Difference Between Softmax Function and Sigmoid Function
Softmax Function Vs Sigmoid Function
While learning the logistic regression concepts, the primary confusion will be on the functions used for calculating the probabilities. As the calculated probabilities are used to predict the target class in logistic regression model. The two principal functions we frequently hear are Softmax and Sigmoid function.
Even though both the functions are same at the functional level. (Helping to predict the target class) many noticeable mathematical differences are playing the vital role in using the functions in deep learning and other fields of areas.
So In this article, we were going to learn more about the fundamental differences between these two function and the usages.
Before we begin, let’s quickly look at the table of contents.
Table of Contents:
- What is Sigmoid Function?
- Properties of Sigmoid Function
- Sigmoid Function Usage
- Implementing Sigmoid Function In Python
- Creating Sigmoid Function Graph
- What is Softmax Function?
- Properties of Softmax Function
- Softmax Function Usage
- Implementing Softmax Function In Python
- Creating Softmax Function Graph
- Difference Between Sigmoid Function and Softmax Function
- Conclusion
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What is Sigmoid Function?
In mathematical definition way of saying the sigmoid function take any range real number and returns the output value which falls in the range of 0 to 1. Based on the convention we can expect the output value in the range of -1 to 1.
The sigmoid function produces the curve which will be in the Shape “S.” These curves used in the statistics too. With the cumulative distribution function (The output will range from 0 to 1)
Properties of Sigmoid Function
- The sigmoid function returns a real-valued output.
- The first derivative of the sigmoid function will be non-negative or non-positive.
- Non-Negative: If a number is greater than or equal to zero.
- Non-Positive: If a number is less than or equal to Zero.
Sigmoid Function Usage
- The Sigmoid function used for binary classification in logistic regression model.
- While creating artificial neurons sigmoid function used as the activation function.
- In statistics, the sigmoid function graphs are common as a cumulative distribution function.
Implementing Sigmoid Function In Python
Now let’s implement the sigmoid function in Python
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# Required Python Package import numpy as np def sigmoid(inputs): """ Calculate the sigmoid for the give inputs (array) :param inputs: :return: """ sigmoid_scores = [1 / float(1 + np.exp(- x)) for x in inputs] return sigmoid_scores sigmoid_inputs = [2, 3, 5, 6] print "Sigmoid Function Output :: {}".format(sigmoid(sigmoid_inputs)) |
The above is the implementation of the sigmoid function.
- The function will take a list of values as an input parameter.
- For each element/value in the list will consider as an input for the sigmoid function and will calculate the output value.
- The code 1 / float(1 + np.exp(- x)) is the fucuntion is used for calcualting the sigmoid scores.
- Next, we take a list sigmiod_inputs having the values 2,3,5,6 as an input the function we implemented to get the sigmoid scores.
Script Output
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Sigmoid Function Output :: [0.8807970779778823, 0.9525741268224334, 0.9933071490757153, 0.9975273768433653] |
Creating Sigmoid Function Graph
Now let’s use the above function to create the graph to understand the nature of the sigmoid function.
- We are going to pass a list which contains numbers in the range 0 to 21.
- Will compute the sigmoid scores for the list we passed.
- Then we will use the outputs values to visualize the graph.
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# Required Python Packages import numpy as np import matplotlib.pyplot as plt def sigmoid(inputs): """ Calculate the sigmoid for the give inputs (array) :param inputs: :return: """ sigmoid_scores = [1 / float(1 + np.exp(- x)) for x in inputs] return sigmoid_scores def line_graph(x, y, x_title, y_title): """ Draw line graph with x and y values :param x: :param y: :param x_title: :param y_title: :return: """ plt.plot(x, y) plt.xlabel(x_title) plt.ylabel(y_title) plt.show() graph_x = range(0, 21) graph_y = sigmoid(graph_x) print "Graph X readings: {}".format(graph_x) print "Graph Y readings: {}".format(graph_y) line_graph(graph_x, graph_y, "Inputs", "Sigmoid Scores") |
- Creating a graph_x list which contains the numbers in the range of 0 to 21.
- Next, in the graph_y list, we are storing the calculated sigmoid scores for the given graph_x inputs.
- Calling the line_graph function, which takes the x, y, and titles of the graph to create the line graph.
