Regression takes a group of random variables, thought to be predicting Y, and tries to find a mathematical relationship between them. This relationship is typically in the form of a straight line (linear regression) that best approximates all the individual data points.

1. Regression
Machine Learning Perspective
BY:
Suyash pratap Singh and Vineet raj Parashar
JUET,GUNA

2. Good understanding of how regression works can help:
• Having a the appropriate model
• Right training algorithm to use
• Good set of hyperparameters for your task.
• Debug issues and perform error analysis more efficiently
• Lastly, most of the topics discussed will be essential in understanding,
building, and training neural networks

3. Agenda
• Linear Regression model
• Closed-form equation
• Normal Equation
• Iterative optimization approach
• Gradient Descent
• Variants

4. Polynomial Regression
• More complex model that can fit nonlinear datasets.
• It has more parameters than Linear Regression so it is more prone to
overfitting the training data
• How to detect overfitting
• Learning curves

5. Regularization and Logistic Regression
• Regularization methods
• Techniques that can reduce the risk of overfitting the training set
• Finally, we will look at two more models that are commonly used for
classification tasks:
• Logistic Regression
• Softmax Regression.

6. Prerequisite: Linear algebra and calculus
• Vectors
• Matrices
• Operations
• Transpose
• Dot product
• Matrix inverse
• Partial derivatives

7. Linear Regression
• Simple regression model of life satisfaction:
life_satisfaction =
• GDP_per_capita is the input feature
• This model is just a linear function of the input feature
• ϴ0 andϴ1 are the model’s parameters
• More generally, a linear model makes a prediction by simply computing a
weighted sum of the input features, plus a constant called the bias term

8. Linear Regression model prediction

9. Linear Regression: Vectorized Form
• This can be written much more concisely using a vectorized form, as
shown below:

10. Performance Measures
• Above given equation is the linear regression model
• Now question is : how to train this model?
• Training a model is to find values its parameters so that the model best fits
the training set.
• What is meaning of best here?
• For this purpose, we first need a measure of how well (or poorly) the model fits the
training data
• Most common performance measure of a regression model is the Root Mean Square
Error (RMSE)
• Therefore, to train a Linear Regression model, you need to find the value of ϴ that
minimizes the RMSE.
• RMSE is more sensitive to outliers

11. RMSE
• RMSE requires MSE
• The MSE of a Linear Regression hypothesis hϴ on a training set X is
calculated using

12. Mean Absolute Error

13. Notations
Age PClaass Gender Fare
67 1 1 1200
54 2 1 1000
8 3 0 600
18 3 1 500
2 1 0 650
• m is the number of instances in the dataset we are measuring the RMSE on
• x(i) is a vector of all the feature values (excluding the label) of the ith instance in the dataset, and
y(i) is its label (the desired output value for that instance)
• For instance : First passenger in the dataset with name “Alan” his age is 67 years and travelled in
1st class and he/she has to pay 1200 dollars
• x (1)=
67
1
1
• y(1) = 1200

14. Vectorized Dataset
67 1 1
54 2 1
8 3 0
18 3 1
2 1 0
𝑦 =
1200
1000
600
500
650

15. Prediction error
• For example, if your system predicts that the fare for the first
passenger is $1110, then ŷ(1) = h(x(1)) = $1110.
• The prediction error for this passenger is:
 ŷ(1) – y(1) = $90

16. The Normal Equation
• To find the value of θ that minimizes the cost function, there is a
closed-form solution
• in other words, a mathematical equation that gives the result directly.
This is called the Normal Equation
• θ is the value of θ that minimizes the cost function.
• y is the vector of target values containing y(1) to y(m).

17. Training Complexity
• The Normal Equation computes the inverse of XT X, which is an (n +
1) × (n + 1) matrix (where n is the number of features). The
computational complexity of inverting such a matrix is typically about
O(n2.4) to O(n3) (depending on the implementation).
• In other words, if you double the number of features, you multiply
the computation time by roughly 22.4 = 5.3 to 23 = 8.
• The SVD approach used by Scikit-Learn’s LinearRegression class is
about O(n2).
• If you double the number of features, you multiply the computation
time by roughly 4.

18. When to use Normal Equation?
• Both the Normal Equation and the SVD approach get very slow when
the number of features grows large (e.g., 100,000).
• On the positive side, both are linear with regards to the number of
instances in the training set (they are O(m)), so they handle large
training sets efficiently, provided they can fit in memory.

19. Prediction Complexity
• Also, once you have trained your Linear Regression model (using the
Normal Equation or any other algorithm), predictions are very fast
• The computational complexity is linear with regards to both the
number of instances you want to make predictions on and the
number of features.
• In other words, making predictions on twice as many instances (or
twice as many features) will just take roughly twice as much time.

