In an ANN, the sigmoid function is a non-linear AF used primarily in feedforward neural networks. It is a differentiable real function, defined for real input values, and containing positive derivatives everywhere with a specific degree of smoothness. The sigmoid function appears in the output layer of the deep learning models and is used for predicting probability-based outputs. The sigmoid function is represented as:

Generally, the derivatives of the sigmoid function are applied to learning algorithms. The graph of the sigmoid function is ‘S’ shaped.

Some of the major drawbacks of the sigmoid function include gradient saturation, slow convergence, sharp damp gradients during backpropagation from within deeper hidden layers to the input layers, and non-zero centered output that causes the gradient updates to propagate in varying directions.

2. Hyperbolic Tangent Function (Tanh)

The hyperbolic tangent function, a.k.a., the tanh function, is another type of AF. It is a smoother, zero-centered function having a range between -1 to 1. As a result, the output of the tanh function is represented by:

The tanh function is much more extensively used than the sigmoid function since it delivers better training performance for multilayer neural networks. The biggest advantage of the tanh function is that it produces a zero-centered output, thereby supporting the backpropagation process. The tanh function has been mostly used in recurrent neural networks for natural language processing and speech recognition tasks.

However, the tanh function, too, has a limitation – just like the sigmoid function, it cannot solve the vanishing gradient problem. Also, the tanh function can only attain a gradient of 1 when the input value is 0 (x is zero). As a result, the function can produce some dead neurons during the computation process.

3. Softmax Function

The softmax function is another type of AF used in neural networks to compute probability distribution from a vector of real numbers. This function generates an output that ranges between values 0 and 1 and with the sum of the probabilities being equal to 1. The softmax function is represented as follows:

This function is mainly used in multi-class models where it returns probabilities of each class, with the target class having the highest probability. It appears in almost all the output layers of the DL architecture where they are used. The primary difference between the sigmoid and softmax AF is that while the former is used in binary classification, the latter is used for multivariate classification.

4. Softsign Function

The softsign function is another AF that is used in neural network computing. Although it is primarily in regression computation problems, nowadays it is also used in DL based text-to-speech applications. It is a quadratic polynomial, represented by:

Here “x” equals the absolute value of the input.

The main difference between the softsign function and the tanh function is that unlike the tanh function that converges exponentially, the softsign function converges in a polynomial form.

5. Rectified Linear Unit (ReLU) Function

One of the most popular AFs in DL models, the rectified linear unit (ReLU) function, is a fast-learning AF that promises to deliver state-of-the-art performance with stellar results. Compared to other AFs like the sigmoid and tanh functions, the ReLU function offers much better performance and generalization in deep learning. The function is a nearly linear function that retains the properties of linear models, which makes them easy to optimize with gradient-descent methods.

The ReLU function performs a threshold operation on each input element where all values less than zero are set to zero. Thus, the ReLU is represented as:

By rectifying the values of the inputs less than zero and setting them to zero, this function eliminates the vanishing gradient problem observed in the earlier types of activation functions (sigmoid and tanh).

The most significant advantage of using the ReLU function in computation is that it guarantees faster computation – it does not compute exponentials and divisions, thereby boosting the overall computation speed. Another critical aspect of the ReLU function is that it introduces sparsity in the hidden units by squishing the values between zero to maximum.

6. Exponential Linear Units (ELUs) Function

The exponential linear units (ELUs) function is an AF that is also used to speed up the training of neural networks (just like ReLU function). The biggest advantage of the ELU function is that it can eliminate the vanishing gradient problem by using identity for positive values and by improving the learning characteristics of the model.

ELUs have negative values that push the mean unit activation closer to zero, thereby reducing computational complexity and improving the learning speed. The ELU is an excellent alternative to the ReLU – it decreases bias shifts by pushing mean activation towards zero during the training process.

The exponential linear unit function is represented as:

The derivative or gradient of the ELU equation is presented as:

Here “α” equals the ELU hyperparameter that controls the saturation point for negative net inputs, which is usually set to 1.0. However, the ELU function has a limitation – it is not zero-centered.

