Sigmoid Function

What is a Sigmoid Function? Sigmoid Function Explained

The sigmoid function, also known as the logistic function, is a mathematical function that maps real-valued numbers to a range between 0 and 1. It is commonly used in various domains, including statistics, machine learning, and neural networks. The sigmoid function is defined as:

σ(x) = 1 / (1 + e^(-x))

where σ(x) represents the output of the sigmoid function for input x, and e is the base of the natural logarithm.

Key characteristics of the sigmoid function:

Range: This function’s output ranges from 0 to 1, meaning that any real number input is transformed into a value between 0 and 1. This property makes it suitable for problems where the output needs to be interpreted as a probability or a value within a bounded range.

S-shaped Curve: It has an S-shaped curve, meaning that its output increases gradually from 0 to 1 as the input increases from negative infinity to positive infinity. The curve is steepest around the origin (x = 0), and as the input moves further away from 0, the rate of change decreases.

Differentiability: It is differentiable, which is important for many optimization algorithms used in machine learning, such as gradient descent. The smoothness of the sigmoid function allows for efficient computation of gradients, making it suitable for training neural networks using backpropagation.

Non-linearity: It introduces non-linearity into models that use it as an activation function. This non-linearity enables the model to capture complex relationships between input features and make more flexible and expressive predictions.

It’s worth noting that while the sigmoid function was commonly used in the past, especially in the context of neural networks, it has some limitations. The sigmoid function suffers from the vanishing gradient problem, where the gradients become very small for extreme input values, leading to slow convergence during training. Consequently, in deep neural networks, other activation functions like ReLU (Rectified Linear Unit) and its variants are often preferred due to their ability to alleviate the vanishing gradient problem.

Despite these limitations, the sigmoid function is still useful in certain scenarios, such as binary classification problems or cases where the output needs to be interpreted as a probability.