20. When not to use Normal Equation:
• Cases where there are a large number of features
• Too many training instances to fit in memory
• Now we have to look at very different ways to train a Linear
Regression model

21. Gradient Descent
• Gradient Descent is a very generic optimization algorithm capable of
finding optimal solutions to a wide range of problems.
• The general idea of Gradient Descent is to tweak parameters
iteratively in order to minimize a cost function.
• How it works?
• It measures the local gradient of the error function with regards to the
parameter vector θ, and it goes in the direction of descending gradient. Once
the gradient is zero, you have reached a minimum!
• Concretely, you start by filling θ with random values (this is called random initialization), and
then you improve it gradually, taking one baby step at a time, each step attempting to
decrease the cost function (e.g., the MSE), until the algorithm converges to a minimum

22. Cost Function

23. Learning Rate: Too small

24. Learning Rate: Too large

25. GD Pitfalls
• Finally, not all cost functions look like nice regular bowls. There may
be holes, ridges, plateaus, and all sorts of irregular terrains, making
convergence to the minimum very difficult.

26. Global minimum
• Fortunately, the MSE cost function for a Linear Regression model
happens to be a convex function, which means that if you pick any
two points on the curve, the line segment joining them never crosses
the curve.
• This implies that there are no local minima, just one global minimum.
It is also a continuous function with a slope that never changes
abruptly.
• These two facts have a great consequence: Gradient Descent is
guaranteed to approach arbitrarily close the global minimum (if you
wait long enough and if the learning rate is not too high).

27. Importance of Feature Scaling
• The cost function has the shape of a bowl, but it can be an elongated bowl if the features have very
different scales.
• Figure shows Gradient Descent on a training set where features 1 and 2 have the same scale (on the left),
and on a training set where feature 1 has such smaller values than feature 2 (on the right).
• When using Gradient Descent, you should ensure that all features have a similar scale (e.g., using Scikit-
Learn’s StandardScaler class), or else it will take much longer to converge.

28. Model’s parameter space
• Above diagram also illustrates the fact that training a model means
searching for a combination of model parameters that minimizes a
cost function (over the training set).
• It is a search in the model’s parameter space: the more parameters a
model has, the more dimensions this space has, and the harder the
search is: searching for a needle in a 300-dimensional haystack is
much trickier than in three dimensions.
• Fortunately, since the cost function is convex in the case of Linear
Regression, the needle is simply at the bottom of the bowl.

29. Batch Gradient Descent
• To implement Gradient Descent, you need to compute the gradient of
the cost function with regards to each model parameter ϴj.
• In other words, you need to calculate how much the cost function will
change if you change ϴj just a little bit.
• This is called a partial derivative.
• Following equation computes the partial derivative of the cost
function with regards to parameter ϴj

30. Gradient Vector of a Cost Function

31. Complexity of Gradient Descent
• Notice that this formula involves calculations over the full training set
X, at each Gradient Descent step!
• This is why the algorithm is called Batch Gradient Descent: it uses the
whole batch of training data at every step
• As a result it is terribly slow on very large training sets
• However, Gradient Descent scales well with the number of features;
training a Linear Regression model when there are hundreds of
thousands of features is much faster using Gradient Descent than
using the Normal Equation or SVD decomposition

32. Learning Rate
• Once you have the gradient vector, which points uphill, just go in the
opposite direction to go downhill.
• This means subtracting ∇θMSE(θ) from θ
• This is where the learning rate η comes into play: multiply the
gradient vector by η to determine the size of the downhill step as
shown in below Equation:

33. Batch Gradient Implementation:
import numpy as np
eta=.1
np.random.seed(0)
n_iter=1000
m=100
x=2*np.random.rand(100,1)
y=4+3*x+np.random.randn(100,1)
x_b=np.c_[np.ones((100,1)),x]
theta=np.random.rand(2,1)
print(theta)
for i in range(n_iter):
gradient=2/m*x_b.T.dot(x_b.dot(theta)-y)
#print(i,"g",gradient)
theta=theta-eta*gradient
#print("iteration no:",i,"theta:",theta)
print(theta)

34. Using different learning rate eta

35. Learning rate selection and stopping criteria
• To find a good learning rate, you can use grid search
• However, you may want to limit the number of iterations so that grid
search can eliminate models that take too long to converge
• How many iterations
• Too large
• Too small
• Gradient Vector value <=Tolerance

36. Convergence Rate
• When the cost function is convex and its slope does not change
abruptly (as is the case for the MSE cost function), Batch Gradient
Descent with a fixed learning rate will eventually converge to the
optimal solution, but you may have to wait a while:
• It can take O(1/ϵ) iterations to reach the optimum within a range of ϵ
depending on the shape of the cost function.
• If you divide the tolerance by 10 to have a more precise solution, then
the algorithm may have to run about 10 times longer.

37. Feature Scaling
• When using Gradient Descent, you should ensure that all features
have a similar scale (e.g., using Scikit-Learn’s StandardScaler class), or
else it will take much longer to converge.

