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In the last layer of CNNs and MLPs it is common to use softmax layer or units with sigmoid activation functions for multi-class classification. I have seen somewhere, I don't remember where, that softmax is used whenever the classes are mutually exclusive and the layer with units containing sigmoid activation function are used in tasks with multiple labels, e.g. recognizing animals in an image which can contain numerous animals. Am I right? Is there any other deduction for distinguishing between them?

I have seen here and here but they do not contain what I want.

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  • $\begingroup$ Can any one tell me how CNN knows the class probability? Specifically, how does CNN know the higher probability in the output will lead to a certain class? $\endgroup$ – Sabir Ullah Jan 10 '19 at 11:01
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Yes, you are right. The soft-max layer outputs a probability distribution, i.e. the values of the output sum to 1. The sigmoid function outputs marginal probabilities and therefore can be used for multiple-class classification, when the classes are not mutually exclusive. Additionally the soft-max layer is soft version of the max-output layer so it is differentiable and also resilient to outliers. A problem with sigmoids is that as you reach saturation (values get close to 1 or 0), the gradients vanish. This is detrimental to optimization speed and soft-max doesn't have this problem. Another interpretation is soft-max as a generalization of sigmoid, actually when there are two classes they are the same.

Wrapping up you should use soft-max when the classes are mutually exclusive and sigmoid when the classes are independent. This can be summarized in the following table:

Soft-Max                            |                 Sigmoid
-------------------------------------------------------------------------
Used for multi-classification       |  Used for binary classification in 
in logistic regression model.       |  logistic regression model.
-------------------------------------------------------------------------
The probabilities sum will be 1     | The probabilities sum need not be 1
-------------------------------------------------------------------------
Used in the different layers of     | Used as activation function while 
neural networks.                    | building neural networks
-------------------------------------------------------------------------
The high value will have the higher | The high value will have the high 
probability than other values.      | probability but not the higher

For more information on soft-max look at the following links: 1, 2 and 3. For a step-by-step guide, including usage in Python see this reference 4.

For more on soft-max vs sigmoid check this: 5, 6 and this 7.

If you want a more reliable source on why to use soft-max regression for mutually exclusive classes you can look here. This page is part of Unsupervised Feature Learning and Deep Learning Tutorial of Stanford University, it contains material contributed by Andrew Ng and others.

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  • $\begingroup$ Have you ever seen this mutually exclusive point in any paper sir? $\endgroup$ – Media Dec 6 '17 at 10:47
  • $\begingroup$ I updated my answer now includes a link to a source I think is more reliable $\endgroup$ – Dani Mesejo Dec 6 '17 at 11:01

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