Script Output
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Graph X readings: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] Graph Y readings: [0.5, 0.7310585786300049, 0.8807970779778823, 0.9525741268224334, 0.9820137900379085, 0.9933071490757153, 0.9975273768433653, 0.9990889488055994, 0.9996646498695336, 0.9998766054240137, 0.9999546021312976, 0.999983298578152, 0.9999938558253978, 0.999997739675702, 0.9999991684719722, 0.999999694097773, 0.9999998874648379, 0.9999999586006244, 0.9999999847700205, 0.9999999943972036, 0.9999999979388463] |
Graph
On successfully running the above code the below image will appear on your screen. If the above code failed in your system. Check the machine learning packages setup.
From the above graph, we can observe that with the increase in the input value the sigmoid score increase till 1. The values which are touching at the top of the graph are the values in the range of 0.9 to 0.99
What is Softmax Function?
Softmax function calculates the probabilities distribution of the event over ‘n’ different events. In general way of saying, this function will calculate the probabilities of each target class over all possible target classes. Later the calculated probabilities will be helpful for determining the target class for the given inputs.
The main advantage of using Softmax is the output probabilities range. The range will 0 to 1, and the sum of all the probabilities will be equal to one. If the softmax function used for multi-classification model it returns the probabilities of each class and the target class will have the high probability.
The formula computes the exponential (e-power) of the given input value and the sum of exponential values of all the values in the inputs. Then the ratio of the exponential of the input value and the sum of exponential values is the output of the softmax function.
Properties of Softmax Function
Below are the few properties of softmax function.
- The calculated probabilities will be in the range of 0 to 1.
- The sum of all the probabilities is equals to 1.
Softmax Function Usage
- Used in multiple classification logistic regression model.
- In building neural networks softmax functions used in different layer level.
Implementing Softmax Function In Python
Now let’s implement the softmax function in Python
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# Required Python Package import numpy as np def softmax(inputs): """ Calculate the softmax for the give inputs (array) :param inputs: :return: """ return np.exp(inputs) / float(sum(np.exp(inputs))) softmax_inputs = [2, 3, 5, 6] print "Softmax Function Output :: {}".format(softmax(softmax_inputs)) |
Script Output
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Softmax Function Output :: [ 0.01275478 0.03467109 0.25618664 0.69638749] |
If we observe the function output for the input value 6 we are getting the high probabilities. This is what we can expect from the softmax function. Later in classification task, we can use the high probability value for predicting the target class for the given input features.
Creating Softmax Function Graph
Now let’s use the implemented Softmax function to create the graph to understand the behavior of this function.
- We are going to create a list which contains values in the range of 0 to 21.
- Next, we are going to pass this list to calculate the scores from the implemented function.
- To create the graph we are going to use the list and the estimated scores.
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# Required Python Packages import numpy as np import matplotlib.pyplot as plt def softmax(inputs): """ Calculate the softmax for the give inputs (array) :param inputs: :return: """ return np.exp(inputs) / float(sum(np.exp(inputs))) def line_graph(x, y, x_title, y_title): """ Draw line graph with x and y values :param x: :param y: :param x_title: :param y_title: :return: """ plt.plot(x, y) plt.xlabel(x_title) plt.ylabel(y_title) plt.show() graph_x = range(0, 21) graph_y = softmax(graph_x) print "Graph X readings: {}".format(graph_x) print "Graph Y readings: {}".format(graph_y) line_graph(graph_x, graph_y, "Inputs", "Softmax Scores") |
Script Output
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Graph X readings: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] Graph Y readings: [ 1.30289758e-09 3.54164282e-09 9.62718331e-09 2.61693975e-08 7.11357976e-08 1.93367146e-07 5.25626399e-07 1.42880069e-06 3.88388295e-06 1.05574884e-05 2.86982290e-05 7.80098744e-05 2.12052824e-04 5.76419338e-04 1.56687021e-03 4.25919483e-03 1.15776919e-02 3.14714295e-02 8.55482149e-02 2.32544158e-01 6.32120559e-01] |
Graph
The figure shows the fundamental property of softmax function. The high value will have the high probability.
Difference Between Sigmoid Function and Softmax Function
The below are the tabular differences between Sigmoid and Softmax function.
Softmax Function | Sigmoid Function | |
1 | Used for multi-classification in logistic regression model. | Used for binary classification in logistic regression model. |
2 | The probabilities sum will be 1 | The probabilities sum need not be 1. |
3 | Used in the different layers of neural networks. | Used as activation function while building neural networks. |
4 | The high value will have the higher probability than other values. | The high value will have the high probability but not the higher probability. |
Conclusion
In this article, you learn in details about two functions which determine the logistic regression model. Just for a glance.
- Softmax: Used for the multi-classification task.
- Sigmoid: Used for the binary classification task.
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good post. easy to follow and understand
Hi Anon,
Thanks for your compliment.