The softmax function and sigmoid function are both activation functions used in neural networks, but they serve different purposes and have distinct characteristics.

Sigmoid Function:

Formula: ( sigma(x) = frac{1}{1 + e^{-x}} )
Range: Outputs values between 0 and 1.
Usage: Commonly used in binary classification problems where the output represents the probability of a single class.
Characteristics: The sigmoid function squashes input values to a range between 0 and 1, making it useful for probabilistic interpretations. However, it can suffer from vanishing gradients, which slow down training for deep networks.
Softmax Function:

Formula: ( text{softmax}(x_i) = frac{e^{xi}}{sum{j} e^{x_j}} ) for each ( x_i ) in the input vector.
Range: Outputs a probability distribution with values between 0 and 1 that sum to 100%.
Usage: Used in multi-class classification problems where the output represents probabilities of multiple classes.
Characteristics: The softmax function converts a vector of values into a probability distribution, making it ideal for tasks where multiple classes are involved. It ensures that the sum of all probabilities is 1, facilitating the interpretation of the output as a probability distribution over classes.