The Sigmoid Activation Function - Python Implementation

An activation function is a mathematical function that controls the output of a neural network. Activation functions help in determining whether a neuron is to be fired or not.

Some of the popular activation functions are :

• Binary Step
• Linear
• Sigmoid
• Tanh
• ReLU
• Leaky ReLU
• Softmax

Activation is responsible for adding non-linearity to the output of a neural network model. Without an activation function, a neural network is simply a linear regression.

The mathematical equation for calculating the output of a neural network is:

In this tutorial, we will focus on the sigmoid activation function. This function comes from the sigmoid function in maths.

Let’s start by discussing the formula for the function.

Mathematically you can represent the sigmoid activation function as:

You can see that the denominator will always be greater than 1, therefore the output will always be between 0 and 1.

In this section, we will learn how to implement the sigmoid activation function in Python.

We can define the function in python as:

import numpy as np 
def sig(x):
 return 1/(1 + np.exp(-x))

Let’s try running the function on some inputs.

import numpy as np 
def sig(x):
 return 1/(1 + np.exp(-x))


x = 1.0
print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x)))

x = -10.0
print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x)))

x = 0.0
print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x)))

x = 15.0
print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x)))

x = -2.0
print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x)))

Output :

Applying Sigmoid Activation on (1.0) gives 0.7
Applying Sigmoid Activation on (-10.0) gives 0.0
Applying Sigmoid Activation on (0.0) gives 0.5
Applying Sigmoid Activation on (15.0) gives 1.0
Applying Sigmoid Activation on (-2.0) gives 0.1

To plot sigmoid activation we’ll use the Numpy library:

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 50)   
p = sig(x)
plt.xlabel("x") 
plt.ylabel("Sigmoid(x)")  
plt.plot(x, p) 
plt.show()

Output :

We can see that the output is between 0 and 1.

The sigmoid function is commonly used for predicting probabilities since the probability is always between 0 and 1.

One of the disadvantages of the sigmoid function is that towards the end regions the Y values respond very less to the change in X values.

This results in a problem known as the vanishing gradient problem.

Vanishing gradient slows down the learning process and hence is undesirable.

Let’s discuss some alternatives that overcome this problem.

A better alternative that solves this problem of vanishing gradient is the ReLu activation function.

The ReLu activation function returns 0 if the input is negative otherwise return the input as it is.

Mathematically it is represented as:

You can implement it in Python as follows:

def relu(x):
    return max(0.0, x)

Let’s see how it works on some inputs.

def relu(x):
    return max(0.0, x)
 
x = 1.0
print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x)))
x = -10.0
print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x)))
x = 0.0
print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x)))
x = 15.0
print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x)))
x = -20.0
print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x)))

Output:

Applying Relu on (1.0) gives 1.0
Applying Relu on (-10.0) gives 0.0
Applying Relu on (0.0) gives 0.0
Applying Relu on (15.0) gives 15.0
Applying Relu on (-20.0) gives 0.0

The problem with ReLu is that the gradient for negative inputs comes out to be zero.

This again leads to the problem of vanishing gradient (zero-gradient) for negative inputs.

To solve this problem we have another alternative known as the Leaky ReLu activation function.

The leaky ReLu addresses the problem of zero gradients for negative value, by giving an extremely small linear component of x to negative inputs.

Mathematically we can define it as:

f(x)= 0.01x, x<0
    = x,   x>=0

You can implement it in Python using:

def leaky_relu(x):
  if x>0 :
    return x
  else :
    return 0.01*x
  
x = 1.0
print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x)))

x = -10.0
print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x)))

x = 0.0
print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x)))

x = 15.0
print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x)))

x = -20.0
print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x)))

Output :

Applying Leaky Relu on (1.0) gives 1.0
Applying Leaky Relu on (-10.0) gives -0.1
Applying Leaky Relu on (0.0) gives 0.0
Applying Leaky Relu on (15.0) gives 15.0
Applying Leaky Relu on (-20.0) gives -0.2

This tutorial was about the Sigmoid activation function. We learned how to implement and plot the function in python.