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Gated recurrent units (GRUs) are a gating mechanism in recurrent neural networks, introduced in 2014 by Kyunghyun Cho et al.^[1] The GRU is like a long short-term memory (LSTM) with a gating mechanism to input or forget certain features,^[2] but lacks a context vector or output gate, resulting in fewer parameters than LSTM.^[3] GRU's performance on certain tasks of polyphonic music modeling, speech signal modeling and natural language processing was found to be similar to that of LSTM.^[4]^[5] GRUs showed that gating is indeed helpful in general, and Bengio's team came to no concrete conclusion on which of the two gating units was better.^[6]^[7]

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 $\odot$ denotes the Hadamard product in the following.

Fully gated unit

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

{\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 ( $d$ denotes the number of input features and $e$ 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

$\sigma$ : The original is a logistic function.
$\phi$ : The original is a hyperbolic tangent.

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

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.
${\begin{aligned}z_{t}&=\sigma (U_{z}h_{t-1}+b_{z})\\r_{t}&=\sigma (U_{r}h_{t-1}+b_{r})\\\end{aligned))$
Type 2, each gate depends only on the previous hidden state.
${\begin{aligned}z_{t}&=\sigma (U_{z}h_{t-1})\\r_{t}&=\sigma (U_{r}h_{t-1})\\\end{aligned))$
Type 3, each gate is computed using only the bias.
${\begin{aligned}z_{t}&=\sigma (b_{z})\\r_{t}&=\sigma (b_{r})\\\end{aligned))$

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]

{\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
$W$ , $U$ and $b$ : parameter matrices and vector

Light gated recurrent unit

The light gated recurrent unit (LiGRU)^[4] removes the reset gate altogether, replaces tanh with the ReLU activation, and applies batch normalization (BN):

{\begin{aligned}z_{t}&=\sigma (\operatorname {BN} (W_{z}x_{t})+U_{z}h_{t-1})\\{\tilde {h))_{t}&=\operatorname {ReLU} (\operatorname {BN} (W_{h}x_{t})+U_{h}h_{t-1})\\h_{t}&=z_{t}\odot h_{t-1}+(1-z_{t})\odot {\tilde {h))_{t}\end{aligned))

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.

Architecture

Fully gated unit

Minimal gated unit

Light gated recurrent unit

References