# Gated Recurrent Unit (GRU)¶

In the previous section, we discussed gradient calculation methods in recurrent neural networks. We found that, when the number of time steps is large or the time step is small, the gradients in recurrent neural networks are prone to vanishing or explosion. Although gradient clipping can cope with gradient explosion, it cannot solve the vanishing gradient problem. Therefore, it is generally quite difficult to capture dependencies for time series with large time step distances during the actual use of recurrent neural networks.

Gated recurrent neural networks were proposed as a way to better capture dependencies for time series with large time step distances. Such a network uses learnable gates to control the flow of information. One common type of gated recurrent neural network is a gated recurrent unit (GRU) [1, 2]. Another common type of gated recurrent neural network is discussed in the next section.

## Gated Recurrent Units¶

In what follows, we will discuss the design of GRUs. These networks introduce the reset gate and update gate concepts to change the method used to calculate hidden states in recurrent neural networks.

### Reset Gates and Update Gates¶

As shown in Figure 6.4, the inputs for both reset gates and update gates in GRU are the current time step input $$\boldsymbol{X}_t$$ and the hidden state of the previous time step $$\boldsymbol{H}_{t-1}$$. The output is computed by the fully connected layer with a sigmoid function as its activation function.

Here, we assume there are $$h$$ hidden units and, for a given time step $$t$$, the mini-batch input is $$\boldsymbol{X}_t \in \mathbb{R}^{n \times d}$$ (number of examples: $$n$$, number of inputs: $$d$$）and the hidden state of the last time step is $$\boldsymbol{H}_{t-1} \in \mathbb{R}^{n \times h}$$. Then, the reset gate $$\boldsymbol{R}_t \in \mathbb{R}^{n \times h}$$ and update gate $$\boldsymbol{Z}_t \in \mathbb{R}^{n \times h}$$ computation is as follows:

\begin{split}\begin{aligned} \boldsymbol{R}_t = \sigma(\boldsymbol{X}_t \boldsymbol{W}_{xr} + \boldsymbol{H}_{t-1} \boldsymbol{W}_{hr} + \boldsymbol{b}_r),\\ \boldsymbol{Z}_t = \sigma(\boldsymbol{X}_t \boldsymbol{W}_{xz} + \boldsymbol{H}_{t-1} \boldsymbol{W}_{hz} + \boldsymbol{b}_z), \end{aligned}\end{split}

Here, $$\boldsymbol{W}_{xr}, \boldsymbol{W}_{xz} \in \mathbb{R}^{d \times h}$$ and $$\boldsymbol{W}_{hr}, \boldsymbol{W}_{hz} \in \mathbb{R}^{h \times h}$$ are weight parameters and $$\boldsymbol{b}_r, \boldsymbol{b}_z \in \mathbb{R}^{1 \times h}$$ is a bias parameter. As described in the “Multilayer Perceptron” section, a sigmoid function can transform element values between 0 and 1. Therefore, the range of each element in the reset gate $$\boldsymbol{R}_t$$ and update gate $$\boldsymbol{Z}_t$$ is $$[0, 1]$$.

### Candidate Hidden States¶

Next, the GRU computes candidate hidden states to facilitate subsequent hidden state computation. As shown in Figure 6.5, we perform multiplication by element between the current time step reset gate output and previous time step hidden state (symbol: $$\odot$$). If the element value in the reset gate approaches 0, this means that it resets the value of the corresponding hidden state element to 0, discarding the hidden state from the previous time step. If the element value approaches 1, this indicates that the hidden state from the previous time step is retained. Then, the result of multiplication by element is concatenated with the current time step input to compute candidate hidden states in a fully connected layer with a tanh activation function. The range of all element values is $$[-1, 1]$$.

For time step $$t$$, the candidate hidden state $$\tilde{\boldsymbol{H}}_t \in \mathbb{R}^{n \times h}$$ is computed by the following formula:

$\tilde{\boldsymbol{H}}_t = \text{tanh}(\boldsymbol{X}_t \boldsymbol{W}_{xh} + \left(\boldsymbol{R}_t \odot \boldsymbol{H}_{t-1}\right) \boldsymbol{W}_{hh} + \boldsymbol{b}_h),$

Here, $$\boldsymbol{W}_{xh} \in \mathbb{R}^{d \times h}$$ and $$\boldsymbol{W}_{hh} \in \mathbb{R}^{h \times h}$$ are weight parameters and $$\boldsymbol{b}_h \in \mathbb{R}^{1 \times h}$$ is a bias parameter. From the formula above, we can see that the reset gate controls how the hidden state of the previous time step enters into the candidate hidden state of the current time step. In addition, the hidden state of the previous time step may contain all historical information of the time series up to the previous time step. Thus, the reset gate can be used to discard historical information that has no bearing on predictions.

