Activation functions are very important for how well neural networks work. They help decide the structure of the network for different tasks. Choosing the right activation function can make learning faster and help the network understand complicated patterns.

Types of Activation Functions

1. Linear Activation Function:

   • What it is: This function is simply $f(x) = x$.
   • Where it's used: Mainly in output layers for tasks that predict numbers (like regression).
   • Drawback: It does not add any non-linear behavior, which makes it less suitable for deep networks.

2. Sigmoid Activation Function:

   • What it is: This function looks like this: $f(x) = \frac{1}{1 + e^{-x}}$.
   • What it does: It gives outputs between 0 and 1.
   • Drawback: It can slow down learning in deep networks because it has trouble with small gradients.

3. Tanh Activation Function:

   • What it is: This function is $f(x) = \tanh(x) = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}$.
   • What it does: It gives outputs between -1 and 1.
   • Drawback: It still has issues with small gradients for larger input values.

4. ReLU (Rectified Linear Unit):

   • What it is: This function is $f(x) = \max(0, x)$.
   • Why it's popular: It's often used in hidden layers because it helps fix the gradient problem.
   • Benefits: It can speed up training by about 6%, according to studies.

5. Leaky ReLU:

   • What it is: This function is $f(x) = \begin{cases} x & \text{if } x > 0 \\ 0.01x & \text{if } x \leq 0 \end{cases}$.
   • Why it's better: It tackles the "dying ReLU" issue by allowing a small gradient when the input is negative.

6. Softmax:

   • What it is: This function is $f(x_j) = \frac{e^{x_j}}{\sum_{k} e^{x_k}}$.
   • Where it's used: It's great for problems where there are multiple classes to choose from.
   • What it does: It turns raw scores into probabilities, making the output easier to understand.

How Activation Functions Affect Network Architecture

The choice of activation function can change the network in several ways:

• Depth: Functions like ReLU and similar ones allow for deeper networks. They help keep track of gradients, so networks with more than 100 layers can work better.

• Width: Wider networks (with more neurons in each layer) can benefit from functions that add non-linearity, like sigmoid or tanh, to capture complex patterns.

• Initialization: Functions like ReLU need careful setup of weights (like He initialization). This helps avoid problems like dead neurons and leads to better training results.

Conclusion

To sum it up, the choice of activation function is very important for the performance of a neural network. It can affect how fast a network learns and how well it can handle different types of data. Picking the right activation function is key to building a network that works effectively. Each function has its place, and finding the best one often involves testing and adjusting based on what you need the model to do.