In the context of artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function[1][2] is an activation function defined as the non-negative part of its argument:
where is the input to a neuron. This is also known as a ramp function and is analogous to half-wave rectification in electrical engineering.
As of 2017[update], it is the most popular activation function for deep neural networks.[3] Rectified linear units find applications in computer vision[4] and speech recognition[5][6] using deep neural nets and computational neuroscience.[7][8][9]
It was first used by Alston Householder in 1941 as a mathematical abstraction of biological neural networks.[10] It was introduced by Kunihiko Fukushima in 1969 in the context of visual feature extraction in hierarchical neural networks.[11][12] It was later argued that it has strong biological motivations and mathematical justifications.[13][14] In 2011,[4] ReLU activation enabled training deep supervised neural networks without unsupervised pre-training, compared to the widely used activation functions prior to 2011, e.g., the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more practical[15] counterpart, the hyperbolic tangent.
Leaky ReLUs allow a small, positive gradient when the unit is not active,[6] helping to mitigate the vanishing gradient problem.
Parametric ReLUs (PReLUs) take this idea further by making the coefficient of leakage into a parameter that is learned along with the other neural-network parameters.[16]
Note that for a ≤ 1, this is equivalent to
and thus has a relation to "maxout" networks.[16]
Concatenated ReLU (CReLU) preserves positive and negative phase information.[17]
GELU is a smooth approximation to the rectifier:
where is the cumulative distribution function of the standard normal distribution.
This activation function is illustrated in the figure at the start of this article. It has a "bump" to the left of x < 0 and serves as the default activation for models such as BERT.[18]
The SiLU (sigmoid linear unit) or swish function[19] is another smooth approximation, first coined in the GELU paper:[18]
where is the sigmoid function.
A smooth approximation to the rectifier is the analytic function
which is called the softplus[20][4] or SmoothReLU function.[21] For large negative it is roughly , so just above 0, while for large positive it is roughly , so just above .
This function can be approximated as:
By making the change of variables , this is equivalent to
A sharpness parameter may be included:
The derivative of softplus is the logistic function.
The logistic sigmoid function is a smooth approximation of the derivative of the rectifier, the Heaviside step function.
The multivariable generalization of single-variable softplus is the LogSumExp with the first argument set to zero:
The LogSumExp function is
and its gradient is the softmax; the softmax with the first argument set to zero is the multivariable generalization of the logistic function. Both LogSumExp and softmax are used in machine learning.
Exponential linear units try to make the mean activations closer to zero, which speeds up learning. It has been shown that ELUs can obtain higher classification accuracy than ReLUs.[22]
In these formulas, is a hyper-parameter to be tuned with the constraint .
The ELU can be viewed as a smoothed version of a shifted ReLU (SReLU), which has the form , given the same interpretation of .
The mish function can also be used as a smooth approximation of the rectifier.[19] It is defined as
where is the hyperbolic tangent, and is the softplus function.
Mish is non-monotonic and self-gated.[23] It was inspired by Swish, itself a variant of ReLU.[23]
Squareplus[24] is the function
where is a hyperparameter that determines the "size" of the curved region near . (For example, letting yields ReLU, and letting yields the metallic mean function.) Squareplus shares many properties with softplus: It is monotonic, strictly positive, approaches 0 as , approaches the identity as , and is smooth. However, squareplus can be computed using only algebraic functions, making it well-suited for settings where computational resources or instruction sets are limited. Additionally, squareplus requires no special consideration to ensure numerical stability when is large.
Rectifier and softplus activation functions. The second one is a smooth version of the first.
{{cite book}}
: |journal=
ignored (help)
Since the sigmoid h has a positive first derivative, its primitive, which we call softplus, is convex.