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

• ${\displaystyle x^{(t)}}$: It first projects the current word ${\displaystyle v^{(t)}}$ into ${\displaystyle x^{(t)}}$ as the input feature. This result would be used in the next hidden state and passed to the proposed content-adaptive gate.
• ${\displaystyle n^{(t)}}$: Compare to GRU, the reset gate has been taken out. It just combines the parameters related to ${\displaystyle h^{(t)}}$ and ${\displaystyle x^{(t)}}$ to produce a new hidden state ${\displaystyle n^{(t)}}$.
• ${\displaystyle z^{(t)}}$: It is the same as the update gate in GRU and is used to the transition of the hidden state.
• ${\displaystyle l^{(t)}}$: There is a Hadamard operator to combine the update gate with the weight of current feature. This gate had been named as content-adaptive gate, which will influence the amount of gradual transition, rather than diluting the current hidden state.
• ${\displaystyle h^{(t+1)}}$: The next hidden state is combined with ${\displaystyle h^{(t)}}$ and ${\displaystyle n^{(t)}}$.

Another feature of CARU is the weighting of hidden states according to the current word and the introduction of content-adaptive gates, instead of using reset gates to alleviate the dependence on long-term content. There are three trunks of data flow to be processed by CARU:

• content-state: It produces a new hidden state ${\displaystyle n^{(t)}}$ achieved by a linear layer, this part is equivalent to simple RNN networks.

• word-weight: It produces the weight ${\displaystyle \sigma (x^{(t)})}$ of the current word, it has the capability like a GRU reset gate but is only based on the current word instead of the entire content. More specifically, it can be considered as the tagging task that connects the relation between the weight and parts-of-speech.
• content-weight: It produces the weight ${\displaystyle z^{(t)}}$ of the current content, the form is the same as a GRU update gate but with the purpose to overcome the long-term dependence.

In contrast to GRU, CARU does not intend to process those data flow, instead dispatch the word-weight to the content-adaptive gate and multiplies it with the content-weight. In this way, the content-adaptive gate considers of both the word and the content.