Gated recurrent unit

Gated Recurrent Unit, fully gated version

Initially, for ${\displaystyle t=0}$, the output vector is ${\displaystyle h_{0}=0}$.

${\displaystyle {\begin{aligned}z_{t}&=\sigma _{g}(W_{z}x_{t}+U_{z}h_{t-1}+b_{z})\\r_{t}&=\sigma _{g}(W_{r}x_{t}+U_{r}h_{t-1}+b_{r})\\{\hat {h}}_{t}&=\phi _{h}(W_{h}x_{t}+U_{h}(r_{t}\odot h_{t-1})+b_{h})\\h_{t}&=(1-z_{t})\odot h_{t-1}+z_{t}\odot {\hat {h}}_{t}\end{aligned}}}$

Variables

• ${\displaystyle x_{t}}$: input vector
• ${\displaystyle h_{t}}$: output vector
• ${\displaystyle {\hat {h}}_{t}}$: candidate activation vector
• ${\displaystyle z_{t}}$: update gate vector
• ${\displaystyle r_{t}}$: reset gate vector
• ${\displaystyle W}$, ${\displaystyle U}$ and ${\displaystyle b}$: parameter matrices and vector

Activation functions

• ${\displaystyle \sigma _{g}}$: The original is a sigmoid function.
• ${\displaystyle \phi _{h}}$: The original is a hyperbolic tangent.

Alternative activation functions are possible, provided that ${\displaystyle \sigma _{g}(x)\in [0,1]}$.

Type 1

Type 2

Type 3

Alternate forms can be created by changing ${\displaystyle z_{t}}$ and ${\displaystyle r_{t}}$[9]

• Type 1, each gate depends only on the previous hidden state and the bias.

  ${\displaystyle {\begin{aligned}z_{t}&=\sigma _{g}(U_{z}h_{t-1}+b_{z})\\r_{t}&=\sigma _{g}(U_{r}h_{t-1}+b_{r})\\\end{aligned}}}$

• Type 2, each gate depends only on the previous hidden state.

  ${\displaystyle {\begin{aligned}z_{t}&=\sigma _{g}(U_{z}h_{t-1})\\r_{t}&=\sigma _{g}(U_{r}h_{t-1})\\\end{aligned}}}$

• Type 3, each gate is computed using only the bias.

  ${\displaystyle {\begin{aligned}z_{t}&=\sigma _{g}(b_{z})\\r_{t}&=\sigma _{g}(b_{r})\\\end{aligned}}}$

The minimal gated unit is similar to the fully gated unit, except the update and reset gate vector is merged into a forget gate. This also implies that the equation for the output vector must be changed:[10]

${\displaystyle {\begin{aligned}f_{t}&=\sigma _{g}(W_{f}x_{t}+U_{f}h_{t-1}+b_{f})\\{\hat {h}}_{t}&=\phi _{h}(W_{h}x_{t}+U_{h}(f_{t}\odot h_{t-1})+b_{h})\\h_{t}&=(1-f_{t})\odot h_{t-1}+f_{t}\odot {\hat {h}}_{t}\end{aligned}}}$

Variables

• ${\displaystyle x_{t}}$: input vector
• ${\displaystyle h_{t}}$: output vector
• ${\displaystyle {\hat {h}}_{t}}$: candidate activation vector
• ${\displaystyle f_{t}}$: forget vector
• ${\displaystyle W}$, ${\displaystyle U}$ and ${\displaystyle b}$: parameter matrices and vector