The Differences between Sigmoid and Softmax Activation Functions

In deep learning and neural networks, activation functions are essential to a model’s output. The Sigmoid and Softmax activation functions (this link will take the user offsite outside of Medium) are among the most often utilized ones. Classification jobs require both, but they are used for different things and in various situations. This paper explores the variations, mathematical definitions, and particular applications of these two activation functions.

Sigmoid Activation Function

The Sigmoid function, also known as the logistic function, is defined by the formula:

This function maps any real-valued number into a value between 0 and 1, making it particularly useful for binary classification problems. The output of the Sigmoid function can be interpreted as a probability, which is why it is often used in the output layer of binary classifiers.

Characteristics of Sigmoid Function:

• Range: The output values lie between 0 and 1.
• Monotonic: The function is monotonically increasing.
• Differentiable: The function is smooth and differentiable, which is essential for gradient-based optimization methods.
• Vanishing Gradient Problem: For very large or very small input values, the gradient of the Sigmoid function approaches zero, which can slow down the training process in deep networks.

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Use Cases:

• Binary Classification: The Sigmoid function is ideal for problems where the output is binary (e.g., yes/no, true/false).
• Logistic Regression: It is used in logistic regression models to predict the probability of a binary outcome.
• Hidden Layers: Occasionally used in hidden layers of neural networks, although other functions like ReLU are more common due to better performance in deep networks.

Softmax Activation Function

The Softmax function is an extension of the Sigmoid function for multi-class classification problems. It converts a vector of raw scores (logits) into a probability distribution. The mathematical formulation of the Softmax function for a vector z with K elements is:

Here, each element of the input vector is exponentiated and then normalized by the sum of all exponentiated elements, ensuring that the output values sum to 1.

Characteristics of Softmax Function:

• Range: The output values lie between 0 and 1.
• Probability Distribution: The sum of the output values is 1, making them interpretable as probabilities.
• Multi-class Classification: Suitable for problems with more than two classes.
• Differentiable: Like the Sigmoid function, Softmax is also differentiable, which is crucial for backpropagation.

Use Cases:

• Multi-class Classification: The Softmax function is predominantly used in the output layer of neural networks for multi-class classification tasks.
• Neural Networks: Commonly used in the final layer of neural networks to normalize the output into a probability distribution over multiple classes.
• Natural Language Processing (NLP): Widely used in NLP tasks such as text classification and language modeling.

Key Differences

1. Application:

• Sigmoid: Used for binary classification problems.
• Softmax: Used for multi-class classification problems.

2. Output:

• Sigmoid: Produces a single probability value between 0 and 1.
• Softmax: Produces a probability distribution over multiple classes, with the sum of probabilities equal to 1.

3. Mathematical Formulation:

4. Use in Neural Networks:

• Sigmoid: Often used in the output layer for binary classification and sometimes in hidden layers.
• Softmax: Typically used in the output layer for multi-class classification.

5. Interpretability:

• Sigmoid: The output can be interpreted as the probability of the positive class in binary classification.
• Softmax: The output can be interpreted as the probability distribution over multiple classes.