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

- Activation function
- Cost function

Table of Contents

## 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:

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

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:

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

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 function**, * J(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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