A linear activation function has two major problems:

1. Not possible to use backpropagation  (gradient descent) to train the model—the derivative of the function is a constant, and has no relation to the input, X. So it’s not possible to go back and understand which weights in the input neurons can provide a better prediction.

2. All layers of the neural network collapse into one—with linear activation functions, no matter how many layers in the neural network, the last layer will be a linear function of the first layer (because a linear combination of linear functions is still a linear function). So a linear activation function turns the neural network into just one layer.

A neural network with a linear activation function is simply a linear regression model. It has limited power and ability to handle complexity varying parameters of input data.

Problems with Linear Activation Function

Non-linear functions address the problems of a linear activation function:

 - They allow backpropagation because they have a derivative function which is related to the inputs.
 - They allow “stacking” of multiple layers of neurons to create a deep neural network. Multiple hidden layers of neurons are needed to learn complex data sets with high levels of accuracy.

Linear vs. Non-Linear Activation Functions

A linear activation function takes the form: $y = cx$.

It takes the inputs, multiplied by the weights for each neuron, and creates an output signal proportional to the input. In one sense, a linear function is better than a step function because it allows multiple outputs, not just yes and no.


University of Michigan - Ann Arbor

There are 12 types of activation functions:

- Binary Step
- Linear
- Sigmoid
- Tanh
- ReLU
- Leaky ReLU
- Parameterised ReLU
- Exponential Linear Unit
- Swish
- Softmax

https://www.analyticsvidhya.com/blog/2020/01/fundamentals-deep-learning-activation-functions-when-to-use-them/

Types of Activation Functions

A helpful website that introduces neural networks:
https://missinglink.ai/guides/neural-network-concepts/

Neural Network Reference

A binary step function is a threshold-based activation function. If the input value is above or below a certain threshold, the neuron is activated and sends exactly the same signal to the next layer.

Binary Step Function

Linear Activation Function

Modern neural network models use non-linear activation functions. They allow the model to create complex mappings between the network’s inputs and outputs, which are essential for learning and modeling complex data, such as images, video, audio, and data sets which are non-linear or have high dimensionality.

There are 8 common non-linear activation functions:
 - Sigmoid / Logistic
 - TanH / Hyperbolic Tangent
 - ELU (Exponential Linear Unit)
 - ReLU (Rectified Linear Unit)
 - Leaky ReLU
 - Parametric ReLU
 - Softmax
 - Swish

Non-Linear Activation Functions

Sigmoid 
Pros: useful for binary classification, output layer 
Cons: shifted from center

Tanh 
Pros: has centering effect, almost always superior to sigmoid 
Cons: complex in comparison to ReLU 

ReLU
Pros: simpler, default
Cons: Derivative is 0 when z is negative --> leaky reLU

Pros and Cons of each Activation Function

Activation functions are a critical part of the design of a neural network- A hidden layer in a neural network is a layer that receives input from another layer (such as another hidden layer or an input layer) and provides output to another layer (such as another hidden layer or an output layer).

There are perhaps three activation functions you may want to consider for use in hidden layers; they are:

- Rectified Linear Activation (ReLU)
The rectified linear activation function calculated as max(0.0, x)
It is common because it is both simple to implement and effective at overcoming the limitations of other previously popular activation functions, such as Sigmoid and Tanh.


- Logistic (Sigmoid)
The sigmoid activation function is also called the logistic function, and calculated as 1.0 / (1.0 + e^-x)
It is used in the logistic regression classification algorithm.

- Hyperbolic Tangent (Tanh)
The function takes any real value as input and outputs values in the range -1 to 1. The larger the input (more positive), the closer the output value will be to 1.0, whereas the smaller the input (more negative), the closer the output will be to -1.0.
IT is calculated as (e^x – e^-x) / (e^x + e^-x)

It is common practive to choose activation function based on the type of model you work on.
Multilayer Perceptron (MLP): ReLU activation function.
Convolutional Neural Network (CNN): ReLU activation function.
Recurrent Neural Network: Tanh and/or Sigmoid activation function.



How to Choose an Activation Function for Deep Learning

Which of the following are activation functions?

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