w,b $ e.g. $ (0,0) $
• keep changing $ w,b $ to reduce $ J(w,b) $
• until we settle at or near a minimume

• Example of a more comples $ J $
• not a squared error cost
• not linear regression
• we want to get to the lowest point in this topography
• pick a direction and take a step that is slightly lower
  • repeat until you're at lowest point
• However, depending on starting point and direction, you will end up at a different "lowest point"
  • Known as local mimina

local minima may not be the true lowest point

C1_W1_M3_2 Implementing gradient descent

• The Gradient Descent algorithm
• $ w = w - \alpha \frac{\partial}{\partial w} J_{(w,b)} $
  • $ \alpha $ == learning rate. ie How "big a step" you take down the hill
  • $ \frac{\partial}{\partial w} J_{(w,b)} $ == derivative. ie which direction
• $ b = b - \alpha \frac{\partial}{\partial b} J_{(w,b)} $
• We repeat these 2 steps for $ w,b $ until the algorithm converges
  • ie each subsequent step doesn't change the value
• We want to simultaneously update w and b at each step
  • tmp_w = $ w - \alpha \frac{\partial}{\partial w} J_{(w,b)} $
  • tmp_b = $ b - \alpha \frac{\partial}{\partial b} J_{(w,b)} $
  • w = tmp_w && b = tmp_b

C1_W1_M3_3 Gradient descent intuition

• We want to find minimum w,b

$$ \begin{aligned} \text{repeat until convergence {} \ &w = w - \alpha \frac{\partial}{\partial w} J_{(w,b)}\ &b = b - \alpha \frac{\partial}{\partial b} J_{(w,b)}\ } \end{aligned} $$

• Starting with finding min w we can simplify to just $ J(w) $
• Gradient descent with $ w = w - \alpha \frac{\partial}{\partial w} J_{(w)} $
• minimize cost by adjusting just w: $ \min J(w) $

• Recall previous example where we set b = 0
• Initialize w at a random location
• $ \frac{\partial}{\partial w} J(w) $ is the slope
  • we want to find slopes that take us to minimum w
• In the first case, we get $ w - \alpha (positive number) $ which is the correct direction
• However (2nd graph), slope is negative, and therefore also in the correct direction

C1_W1_M3_4 Learning rate

• $ \alpha $ is the learning rate ie how big a step to take
  • If too small then you take small steps and will take a long time to find minimum
  • If too big then you might miss true minimum ie diverge instead of converge

• If you're already at local minimum...
• slope = 0 and therefore $ \frac{\partial}{\partial w} J(w) = 0 $
  • ie w = w * 0
  • further steps will bring you back here

• As we get closer to local minimum, gradient descent (derivative function) will automatically take smaller steps

C1_W1_M3_5 Gradient descent for linear regression

• linear regression model $ f_{w,b}(x) = wx + b $
• cost function $ J_{(w,b)} = \frac{1}{2m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)} - y^{(i)})^{2} $
• gradient descent algorithm
  • repeat until convergence {
  • $ w = w - \alpha \frac{\partial}{\partial w} J_{(w,b)} $
  • $ b = b - \alpha \frac{\partial}{\partial b} J_{(w,b)} $
  • }
• where $ \frac{\partial}{\partial w} J_{(w,b)} $ = $ \frac{1}{m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)} - y^{(i)})x^{(i)} $
• and $ \frac{\partial}{\partial b} J_{(w,b)} $ = $ \frac{1}{m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)} - y^{(i)}) $

• We can simplify for w

$$ \begin{align} \frac{\partial}{\partial w} J_{(w,b)} \ &= \frac{\partial}{\partial w} \frac{1}{2m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)} - y^{(i)})^2 \ &= \frac{\partial}{\partial w} \frac{1}{2m} \sum\limits_{i=1}^{m} (wx^{(i)} + b - y^{(i)})^2 \ &= \frac{1}{2m} \sum\limits_{i=1}^{m} (wx^{(i)} + b - y^{(i)}) 2x^{(i)} \ &= \frac{1}{m} \sum\limits_{i=1}^{m} (wx^{(i)} + b - y^{(i)}) x^{(i)} \ &= \frac{1}{m} \sum\limits_{i=1}^{m} ((f_{w,b}(x^{(i)}) - y^{(i)}) x^{(i)} \end{align} $$

• and for b

$$ \begin{align} \frac{\partial}{\partial b} J_{(w,b)} \ &= \frac{\partial}{\partial b} \frac{1}{2m} \sum\limits_{i=1}^{m} (f_{w,b}(x^{(i)} - y^{(i)})^2 \ &= \frac{\partial}{\partial b} \frac{1}{2m} \sum\limits_{i=1}^{m} (wx^{(i)} + b - y^{(i)})^2 \ &= \frac{1}{2m} \sum\limits_{i=1}^{m} (wx^{(i)} + b - y^{(i)}) 2 \ &= \frac{1}{m} \sum\limits_{i=1}^{m} (wx^{(i)} + b - y^{(i)}) \ &= \frac{1}{m} \sum\limits_{i=1}^{m} ((f_{w,b}(x^{(i)}) - y^{(i)}) \end{align} $$

• a convex function will have a single global minimum

C1_W1_M3_6 Running gradient descent

• left is plot of the model
• right is contour plot of the cost function
• bottom is the surface plot of the cost function
• for this example, w = -0.1, b = 900
• as we take each step we get closer to the global minimum
  • the yellow line is the best line fit
• Given a house with 1250 sq ft, we can predict it should sell for $250k per the model

• Batch Gradient Descent => each step of the gradient descent uses all the training examples
• DeepLearning.AI newsletter: The Batch

Lab 04: Gradient descent

• Coursera Jupyter: Gradient descent
• Local: Gradient Descent

Quiz: Train the Model with Gradient Descent

1. Gradient descent is an algorithm for finding values of parameters w and b that minimize the cost function J.

$$ \begin{aligned} \text{repeat until convergence {} \ &w = w - \alpha \frac{\partial}{\partial w} J_{(w,b)}\ &b = b - \alpha \frac{\partial}{\partial b} J_{(w,b)}\ } \end{aligned} $$

When $ \frac{\partial}{\partial w} J_{(w,b)} $ is a negative number, what happens to w after one update step?

