Gated recurrent unit

Gated recurrent units (GRUs) are a gating mechanism in

Architecture

There are several variations on the full gated unit, with gating done using the previous hidden state and the bias in various combinations, and a simplified form called minimal gated unit.^[8]

The operator ${\displaystyle \odot }$ denotes the Hadamard product in the following.

Fully gated unit

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

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

Variables (${\displaystyle d}$ denotes the number of input features and ${\displaystyle e}$ the number of output features):

• ${\displaystyle x_{t}\in \mathbb {R} ^{d}}$: input vector
• ${\displaystyle h_{t}\in \mathbb {R} ^{e}}$: output vector
• ${\displaystyle {\hat {h}}_{t}\in \mathbb {R} ^{e}}$: candidate activation vector
• ${\displaystyle z_{t}\in (0,1)^{e}}$: update gate vector
• ${\displaystyle r_{t}\in (0,1)^{e}}$: reset gate vector
• ${\displaystyle W\in \mathbb {R} ^{d\times e}}$, ${\displaystyle U\in \mathbb {R} ^{e\times e}}$ and ${\displaystyle b\in \mathbb {R} ^{e}}$: parameter matrices and vector which need to be learned during training

Activation functions

• ${\displaystyle \sigma }$: The original is a logistic function.
• ${\displaystyle \phi }$: The original is a
  hyperbolic tangent
  .

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

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.

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

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

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

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

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

Minimal gated unit

The minimal gated unit (MGU) 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]

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

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

Light gated recurrent unit

The light gated recurrent unit (LiGRU)

LiGRU has been studied from a Bayesian perspective.[11] This analysis yielded a variant called light Bayesian recurrent unit (LiBRU), which showed slight improvements over the LiGRU on speech recognition tasks.

References