Swish function

 ( Functions And Mappings ) 

The swish function is a family of mathematical function defined as follows:

$\backslash operatorname\{swish\}\_\backslash beta(x)\; =\; x\; \backslash operatorname\{sigmoid\}(\backslash beta\; x)\; =\; \backslash frac\{x\}\{1+e^\{-\backslash beta\; x\}\}.$

where $\backslash beta$ can be constant (usually set to 1) or trainable and "sigmoid" refers to the logistic function.

The swish family was designed to smoothly Interpolation between a linear function and the ReLU function.

When considering positive values, Swish is a particular case of doubly parameterized sigmoid shrinkage function defined in

. Variants of the swish function include Mish.

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Special values For β = 0, the function is linear: f( x) = x/2.

For β = 1, the function is the Sigmoid Linear Unit (SiLU).

With β → ∞, the function converges to ReLU.

Thus, the swish family smoothly between a linear function and the ReLU function.

Since $\backslash operatorname\{swish\}\_\backslash beta(x)\; =\; \backslash operatorname\{swish\}\_1(\backslash beta\; x)\; /\; \backslash beta$, all instances of swish have the same shape as the default $\backslash operatorname\{swish\}\_1$, zoomed by $\backslash beta$. One usually sets $\backslash beta\; >\; 0$. When $\backslash beta$ is trainable, this constraint can be enforced by $\backslash beta\; =\; e^b$, where $b$ is trainable.

$$\backslash operatorname\{swish\}\_1(x)\; =\; \backslash frac\; x2\; +\; \backslash frac\{x^2\}\{4\}-\backslash frac\{x^4\}\{48\}+\backslash frac\{x^6\}\{480\}+O\backslash left(x^8\backslash right)$$

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Derivatives Because $\backslash operatorname\{swish\}\_\backslash beta(x)\; =\; \backslash operatorname\{swish\}\_1(\backslash beta\; x)\; /\; \backslash beta$, it suffices to calculate its derivatives for the default case.$$\backslash operatorname\{swish\}\_1\text{'}(x)\; =\; \backslash frac\{x+\backslash sinh\; (x)\}\{4\; \backslash cosh\; ^2\backslash left(\backslash frac\{x\}\{2\}\backslash right)\}\; +\; \backslash frac\; 12$$so $\backslash operatorname\{swish\}\_1\text{'}(x)\; -\; \backslash frac\; 12$ is odd.$$\backslash operatorname\{swish\}\_1$$(x) = \frac{1- \frac x2 \tanh \left(\frac{x}{2}\right)}{2 \cosh ^2\left(\frac{x}{2}\right)} so $\backslash operatorname\{swish\}\_1$(x) is even.

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History SiLU was first proposed alongside the GELU in 2016, then again proposed in 2017 as the Sigmoid-weighted Linear Unit (SiL) in reinforcement learning. The SiLU/SiL was then again proposed as the SWISH over a year after its initial discovery, originally proposed without the learnable parameter β, so that β implicitly equaled 1. 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. It is believed that one reason for the improvement is that the swish function helps alleviate the vanishing gradient problem during backpropagation.

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See also

• Activation function
• Gating mechanism

Categories: Functions And Mappings, Artificial Neural Networks

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