• [ ] It is not possible to tell is w will increase or decrease
• [ ] w increases
• [ ] w stays the same
• [ ] w decreases

2. For linear regression, what is the update step for parameter b?

• [ ] $ b = b - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} ((f_{w,b}(x^{(i)}) - y^{(i)}) $
• [ ] $ b = b - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} ((f_{w,b}(x^{(i)}) - y^{(i)}) x^{(i)} $

Ans

2, 2

C1_W2: Regression with Multiple Input Variables

This week, you'll extend linear regression to handle multiple input features. You'll also learn some methods for improving your model's training and performance, such as vectorization, feature scaling, feature engineering and polynomial regression. At the end of the week, you'll get to practice implementing linear regression in code.

C1_W2_M1 Multiple Linear Regression

C1_W2_M1_1 Multiple features

• $ \vec{x}^{(i)} $ = vector of 4 parameters for $ i^{th} $ row = $ [1416 3 2 40] $

• In this example, house price increase by (multiply 1k)
  • 0.1 per square foot
  • 4 per bedroom
  • 10 per floor
  • -2 per year old
  • add 80 base price

• We can simplify the model
• From linear algebra, this is a row vector as opposed to column vector
• this is multiple linear regression
  • Not multivariate regression

Quiz

In the training set below (see slide: C1_W2_M1_1 Multiple features), what is $ x_{1}^{(4)} $?

Ans

852

C1_W2_M1_2 Vectorization part 1

Learning to write vectorized code allows you to take advantage of modern numberical linear algebra libraries, as well as maybe GPU hardware.

• Vector can be represented in Python as np.array([1.0, 2.5, -3.3])
• if n is large, this code (on left) is inefficient
• for loop is more concise, but still not efficient
• np.dot(w,x) + b is most efficient using vectorization
• Vectorization has 2 benefits: concise and efficient
• np.dot can use parallel hardware

C1_W2_M1_3 Vectorization part 2

How does vectorized algorithm works...

• Without vectorization, we run calculations linearly
• np.dot works in multiple steps:
  • get values of the vectors w, x
  • In parallel run w[i] * x[i]

C1_W2_Lab01: Python Numpy Vectorization

• Coursera
• Local
  • $ a \cdot b $ returns a scalar
  • e.g. $ [1, 2, 3, 4] \cdot [-1, 4, 3, 2] = 24 $

C1_W2_M1_4 Gradient descent for multiple linear regression

• w & x are now vectors
• have to update all the parameters simultaneously for $ w_{1} .. w_{n} $ as well as $ b $

• Normal Equation

C1_W2_Lab02: Muliple linear regression

• Optional Lab: Multiple linear regression | Coursera
• Local

Quiz: Multiple linear regression

1. In the training set below, what is $ x_4^{(3)} $?

Size	Rooms	Floors	Age	Price
2104	5	1	45	460
1416	3	2	40	232
1534	3	2	30	315
852	2	1	36	178

2. Which of the following are potential benefits of vectorization?

• [ ] It makes your code run faster
• [ ] It makes your code shorter
• [ ] It allows your code to run more easily on parallel compute hardware
• [ ] All of the above

3. To make a gradient descent converge about twice as fast, a technique that almost always works is to double the learning rate $ alpha $

• [ ] True
• [ ] False

Ans

30, 4, F

C1_W2_M2 Gradient Descent in Practice

C1_W2_M2_01 Feature scaling part 1

• Use Feature Scaling to enable gradient descent to run faster

• when we scatterplot size vs bedrooms, we see x has a much larger range than y
• when we contour plot we see an oval
• ie small w(size) has a large change & large w(bedrooms has a small change

• since contour is tall & skinny, gradeient descent may end up bounding back and forth for a long time
• a technique is to scale the data to get a more circular contour plot

We can speed up gradient descent by scaling our features

C1_W2_M2_02 Feature scaling part 2

• scale by dividing $ x_i^{(j)} / \max_x $

• Mean Normalization

• Z-score Normalization also called Gaussian Distribution

• When Feature Scaling we want to range somewhere between -1 <==> 1
• but the range is ok if it's relatively close
• rescale if range is too large or too small

Quiz:

Which of the following is a valid step used during feature scaling? (see bedrooms vs size scatterplot)

• [ ] Multiply each value by the maximum value for that feature
• [ ] Divide each value by the maximum value for that feature

Ans

2

C1_W2_M2_03 Checking gradient descent for convergence

• We can choose $ \alpha $

• Want to minimize cost function $ \min\limits_{\vec{w}, b} J(\vec{w}, b) $

C1_W2_M2_04 Choosing the learning rate

C1_W2_M2_05 Optional Lab: Feature scaling and learning rate

C1_W2_M2_06 Feature engineering

C1_W2_M2_07 Polynomial regression

C1_W2_M2_08 Optional lab: Feature engineering and Polynomial regression

C1_W2_M2_09 Optional lab: Linear regression with scikit-learn

C1_W2_M2_10 Practice quiz: Gradient descent in practice

C1_W2_M2_11 Week 2 practice lab: Linear regression