38. Stochatic GD
• The main problem with Batch Gradient Descent is the fact that it uses the whole training set to
compute the gradients at every step, which makes it very slow when the training set is large.
• At the opposite extreme, Stochastic Gradient Descent just picks a random instance in the training
set at every step and computes the gradients based only on that single instance.
• Obviously this makes the algorithm much faster since it has very little data to manipulate at every
iteration.
• It also makes it possible to train on huge training sets, since only one instance needs to be in
memory at each iteration (SGD can be implemented as an out-of-core algorithm)
• On the other hand, due to its stochastic (i.e., random) nature, this algorithm is much less regular
than Batch Gradient Descent: instead of gently decreasing until it reaches the minimum, the cost
function will bounce up and down, decreasing only on average.
• Over time it will end up very close to the minimum, but once it gets there it will continue to
bounce around, never settling down
• So once the algorithm stops, the final parameter values are good, but not optimal.

39. SGD Implementationimport numpy as np
np.random.seed(0)
n_epochs=50
m=100
x=2*np.random.rand(100,1)
y=4+3*x+np.random.randn(100,1)
x_b=np.c_[np.ones((100,1)),x]
t,t1=5,50
def learning_schedule(t):
return t/(t+t1)
theta=np.random.randn(2,1)
for epoch in range(n_epochs):
for i in range(m):
random_index=np.random.randint(m)
xi=x_b[random_index:random_index+1]
yi=y[random_index:random_index+1]
gradients=2*xi.T.dot(xi.dot(theta)-yi)
eta=learning_schedule(epoch*m+i)
theta=theta-eta*gradients
print(theta)

40. Linear Regression Algorithms: Comparision

41. Graph
Problem of Overfitting
&
Learning Curves

43. Regularization
• As we saw in Chapters 1 and 2, a good way to reduce overfitting is to
regularize the model (i.e., to constrain it): the fewer degrees of
freedom it has, the harder it will be for it to overfit the data
• For example, a simple way to regularize a polynomial model is to the
reduce number of polynomial degrees.
• Three different ways to constrain the weights (i.e. the number of
polynomial degrees):
• Ridge Regression, Lasso Regression, and Elastic Net

44. Ridge cost function Ridge closed form equation
Lasso cost function Lasso gradient calculation
Elastic-Net cost function
Elastic-Net gradient calculation:
𝑀𝑆𝐸 𝜃 + r ∝ sign 𝜃 + (1 − r) ∝ 𝜃
Ridge Gradient calculation
𝑀𝑆𝐸 𝜃 +∝ 𝜃

45. Sklearn (ridge, lasso, elasticnet)

46. Sklearn – Linear, GD and Polynomial regression

47. Logistic Regression
• Logistic Regression (also called Logit Regression) is commonly used to
estimate the probability that an instance belongs to a particular class
• If the estimated probability is greater than 50%, then the model
predicts that the instance belongs to the positive class else negative
class
• This makes it a binary classifier
• Instead of outputting the result directly like the Linear Regression
model does, it outputs the logistic of this result

48. Logistic Regression
• Logistic Regression model estimated probability (vectorized form):
• The logistic—noted σ(・)—is a sigmoid function (i.e., S-shaped) that
outputs a number between 0 and 1.
• It is defined as shown in Equation and as shown in figure (matplotlib)

49. Sigmoid Function
• Notice that σ(t) < 0.5 when t < 0, and σ(t) ≥ 0.5 when t ≥ 0, so a
Logistic Regression model predicts 1 if xT θ is positive, and 0 if it is
negative.

50. Prediction
• Once the Logistic Regression model has estimated the probability p =
hθ(x) that an instance x belongs to the positive class, it can make its
prediction ŷ easily

51. Cost function
• Good, now you know how a Logistic Regression model estimates
probabilities and makes predictions.
• But how is it trained?
• The objective of training is to set the parameter vector θ so that the
model estimates high probabilities for positive instances (y =1) and
low probabilities for negative instances (y = 0).
• This idea is captured by the cost function shown in Equation

52. Continue:
• This cost function makes sense because – log(t) grows very large
when t approaches 0, so the cost will be large if the model estimates
a probability close to 0 for a positive instance, and it will also be very
large if the model estimates a probability close to 1 for a negative
instance.
• On the other hand, – log(t) is close to 0 when t is close to 1, so the
cost will be close to 0 if the estimated probability is close to 0 for a
negative instance or close to 1 for a positive instance, which is
precisely what we want.

53. Logistic cost function partial derivatives

54. Comment:
• It is often the case that a learning algorithm will try to optimize a
different function than the performance
• measure used to evaluate the final model. This is generally because
that function is easier to compute, because
• it has useful differentiation properties that the performance measure
lacks, or because we want to constrain
• the model during training, as we will see when we discuss
regularization.

55. Knowledge:
• Gradient Descent (GD), that gradually tweaks the model parameters
to minimize the cost function over the training set, eventually
converging to the same set of parameters as the first method.
• Variants of Gradient Descent that will be used in neural networks:
Batch GD, Mini-batch GD, and Stochastic GD.

56. Thank You!!!
"Success is not an accident, it is hard work,
perseverance, learning, sacrifice and most of all
determination towards what you are doing."