8.8. Gated Recurrent Units (GRU)¶

In the previous section we discussed how gradients are calculated in a recurrent neural network. In particular we found that long products of matrices can lead to vanishing or divergent gradients. Let’s briefly think about what such gradient anomalies mean in practice:

• We might encounter a situation where an early observation is highly significant for predicting all future observations. Consider the somewhat contrived case where the first observation contains a checksum and the goal is to discern whether the checksum is correct at the end of the sequence. In this case the influence of the first token is vital. We would like to have some mechanism for storing vital early information in a memory cell. Without such a mechanism we will have to assign a very large gradient to this observation, since it affects all subsequent observations.
• We might encounter situations where some symbols carry no pertinent observation. For instance, when parsing a webpage there might be auxiliary HTML code that is irrelevant for the purpose of assessing the sentiment conveyed on the page. We would like to have some mechanism for skipping such symbols in the latent state representation.
• We might encounter situations where there is a logical break between parts of a sequence. For instance there might be a transition between chapters in a book, a transition between a bear and a bull market for securities, etc.; In this case it would be nice to have a means of resetting our internal state representation.

A number of methods have been proposed to address this. One of the earliest is the Long Short Term Memory (LSTM)

which we will discuss in

Section 8.9. The Gated Recurrent Unit (GRU) [10] is a slightly more streamlined variant that often offers comparable performance and is significantly faster to compute. See also [12] for more details. Due to its simplicity we start with the GRU.

8.8.1. Gating the Hidden State¶

The key distinction between regular RNNs and GRUs is that the latter support gating of the hidden state. This means that we have dedicated mechanisms for when the hidden state should be updated and also when it should be reset. These mechanisms are learned and they address the concerns listed above. For instance, if the first symbol is of great importance we will learn not to update the hidden state after the first observation. Likewise, we will learn to skip irrelevant temporary observations. Lastly, we will learn to reset the latent state whenever needed. We discuss this in detail below.

8.8.1.1. Reset Gates and Update Gates¶

The first thing we need to introduce are reset and update gates. We engineer them to be vectors with entries in $$(0,1)$$ such that we can perform convex combinations, e.g. of a hidden state and an alternative. For instance, a reset variable would allow us to control how much of the previous state we might still want to remember. Likewise, an update variable would allow us to control how much of the new state is just a copy of the old state.

We begin by engineering gates to generate these variables. The figure below illustrates the inputs for both reset and update gates in a GRU, given the current time step input $$\mathbf{X}_t$$ and the hidden state of the previous time step $$\mathbf{H}_{t-1}$$. The output is given by a fully connected layer with a sigmoid as its activation function.

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

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

Here, $$\mathbf{W}_{xr}, \mathbf{W}_{xz} \in \mathbb{R}^{d \times h}$$ and $$\mathbf{W}_{hr}, \mathbf{W}_{hz} \in \mathbb{R}^{h \times h}$$ are weight parameters and $$\mathbf{b}_r, \mathbf{b}_z \in \mathbb{R}^{1 \times h}$$ are biases. We use a sigmoid function (see e.g. refer to Section 4.1 for a description) to transform values to the interval $$(0,1)$$.

8.8.1.2. Reset Gate in Action¶

We begin by integrating the reset gate with a regular latent state updating mechanism. In a conventional deep RNN we would have an update of the form

(8.8.2)$\mathbf{H}_t = \tanh(\mathbf{X}_t \mathbf{W}_{xh} + \mathbf{H}_{t-1}\mathbf{W}_{hh} + \mathbf{b}_h).$

This is essentially identical to the discussion of the previous section, albeit with a nonlinearity in the form of $$\tanh$$ to ensure that the values of the hidden state remain in the interval $$(-1, 1)$$. If we want to be able to reduce the influence of previous states we can multiply $$\mathbf{H}_{t-1}$$ with $$\mathbf{R}_t$$ elementwise. Whenever the entries in $$\mathbf{R}_t$$ are close to $$1$$ we recover a conventional deep RNN. For all entries of $$\mathbf{R}_t$$ that are close to $$0$$ the hidden state is the result of an MLP with $$\mathbf{X}_t$$ as input. Any pre-existing hidden state is thus ‘reset’ to defaults. This leads to the following candidate for a new hidden state (it is a candidate since we still need to incorporate the action of the update gate).

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

The figure below illustrates the computational flow after applying the reset gate. The symbol $$\odot$$ indicates pointwise multiplication between tensors.

8.8.1.3. Update Gate in Action¶

Next we need to incorporate the effect of the update gate. This determines the extent to which the new state $$\mathbf{H}_t$$ is just the old state $$\mathbf{H}_{t-1}$$ and by how much the new candidate state $$\tilde{\mathbf{H}}_t$$ is used. The gating variable $$\mathbf{Z}_t$$ can be used for this purpose, simply by taking elementwise convex combinations between both candidates. This leads to the final update equation for the GRU.

