Introduction to the Softmax Function

I first encountered the softmax function while learning about convolutional neural networks and transformers , where it’s used as a crucial activation function. At the time, I didn’t fully grasp what this function does . In this post, I aim to break down the softmax function, explaining what it is, how it works, and why it’s important.

What is the Softmax Function?

The softmax function is a mathematical function that transforms a vector of \(K\) real values into a vector of \(K\) real values that sum to 1, effectively turning them into a probability distribution . The formula for the softmax function is:

Here, \(z\) is a vector of \(K\) real values. The softmax of an entry \(z_{i}\) in the vector \(z\) is calculated by taking the exponential of \(z_{i}\) and dividing it by the sum of the exponentials of all entries in the vector. This operation ensures that:

1. The output for each entry lies between 0 and 1.
2. The sum of all output values equals 1, making it possible to interpret them as probabilities.

For example, if we apply the softmax function to a vector of scores generated by a neural networks , the result is a probability distribution over possible classes, with each entry representing the probability that the input belongs to that class.

Properties of the Softmax Function

The softmax function has several important properties:

1. Normalization: The output values are normalized to sum to 1, making them interpretable as probabilities.
2. Differentiability: The function is differentiable, allowing it to be used in gradient-based optimization methods like backpropagation.
3. Exponentiation: The use of the exponential function means that higher values in the input vector have a disproportionately higher influence on the output probabilities, which can be useful in certain contexts.

Where is Softmax Used?

Given its desirable properties, the softmax function is widely used in machine learning, particularly in multi-class classification problems. For instance, consider a neural network model designed to classify car brands. The network might produce raw scores for each brand, and by applying the softmax function to these scores, we can convert them into probabilities. The brand with the highest probability is then chosen as the classification result.

The softmax function is also a key component of the attention mechanism in transformer models, where it helps distribute attention across different parts of the input sequence.