1. Sigmoid FunctionIn an ANN, the sigmoid function is a non-linear AF used primarily in feedforward neural networks. It is a differentiable real function, defined for real input values, and containing positive derivatives everywhere with a specific degree of smoothness. The sigmoid function appears in the output layer of the deep learning models and is used for predicting probability-based outputs. The sigmoid function is represented as:

Generally, the derivatives of the sigmoid function are applied to learning algorithms. The graph of the sigmoid function is ‘S’ shaped.

Some of the major drawbacks of the sigmoid function include gradient saturation, slow convergence, sharp damp gradients during backpropagation from within deeper hidden layers to the input layers, and non-zero centered output that causes the gradient updates to propagate in varying directions.

2. Hyperbolic Tangent Function (Tanh)The hyperbolic tangent function, a.k.a., the tanh function, is another type of AF. It is a smoother, zero-centered function having a range between -1 to 1. As a result, the output of the tanh function is represented by:

The tanh function is much more extensively used than the sigmoid function since it delivers better training performance for multilayer neural networks. The biggest advantage of the tanh function is that it produces a zero-centered output, thereby supporting the backpropagation process. The tanh function has been mostly used in recurrent neural networks for natural language processing and speech recognition tasks.

However, the tanh function, too, has a limitation – just like the sigmoid function, it cannot solve the vanishing gradient problem. Also, the tanh function can only attain a gradient of 1 when the input value is 0 (x is zero). As a result, the function can produce some dead neurons during the computation process.

3. Softmax FunctionThe softmax function is another type of AF used in neural networks to compute probability distribution from a vector of real numbers. This function generates an output that ranges between values 0 and 1 and with the sum of the probabilities being equal to 1. The softmax function is represented as follows:

This function is mainly used in multi-class models where it returns probabilities of each class, with the target class having the highest probability. It appears in almost all the output layers of the DL architecture where they are used. The primary difference between the sigmoid and softmax AF is that while the former is used in binary classification, the latter is used for multivariate classification.

4. Softsign FunctionThe softsign function is another AF that is used in neural network computing. Although it is primarily in regression computation problems, nowadays it is also used in DL based text-to-speech applications. It is a quadratic polynomial, represented by:

Here “x” equals the absolute value of the input.

The main difference between the softsign function and the tanh function is that unlike the tanh function that converges exponentially, the softsign function converges in a polynomial form.

5. Rectified Linear Unit (ReLU) FunctionOne of the most popular AFs in DL models, the rectified linear unit (ReLU) function, is a fast-learning AF that promises to deliver state-of-the-art performance with stellar results. Compared to other AFs like the sigmoid and tanh functions, the ReLU function offers much better performance and generalization in deep learning. The function is a nearly linear function that retains the properties of linear models, which makes them easy to optimize with gradient-descent methods.

The ReLU function performs a threshold operation on each input element where all values less than zero are set to zero. Thus, the ReLU is represented as:

By rectifying the values of the inputs less than zero and setting them to zero, this function eliminates the vanishing gradient problem observed in the earlier types of activation functions (sigmoid and tanh).

The most significant advantage of using the ReLU function in computation is that it guarantees faster computation – it does not compute exponentials and divisions, thereby boosting the overall computation speed. Another critical aspect of the ReLU function is that it introduces sparsity in the hidden units by squishing the values between zero to maximum.

6. Exponential Linear Units (ELUs) FunctionThe exponential linear units (ELUs) function is an AF that is also used to speed up the training of neural networks (just like ReLU function). The biggest advantage of the ELU function is that it can eliminate the vanishing gradient problem by using identity for positive values and by improving the learning characteristics of the model.

ELUs have negative values that push the mean unit activation closer to zero, thereby reducing computational complexity and improving the learning speed. The ELU is an excellent alternative to the ReLU – it decreases bias shifts by pushing mean activation towards zero during the training process.

The exponential linear unit function is represented as:

The derivative or gradient of the ELU equation is presented as:

Here “α” equals the ELU hyperparameter that controls the saturation point for negative net inputs, which is usually set to 1.0. However, the ELU function has a limitation – it is not zero-centered.