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

Whenever the update gate is close to $$1$$ we simply retain the old state. In this case the information from $$\mathbf{X}_t$$ is essentially ignored, effectively skipping time step $$t$$ in the dependency chain. Whenever it is close to $$1$$ the new latent state $$\mathbf{H}_t$$ approaches the candidate latent state $$\tilde{\mathbf{H}}_t$$. These designs can help cope with the vanishing gradient problem in RNNs and better capture dependencies for time series with large time step distances. In summary GRUs have the following two distinguishing features:

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

8.8.2. Implementation from Scratch¶

To gain a better understanding of the model let us implement a GRU from scratch.

We begin by reading The Time Machine corpus that we used in Section 8.5. The code for reading the data set is given below:

import d2l
from mxnet import np, npx
from mxnet.gluon import rnn
npx.set_np()

batch_size, num_steps = 32, 35


8.8.2.2. Initialize Model Parameters¶

The next step is to initialize the model parameters. We draw the weights from a Gaussian with variance $$0.01$$ and set the bias to $$0$$. The hyper-parameter num_hiddens defines the number of hidden units. We instantiate all terms relating to update and reset gate and the candidate hidden state itself. Subsequently we attach gradients to all parameters.

def get_params(vocab_size, num_hiddens, ctx):
num_inputs = num_outputs = vocab_size
normal = lambda shape : np.random.normal(scale=0.01, size=shape, ctx=ctx)
three = lambda : (normal((num_inputs, num_hiddens)),
normal((num_hiddens, num_hiddens)),
np.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 = normal((num_hiddens, num_outputs))
b_q = np.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


8.8.2.3. Define the Model¶

Now we will define the hidden state initialization function init_gru_state. Just like the init_rnn_state function defined in Section 8.5, this function returns an ndarray with a shape (batch size, number of hidden units) whose values are all zeros.

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


Now we are ready to define the actual model. Its structure is the same as the basic RNN cell, just that the update equations are more complex.

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 = npx.sigmoid(np.dot(X, W_xz) + np.dot(H, W_hz) + b_z)
R = npx.sigmoid(np.dot(X, W_xr) + np.dot(H, W_hr) + b_r)
H_tilda = np.tanh(np.dot(X, W_xh) + np.dot(R * H, W_hh) + b_h)
H = Z * H + (1 - Z) * H_tilda
Y = np.dot(H, W_hq) + b_q
outputs.append(Y)
return np.concatenate(outputs, axis=0), (H,)


8.8.2.4. Training and Prediction¶

Training and prediction work in exactly the same manner as before.

vocab_size, num_hiddens, ctx = len(vocab), 256, d2l.try_gpu()
num_epochs, lr = 500, 1
model = d2l.RNNModelScratch(len(vocab), num_hiddens, ctx, get_params,
init_gru_state, gru)
d2l.train_ch8(model, train_iter, vocab, lr, num_epochs, ctx)

Perplexity 1.1, 12613 tokens/sec on gpu(0)
time traveller  it s against reason said filby  what reason said
traveller  it s against reason said filby  what reason said


8.8.3. Concise Implementation¶

In Gluon, we can directly call the GRU class in the rnn module. This encapsulates all the configuration details that we made explicit above. The code is significantly faster as it uses compiled operators rather than Python for many details that we spelled out in detail before.

gru_layer = rnn.GRU(num_hiddens)
model = d2l.RNNModel(gru_layer, len(vocab))
d2l.train_ch8(model, train_iter, vocab, lr, num_epochs, ctx)

Perplexity 1.2, 156083 tokens/sec on gpu(0)
time traveller smiled round at us then still smiling faintly and
traveller  it s against reason said filby  what reason said


8.8.4. Summary¶

• Gated recurrent neural networks are better at capturing dependencies for time series with large time step distances.
• Reset gates help capture short-term dependencies in time series.
• Update gates help capture long-term dependencies in time series.
• GRUs contain basic RNNs as their extreme case whenever the reset gate is switched on. They can ignore sequences as needed.

8.8.5. Exercises¶

1. Compare runtimes, perplexity and the extracted strings for rnn.RNN and rnn.GRU implementations with each other.
2. Assume that we only want to use the input for time step $$t'$$ to predict the output at time step $$t > t'$$. What are the best values for reset and update gates for each time step?
3. Adjust the hyper-parameters and observe and analyze the impact on running time, perplexity, and the written lyrics.
4. What happens if you implement only parts of a GRU? That is, implement a recurrent cell that only has a reset gate. Likewise, implement a recurrent cell only with an update gate.