### Hidden States¶

Finally, the computation of the hidden state $$\boldsymbol{H}_t \in \mathbb{R}^{n \times h}$$ for time step $$t$$ uses the current time step’s update gate $$\boldsymbol{Z}_t$$ to combine the previous time step hidden state $$\boldsymbol{H}_{t-1}$$ and current time step candidate hidden state $$\tilde{\boldsymbol{H}}_t$$:

$\boldsymbol{H}_t = \boldsymbol{Z}_t \odot \boldsymbol{H}_{t-1} + (1 - \boldsymbol{Z}_t) \odot \tilde{\boldsymbol{H}}_t.$

It should be noted that update gates can control how hidden states should be updated by candidate hidden states containing current time step information, as shown in Figure 6.6. Here, we assume that the update gate is always approximately 1 between the time steps $$t'$$ and $$t$$ ($$t' < t$$). Therefore, the input information between the time steps $$t'$$ and $$t$$ almost never enters the hidden state $$\boldsymbol{H}_t$$ for time step $$t$$. In fact, we can think of it like this: The hidden state of an earlier time $$\boldsymbol{H}_{t'-1}$$ is saved over time and passed to the current time step $$t$$. This design can cope with the vanishing gradient problem in recurrent neural networks and better capture dependencies for time series with large time step distances.

We can summarize the design of GRUs as follows:

• Reset gates help capture short-term dependencies in time series.
• Update gates help capture long-term dependencies in time series.

To implement and display a GRU, we will again use the Jay Chou lyrics data set to train the model to compose song lyrics. The implementation, except for the GRU, has already been described in the “Recurrent Neural Network” section. The code for reading the data set is given below:

In [1]:

import gluonbook as gb
from mxnet import nd
from mxnet.gluon import rnn

(corpus_indices, char_to_idx, idx_to_char,


## Implementation from Scratch¶

We will start by showing how to implement a GRU from scratch.

### Initialize Model Parameters¶

The code below initializes the model parameters. The hyper-parameter num_hiddens defines the number of hidden units.

In [2]:

num_inputs, num_hiddens, num_outputs = vocab_size, 256, vocab_size
ctx = gb.try_gpu()

def get_params():
def _one(shape):
return nd.random.normal(scale=0.01, shape=shape, ctx=ctx)

def _three():
return (_one((num_inputs, num_hiddens)),
_one((num_hiddens, num_hiddens)),
nd.zeros(num_hiddens, ctx=ctx))

W_xz, W_hz, b_z = _three()  # Update gate parameter
W_xr, W_hr, b_r = _three()  # Reset gate parameter
W_xh, W_hh, b_h = _three()  # Candidate hidden state parameter
# Output layer parameters
W_hq = _one((num_hiddens, num_outputs))
b_q = nd.zeros(num_outputs, ctx=ctx)
params = [W_xz, W_hz, b_z, W_xr, W_hr, b_r, W_xh, W_hh, b_h, W_hq, b_q]
for param in params:
return params


### Define the Model¶

Now we will define the hidden state initialization function init_gru_state. Just like the init_rnn_state function defined in the “Implementation of the Recurrent Neural Network from Scratch” section, this function returns a tuple composed of an NDArray with a shape (batch size, number of hidden units) value of 0.

In [3]:

def init_gru_state(batch_size, num_hiddens, ctx):
return (nd.zeros(shape=(batch_size, num_hiddens), ctx=ctx), )


Below, we define the model based on GRU computing expressions.

In [4]:

def gru(inputs, state, params):
W_xz, W_hz, b_z, W_xr, W_hr, b_r, W_xh, W_hh, b_h, W_hq, b_q = params
H, = state
outputs = []
for X in inputs:
Z = nd.sigmoid(nd.dot(X, W_xz) + nd.dot(H, W_hz) + b_z)
R = nd.sigmoid(nd.dot(X, W_xr) + nd.dot(H, W_hr) + b_r)
H_tilda = nd.tanh(nd.dot(X, W_xh) + R * nd.dot(H, W_hh) + b_h)
H = Z * H + (1 - Z) * H_tilda
Y = nd.dot(H, W_hq) + b_q
outputs.append(Y)
return outputs, (H,)


### Train the Model and Write Lyrics¶

During model training, we only use adjacent examples. After setting the hyper-parameters, we train and model and create a 50 character string of lyrics based on the prefixes “separate” and “not separated”.

In [5]:

num_epochs, num_steps, batch_size, lr, clipping_theta = 160, 35, 32, 1e2, 1e-2
pred_period, pred_len, prefixes = 40, 50, ['分开', '不分开']


We create a string of lyrics based on the currently trained model every 40 epochs.

