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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 ]
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.
Logistic activation function. 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. [1]
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
A widely used type of composition is the nonlinear weighted sum, where () = (()), where (commonly referred to as the activation function [3]) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important characteristic of the activation function is that it provides a smooth ...
Traditional activation functions include sigmoid, tanh, and ReLU. Swish , [ 9 ] mish , [ 10 ] and other activation functions have since been proposed as well. The overall network is a combination of function composition and matrix multiplication :
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 ]
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.