Understanding Activation Functions in Neural Networks:

When building neural networks, one of the essential elements that drives their performance is the activation function. Activation functions are crucial for transforming the input into outputs, determining whether a neuron should be activated or not, and introducing non-linearity into the system.

In this blog post, we’ll explore the purpose of activation functions and compare some of the most widely used ones:

• Binary
• Linear
• Sigmoid
• Tanh
• ReLU
• Softmax

What Is an Activation Function?

In a neural network, the input is multiplied by weights and then passed through an activation function to produce the output. Without an activation function, the network behaves like a simple linear model, incapable of solving complex tasks.

The main goals of an activation function are:

1. Introducing non-linearity: Non-linearity allows the model to learn from data that is not just linearly separable, enabling it to capture more complex patterns.
2. Deciding neuron activation: The function determines if the neuron’s output should be passed to the next layer.

Let's now dive into each activation function and explore its use cases, advantages, and limitations.

1. Binary Step Function

The Binary step function is the simplest activation function. It outputs either a 0 or 1 depending on whether the input is less than or greater than a certain threshold, typically 0.

Formula:

1 & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases}

Example:

If an input is positive, the output is 1. If it’s negative, the output is 0.

Pros:

• Easy to compute.
• Suitable for binary classification.

Cons:

• Non-differentiable: This makes it unsuitable for gradient-based optimization techniques like backpropagation.
• No learning of complex patterns: It introduces no non-linearity, meaning the network cannot learn from complex data.

2. Linear Activation Function

The Linear activation function is just a straight line where the output is proportional to the input. Mathematically, it looks like this:

Formula:

f(x)=axf(x) = axf(x)=ax

Where a is a constant.

Example:

For any input x, the output would be ax. For example, if a=2 and x=3, then the output will be 6.

Pros:

• Works well for tasks like linear regression.

Cons:

• No non-linearity: Like the binary step, this function fails to introduce any non-linearity.
• Unbounded output: The output can grow infinitely, making it less useful for deep networks.

3. Sigmoid Activation Function

The Sigmoid function outputs a value between 0 and 1. It is one of the most commonly used activation functions for binary classification tasks.

Formula:

f(x)=11+e?xf(x) = \frac{1}{1 + e^{-x}}f(x)=1+e?x1

Example:

For an input of 2, the sigmoid function would output about 0.88.

Pros:

• Smooth gradient: Ensures small changes in input cause small changes in output.
• Bounded output: The output is always between 0 and 1, making it useful for probabilistic interpretations.

Cons:

• Vanishing gradient problem: At extreme values of x, the gradient approaches zero, which slows down learning.
• Outputs close to 0 or 1: Can saturate and make it hard to learn from errors.

4. Tanh Activation Function

The Tanh (hyperbolic tangent) function is similar to the sigmoid but outputs values between -1 and 1, making it more useful in certain tasks where negative outputs are required.

Formula:

f(x)=tanh(x)=21+e?2x?1f(x) = \tanh(x) = \frac{2}{1 + e^{-2x}} - 1f(x)=tanh(x)=1+e?2x2?1

Example:

For an input of 1, the tanh function would output about 0.76.

Pros:

• Zero-centered output: Unlike the sigmoid, the tanh function produces outputs ranging from -1 to 1, making it easier to model inputs that have strongly negative, neutral, or positive relationships.
• Better gradient than sigmoid: Learning progresses faster due to steeper gradients.

Cons:

• Vanishing gradient problem: Like sigmoid, it can suffer from the vanishing gradient issue at extreme values.

5. ReLU (Rectified Linear Unit)

The ReLU function has become the de facto standard for hidden layers in neural networks. It is defined as the positive part of its input.

Formula:

f(x)=max(0,x)f(x) = \max(0, x)f(x)=max(0,x)

Example:

For an input of -2, the ReLU function would output 0, whereas for an input of 3, the output is 3.

Pros:

• Non-linear: ReLU introduces non-linearity while maintaining simplicity.
• Efficient: ReLU is computationally efficient, only requiring a comparison.
• Sparse activation: Many neurons are deactivated (output is 0), which reduces complexity.

Cons:

• Dying ReLU problem: Neurons can get "stuck" when inputs are always negative, causing them to output zero indefinitely, halting learning.

6. Softmax Activation Function

The Softmax function is used in the output layer for multi-class classification problems. It converts raw input into a probability distribution, where the sum of the outputs is 1.

Formula:

f(xi)=exi∑jexjf(x_i) = \frac{e^{x_i}}{\sum_{j} e^{x_j}}f(xi)=∑jexjexi

Example:

For an input array [2, 1, 0.1], the softmax function would produce something like [0.7, 0.2, 0.1].

Pros:

• Probabilistic output: Outputs can be interpreted as probabilities, ideal for multi-class classification.
• Differentiable: Works well with gradient-based optimization.

Cons:

• Not used in hidden layers: Primarily used for the final output layer.
• Computationally expensive: Slightly more expensive than other functions due to the exponentiation and summation.

Conclusion

Activation functions play a critical role in enabling neural networks to learn complex patterns. Whether you're working with binary classification, multi-class tasks, or regression problems, choosing the right activation function can make or break the model's performance. While simpler functions like the binary step and linear activations serve specific roles, modern deep learning relies heavily on non-linear functions like ReLU, sigmoid, and softmax for effective learning.