In [6]:

gb.train_and_predict_rnn(gru, get_params, init_gru_state, num_hiddens,
vocab_size, ctx, corpus_indices, idx_to_char,
char_to_idx, False, num_epochs, num_steps, lr,
clipping_theta, batch_size, pred_period, pred_len,
prefixes)

epoch 40, perplexity 151.068701, time 0.63 sec
- 分开 我不的让我爱爱人 我想你的让我爱爱人 坏坏的让我爱人 我想我想你的可爱人 我坏的让我爱爱人 我坏的
- 不分开 我想你的让我爱人 我想你的让我爱爱人 坏坏的让我爱人 我想我想你的可爱人 我坏的让我爱爱人 我坏的
epoch 80, perplexity 32.226311, time 0.64 sec
- 分开 我想要你的微笑 像人在美不鸠 让我的让我疯狂的可爱女人 坏坏的让我疯狂的可爱女人 坏坏的让我疯狂的
- 不分开 我想你的爱写在西元前 深埋在美索不达米亚平原 我有你的爱写在西元前 深埋在美索不达米亚平原 我有你
epoch 120, perplexity 5.019609, time 0.63 sec
- 分开我 你说啊 分怎么 什么了 什么的空空 还制茶人豆无 它在人敌的牛肉 我说店小二 三两银够不够 景色
- 不分开 我已经这样 我该好好生活 静静悄悄默默离开 陷入了危险 让我都有你 我不要这些我 这样的假活 我爱
epoch 160, perplexity 1.493144, time 0.63 sec
- 分开 我想大声宣布 对你依依不舍 连隔壁邻居都猜到我现在的感受 河边的风 在吹着头发飘动 牵着你的手 一
- 不分开 你已经离开我 不知不觉 我跟了这节奏 后知后觉 又过了一个秋 后知后觉 我该好好生活 我该好好生活


## Gluon Implementation¶

In Gluon, we can directly call the GRU class in the rnn module.

In [7]:

gru_layer = rnn.GRU(num_hiddens)
model = gb.RNNModel(gru_layer, vocab_size)
gb.train_and_predict_rnn_gluon(model, num_hiddens, vocab_size, ctx,
corpus_indices, idx_to_char, char_to_idx,
num_epochs, num_steps, lr, clipping_theta,
batch_size, pred_period, pred_len, prefixes)

epoch 40, perplexity 152.689235, time 0.18 sec
- 分开 我想你的让我想想想想想想想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想
- 不分开 我想你的让我想想想想想想想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想你想
epoch 80, perplexity 32.467845, time 0.18 sec
- 分开 我想要这样的怒火 我想想你的爱写 一定 没有你的肩笑 一定我不多 你不知不想 我不要再想 我不 我
- 不分开 我不要再想你 我不 我不 我不 我不 我不 我不 我不 我不 我不 我不 我不 我不 我不 我不
epoch 120, perplexity 5.091850, time 0.18 sec
- 分开 我想带这样坦堡 想要和你已经到宙 就是我 说你眼睛看着我 别发抖 快给我抬起头 有话去对医药箱说
- 不分开  没有你烦我有多 我说 你想很久了吧? 别是你 说你眼睛看着我 别发抖 快给我抬起头 有话去对医药
epoch 160, perplexity 1.491004, time 0.18 sec
- 分开 我想带这样布堡 就想是你笑棒久 我想要你的微笑每天都能看到  我知道这里很美但家乡的你更美原来我只
- 不分开 你已经离开我 不知不觉 我跟了这节奏 后知后觉 又过了一个秋 后知后觉 我该好好生活 我该好好生活


## Summary¶

• Gated recurrent neural networks can better capture dependencies for time series with large time step distances.
• GRUs introduce the reset gate and update gate concepts to change the method used to calculate hidden states in recurrent neural networks. They include reset gates, update gates, candidate hidden states, and hidden states.
• Reset gates help capture short-term dependencies in time series.
• Update gates help capture long-term dependencies in time series.

## Problems¶

• Assume that time step $$t' < t$$. If we only want to use the input for time step $$t'$$ to predict the output at time step $$t$$, what are the best values for the reset and update gates for each time step?
• Adjust the hyper-parameters and observe and analyze the impact on running time, perplexity, and the written lyrics.
• Compare the running times of a GRU and ungated recurrent neural network under the same conditions.

## References¶

[1] Cho, K., Van Merri ë nboer, B., Bahdanau, D., & Bengio, Y. (2014). On the properties of neural machine translation: Encoder-decoder approaches. arXiv preprint arXiv:1409.1259.

[2] Chung, J., Gulcehre, C., Cho, K., & Bengio, Y. (2014). Empirical evaluation of gated recurrent neural networks on sequence modeling. arXiv preprint arXiv:1412.3555.