Swish function

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

The swish function

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

where $\beta$ can be constant (usually set to 1) or trainable.

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

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, as , it can be viewed as a smoothing function which nonlinearly interpolates between a linear function and the ReLU function.

Since $\operatorname {swish} _{-\beta }(x)=-\operatorname {swish} _{\beta }(-x)$, swish with negative values of $\beta$ is equivalent to a linear transform of swish with positive values of $\beta$, and do not lead to a different shape. Consequently, one usually sets $\beta >0$. When $\beta$ is trainable, this constraint can be enforced by $\beta =e^{b}$, where $b$ is trainable.

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.

Variants of the swish function include Mish.

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.