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A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function , which is defined by the formula: [ 1 ]
The activation function of a node in an artificial neural network is a function that calculates the output of the node based on its individual inputs and their weights. Nontrivial problems can be solved using only a few nodes if the activation function is nonlinear .
For a concrete example, consider a typical recurrent network defined by = (,,) = + + where = (,) is the network parameter, is the sigmoid activation function [note 2], applied to each vector coordinate separately, and is the bias vector.
Also, certain non-continuous activation functions can be used to approximate a sigmoid function, which then allows the above theorem to apply to those functions. For example, the step function works. In particular, this shows that a perceptron network with a single infinitely wide hidden layer can approximate arbitrary functions.
The swish paper was then updated to propose the activation with the learnable parameter β. In 2017, after performing analysis on ImageNet data, researchers from Google indicated that using this function as an activation function in artificial neural networks improves the performance, compared to ReLU and sigmoid functions. [1]
Non-monotonic, unbounded, and oscillating activation functions with multiple zeros that outperform sigmoidal and ReLU-like activation functions on many tasks have also been recently explored. The threshold function has inspired building logic gates referred to as threshold logic; applicable to building logic circuits resembling brain processing.
The Gudermannian function is a sigmoid function, and as such is sometimes used as an activation function in machine learning. The (scaled and shifted) Gudermannian function is the cumulative distribution function of the hyperbolic secant distribution. A function based on the Gudermannian provides a good model for the shape of spiral galaxy arms ...
More specialized activation functions include radial basis functions (used in radial basis networks, another class of supervised neural network models). In recent developments of deep learning the rectified linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numerical problems related to the sigmoids.