# Logistic Regression as a Neural Network (Neural Network Basics)

Logistic Regression is a classification algorithm. It is commonly used for Binary Classification.

Binary Classification is the categorization of items into one of two categories/classes.

## Some Notations

The number of training examples is denoted by **m**. Training examples are denoted as (x,y) tuples where x is independent (features/attributes) and y is dependent (label).

Say we have a pxq image. It will have pxqx3 pixels (for an RGB image). These pxqx3 values are stored in the form of a single column vector **x** having pxqx3 rows. This is how a training image is represented.

We then create a matrix **X** that stores all the m training images as vectors (column-wise). So it will have m columns, each with pxqx3 rows.

Another vector **Y** is used to store the m training labels, column-wise. So it has dimensions 1xm.

## Logistic Regression Equation

We have a vector of features **x**. Given x, we need to predict the probability $$\hat{y}$$ that y=1.

We also have parameters **w** and **b**.

We denote $$\hat{y}$$ as follows:

$$\hat{y} = \sigma(w^T x + b)$$

where $$\sigma$$ is the sigmoid activation function that restricts the value between 0 and 1.

## Logistic Regression Cost Function

To train the parameters w and b, we need a cost function.

Let us consider the Loss Function **L** that denotes the error in the prediction for a given example as follows:

$$L(\hat{y}, y) = - (y log \hat{y} + (1-y) log (1-\hat{y}))$$

Now, we denote the cost function as follows:

$$J(w, b) = \frac{1}{m}\sum\_{i=1}^{m}L(\hat{y}^{(i)}, y^{(i)})$$

i=1 to m denote the m training examples.

Our aim now is to find w and b which minimize the cost function J. This is done using **Gradient Descent**.

## Gradient Descent

Gradient Descent helps us find the global minimum of a function, thereby letting us find optimal w and b values that minimize J.

It is denoted as follows:

Repeat {

$$w := w - \alpha\frac{\partial J(w,b)}{\partial w}$$

$$b := b - \alpha\frac{\partial J(w,b)}{\partial b}$$

}

where $$\alpha$$ is the **learning rate**.


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