In this post, you will understand the key differences between Adaline (Adaptive Linear Neuron) and Logistic Regression

1. Activation function
2. Cost function

## Difference in Activation Function

The primary difference is the activation function. In Adaline, the activation function is called as linear activation function while in logistic regression, the activation function is called as sigmoid activation function. The diagram below represents the activation functions for Adaline. The  activation function for Adaline, also called as linear activation function, is the identity function which can be represented as the following:

$$\phi(W^TX) = W^TX$$

Fig 1. Adaline Linear Activation Function Representation

The diagram below represents the activation functions for Logistic Regression. The  activation function for Logistic Regression, also called as sigmoid activation function, is the identity function can be represented as the following:

$$\phi(W^TX) = \frac{1}{(1 + e^-Z)}$$

Fig 2: Logistic Sigmoid Activation Function Representation

## Difference in Cost Function

For Adaline, the cost function or loss function looks like below:

$$J(w) = \sum\limits_{i} \frac{1}{2} (\phi(z^{(i)}) – y^{(i)})^2$$

For Adaline, the goal is to minimize the above sum of squared error function.

For Logistic regression, the cost function is created based on likelihood function that looks like below:

$$L(w) = P(y \vert x; w) = \prod\limits_{i=1}^n P(y^{(i)} \vert x^{(i)}; w) = \prod\limits_{i=1}^n (\phi(z^{(i)}))^{y^{(i)}} (1 – \phi(z^{(i)}))^{(1-y^{(i)})}$$

For logistic regression, the idea is to maximize the above likelihood function. For ease of calculation and numerical stability the above equation is converted into log-likelihood function which is then maximized. The below represents the log-likelihood function:

$$\log(L(w)) = \sum\limits_{i=1}^n [y^{(i)}\log(\phi(z^{(i)})) + (i – y^{(i)})\log(1 – \phi(z^{(i)}))]$$

The above log-likelihood function could be written in the following manner as the cost functionJ(w) that can be minimized using the gradient descent

$$J(w) = \sum\limits_{i=1}^n [-y^{(i)}\log(\phi(z^{(i)})) – (i – y^{(i)})\log(1 – \phi(z^{(i)}))]$$

Note that the superscript represents the ith row. Underscript represents the specific feature in that row.

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