10.2  Gated Recurrent Units (GRU)

As RNNs and particularly the LSTM architecture (Section 10.1) rapidly gained popularity during the 2010s, a number of researchers began to experiment with simplified architectures in hopes of retaining the key idea of incorporating an internal state and multiplicative gating mechanisms but with the aim of speeding up computation. The gated recurrent unit (GRU) (Cho et al. 2014) offered a streamlined version of the LSTM memory cell that often achieves comparable performance but with the advantage of being faster to compute (Chung et al. 2014).

from d2l import torch as d2l
import torch
from torch import nn
from d2l import tensorflow as d2l
import tensorflow as tf
from d2l import jax as d2l
from flax import linen as nn
import jax
from jax import numpy as jnp
from d2l import mxnet as d2l
from mxnet import np, npx
from mxnet.gluon import rnn
npx.set_np()

10.2.1 Reset Gate and Update Gate

Here, the LSTM’s three gates are replaced by two: the reset gate and the update gate. As with LSTMs, these gates are given sigmoid activations, forcing their values to lie in the interval \((0, 1)\). Intuitively, the reset gate controls how much of the previous state we might still want to remember. Likewise, an update gate would allow us to control how much of the new state is just a copy of the old one. Figure 10.2.1 illustrates the inputs for both the reset and update gates in a GRU, given the input of the current time step and the hidden state of the previous time step. The outputs of the gates are given by two fully connected layers with a sigmoid activation function.

Figure 10.2.1: Computing the reset gate and the update gate in a GRU model.

Mathematically, for a given time step \(t\), suppose that the input is a minibatch \(\mathbf{X}_t \in \mathbb{R}^{n \times d}\) (number of examples \(=n\); number of inputs \(=d\)) and the hidden state of the previous time step is \(\mathbf{H}_{t-1} \in \mathbb{R}^{n \times h}\) (number of hidden units \(=h\)). 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:

\[ \begin{aligned} \mathbf{R}_t = \sigma(\mathbf{X}_t \mathbf{W}_{\textrm{xr}} + \mathbf{H}_{t-1} \mathbf{W}_{\textrm{hr}} + \mathbf{b}_\textrm{r}),\\ \mathbf{Z}_t = \sigma(\mathbf{X}_t \mathbf{W}_{\textrm{xz}} + \mathbf{H}_{t-1} \mathbf{W}_{\textrm{hz}} + \mathbf{b}_\textrm{z}), \end{aligned} \]

where \(\mathbf{W}_{\textrm{xr}}, \mathbf{W}_{\textrm{xz}} \in \mathbb{R}^{d \times h}\) and \(\mathbf{W}_{\textrm{hr}}, \mathbf{W}_{\textrm{hz}} \in \mathbb{R}^{h \times h}\) are weight parameters and \(\mathbf{b}_\textrm{r}, \mathbf{b}_\textrm{z} \in \mathbb{R}^{1 \times h}\) are bias parameters.

10.2.2 Candidate Hidden State

Next, we integrate the reset gate \(\mathbf{R}_t\) with the regular updating mechanism in Equation 9.4.4, leading to the following candidate hidden state \(\tilde{\mathbf{H}}_t \in \mathbb{R}^{n \times h}\) at time step \(t\):

\[\tilde{\mathbf{H}}_t = \tanh(\mathbf{X}_t \mathbf{W}_{\textrm{xh}} + \left(\mathbf{R}_t \odot \mathbf{H}_{t-1}\right) \mathbf{W}_{\textrm{hh}} + \mathbf{b}_\textrm{h}), \tag{10.2.1}\]

where \(\mathbf{W}_{\textrm{xh}} \in \mathbb{R}^{d \times h}\) and \(\mathbf{W}_{\textrm{hh}} \in \mathbb{R}^{h \times h}\) are weight parameters, \(\mathbf{b}_\textrm{h} \in \mathbb{R}^{1 \times h}\) is the bias, and the symbol \(\odot\) is the Hadamard (elementwise) product operator. Here we use a tanh activation function.

The result is a candidate, since we still need to incorporate the action of the update gate. Comparing with Equation 9.4.4, the influence of the previous states can now be reduced with the elementwise multiplication of \(\mathbf{R}_t\) and \(\mathbf{H}_{t-1}\) in Equation 10.2.1. Whenever the entries in the reset gate \(\mathbf{R}_t\) are close to 1, we recover a vanilla RNN such as that in Equation 9.4.4. For all entries of the reset gate \(\mathbf{R}_t\) that are close to 0, the candidate hidden state is the result of an MLP with \(\mathbf{X}_t\) as input. Any pre-existing hidden state is thus reset to defaults.

Figure 10.2.2 illustrates the computational flow after applying the reset gate.

Figure 10.2.2: Computing the candidate hidden state in a GRU model.

10.2.3 Hidden State

Finally, we need to incorporate the effect of the update gate \(\mathbf{Z}_t\). This determines the extent to which the new hidden state \(\mathbf{H}_t \in \mathbb{R}^{n \times h}\) matches the old state \(\mathbf{H}_{t-1}\) compared with how much it resembles the new candidate state \(\tilde{\mathbf{H}}_t\). The update gate \(\mathbf{Z}_t\) can be used for this purpose, simply by taking elementwise convex combinations of \(\mathbf{H}_{t-1}\) and \(\tilde{\mathbf{H}}_t\). This leads to the final update equation for the GRU:

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

Whenever the update gate \(\mathbf{Z}_t\) is close to 1, we simply retain the old state. In this case the information from \(\mathbf{X}_t\) is ignored, effectively skipping time step \(t\) in the dependency chain. By contrast, whenever \(\mathbf{Z}_t\) is close to 0, the new latent state \(\mathbf{H}_t\) approaches the candidate latent state \(\tilde{\mathbf{H}}_t\). Figure 10.2.3 shows the computational flow after the update gate is in action.

