Sigmoidfunktionen

Sigmoid functions are a class of mathematical functions that have an "S"-shaped curve or sigmoid curve. They are commonly used in machine learning and neural networks to introduce non-linearity into models. The most well-known sigmoid function is the logistic function, defined by the formula:

f(x) = 1 / (1 + e^(-x))

where e is the base of the natural logarithm. This function maps any real-valued number into the range (0, 1), making it useful for binary classification tasks.

Another popular sigmoid function is the hyperbolic tangent (tanh) function, defined by:

f(x) = (e^x - e^(-x)) / (e^x + e^(-x))

The tanh function maps real-valued numbers to the range (-1, 1), which can be beneficial in certain neural network architectures.

Sigmoid functions have several properties that make them useful in machine learning. They are differentiable, which allows for the use of gradient-based optimization algorithms. They are also bounded, which can help prevent the exploding gradient problem. However, sigmoid functions can suffer from vanishing gradient problems, where the gradient becomes very small, slowing down learning.

Despite these drawbacks, sigmoid functions remain a fundamental component in many machine learning models, particularly in the context of neural networks. They are often used in the output layer for binary classification tasks and in hidden layers to introduce non-linearity.