Logistic Regression
Jupyter Demos
▶️ Demo | Logistic Regression With Linear Boundary - predict Iris flower class based on petal_length and petal_width
▶️ Demo | Logistic Regression With Non-Linear Boundary - predict microchip validity based on param_1 and param_2
▶️ Demo | Multivariate Logistic Regression | MNIST - recognize handwritten digits from 28x28 pixel images.
▶️ Demo | Multivariate Logistic Regression | Fashion MNIST - recognize clothes types from 28x28 pixel images.
Definition
Logistic regression is the appropriate regression analysis to conduct when the dependent variable is dichotomous (binary). Like all regression analyses, the logistic regression is a predictive analysis. Logistic regression is used to describe data and to explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.
Logistic Regression is used when the dependent variable (target) is categorical.
For example:
- To predict whether an email is spam (1) or (0).
- Whether online transaction is fraudulent (1) or not (0).
- Whether the tumor is malignant (1) or not (0).
In other words the dependant variable (output) for logistic regression model may be described as:
Logistic Regression Output
Logistic Regression
Logistic Regression
Training Set
Training set is an input data where for every predefined set of features x we have a correct classification y.
Training Set
m - number of training set examples.
Training Set
For convenience of notation, define:
x-zero
Logistic Regression Output
Hypothesis (the Model)
The equation that gets features and parameters as an input and predicts the value as an output (i.e. predict if the email is spam or not based on some email characteristics).
Hypothesis
Where g() is a sigmoid function.
Sigmoid
Sigmoid
Now we my write down the hypothesis as follows:
Hypothesis
Predict 0
Predict 1
Cost Function
Function that shows how accurate the predictions of the hypothesis are with current set of parameters.
Cost Function
Cost Function
Cost Function
Cost function may be simplified to the following one-liner:
Cost Function
Batch Gradient Descent
Gradient descent is an iterative optimization algorithm for finding the minimum of a cost function described above. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or approximate gradient) of the function at the current point.
Picture below illustrates the steps we take going down of the hill to find local minimum.
Gradient Descent
The direction of the step is defined by derivative of the cost function in current point.
Gradient Descent
Once we decided what direction we need to go we need to decide what the size of the step we need to take.
Gradient Descent
We need to simultaneously update 
for j = 0, 1, …, n
Gradient Descent
Gradient Descent
- the learning rate, the constant that defines the size of the gradient descent step
- jth feature value of the ith training example
- input (features) of ith training example
yi - output of ith training example
m - number of training examples
n - number of features
When we use term “batch” for gradient descent it means that each step of gradient descent uses all the training examples (as you might see from the formula above).
Multi-class Classification (One-vs-All)
Very often we need to do not just binary (0/1) classification but rather multi-class ones, like:
- Weather: Sunny, Cloudy, Rain, Snow
- Email tagging: Work, Friends, Family
To handle these type of issues we may train a logistic regression classifier 
several times for each class i to predict the probability that y = i.
One-vs-All
Regularization
Overfitting Problem
If we have too many features, the learned hypothesis may fit the training set very well:
overfitting
But it may fail to generalize to new examples (let’s say predict prices on new example of detecting if new messages are spam).
overfitting
Solution to Overfitting
Here are couple of options that may be addressed:
- Reduce the number of features
- Manually select which features to keep
- Model selection algorithm
- Regularization
- Keep all the features, but reduce magnitude/values of model parameters (thetas).
- Works well when we have a lot of features, each of which contributes a bit to predicting y.
Regularization works by adding regularization parameter to the cost function:
Cost Function
- regularization parameter
Note that you should not regularize the parameter
.
In this case the gradient descent formula will look like the following:
Gradient Descent