Figure 10.2.3: Computing the hidden state in a GRU model.

In summary, GRUs have the following two distinguishing features:

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

10.2.4 Implementation from Scratch

To gain a better understanding of the GRU model, let’s implement it from scratch.

10.2.4.1 Initializing Model Parameters

The first step is to initialize the model parameters. We draw the weights from a Gaussian distribution with standard deviation to be sigma and set the bias to 0. The hyperparameter num_hiddens defines the number of hidden units. We instantiate all weights and biases relating to the update gate, the reset gate, and the candidate hidden state.

class GRUScratch(d2l.Module):
    def __init__(self, num_inputs, num_hiddens, sigma=0.01):
        super().__init__()
        self.save_hyperparameters()
        
        init_weight = lambda *shape: nn.Parameter(d2l.randn(*shape) * sigma)
        triple = lambda: (init_weight(num_inputs, num_hiddens),
                          init_weight(num_hiddens, num_hiddens),
                          nn.Parameter(d2l.zeros(num_hiddens)))
        self.W_xz, self.W_hz, self.b_z = triple()  # Update gate
        self.W_xr, self.W_hr, self.b_r = triple()  # Reset gate
        self.W_xh, self.W_hh, self.b_h = triple()  # Candidate hidden state        
class GRUScratch(d2l.Module):
    def __init__(self, num_inputs, num_hiddens, sigma=0.01):
        super().__init__()
        self.save_hyperparameters()
        
        init_weight = lambda *shape: tf.Variable(d2l.normal(shape) * sigma)
        triple = lambda: (init_weight(num_inputs, num_hiddens),
                          init_weight(num_hiddens, num_hiddens),
                          tf.Variable(d2l.zeros(num_hiddens)))            
            
        self.W_xz, self.W_hz, self.b_z = triple()  # Update gate
        self.W_xr, self.W_hr, self.b_r = triple()  # Reset gate
        self.W_xh, self.W_hh, self.b_h = triple()  # Candidate hidden state        
class GRUScratch(d2l.Module):
    num_inputs: int
    num_hiddens: int
    sigma: float = 0.01

    def setup(self):
        init_weight = lambda name, shape: self.param(name,
                                                     nn.initializers.normal(self.sigma),
                                                     shape)
        triple = lambda name : (
            init_weight(f'W_x{name}', (self.num_inputs, self.num_hiddens)),
            init_weight(f'W_h{name}', (self.num_hiddens, self.num_hiddens)),
            self.param(f'b_{name}', nn.initializers.zeros, (self.num_hiddens)))

        self.W_xz, self.W_hz, self.b_z = triple('z')  # Update gate
        self.W_xr, self.W_hr, self.b_r = triple('r')  # Reset gate
        self.W_xh, self.W_hh, self.b_h = triple('h')  # Candidate hidden state
class GRUScratch(d2l.Module):
    def __init__(self, num_inputs, num_hiddens, sigma=0.01):
        super().__init__()
        self.save_hyperparameters()
        
        init_weight = lambda *shape: d2l.randn(*shape) * sigma
        triple = lambda: (init_weight(num_inputs, num_hiddens),
                          init_weight(num_hiddens, num_hiddens),
                          d2l.zeros(num_hiddens))            
        self.W_xz, self.W_hz, self.b_z = triple()  # Update gate
        self.W_xr, self.W_hr, self.b_r = triple()  # Reset gate
        self.W_xh, self.W_hh, self.b_h = triple()  # Candidate hidden state

10.2.4.2 Defining the Model

Now we are ready to define the GRU forward computation. Its structure is the same as that of the basic RNN cell, except that the update equations are more complex.

@d2l.add_to_class(GRUScratch)
def forward(self, inputs, H=None):
    if H is None:
        # Initial state with shape: (batch_size, num_hiddens)
        H = d2l.zeros((inputs.shape[1], self.num_hiddens),
                      device=inputs.device)
    outputs = []
    for X in inputs:
        Z = d2l.sigmoid(d2l.matmul(X, self.W_xz) +
                        d2l.matmul(H, self.W_hz) + self.b_z)
        R = d2l.sigmoid(d2l.matmul(X, self.W_xr) + 
                        d2l.matmul(H, self.W_hr) + self.b_r)
        H_tilde = d2l.tanh(d2l.matmul(X, self.W_xh) + 
                           d2l.matmul(R * H, self.W_hh) + self.b_h)
        H = Z * H + (1 - Z) * H_tilde
        outputs.append(H)
    return outputs, H
@d2l.add_to_class(GRUScratch)
def forward(self, inputs, H=None):
    if H is None:
        # Initial state with shape: (batch_size, num_hiddens)
        H = tf.zeros((tf.shape(inputs)[1], self.num_hiddens))
    outputs = []
    for X in tf.unstack(inputs):
        Z = d2l.sigmoid(d2l.matmul(X, self.W_xz) +
                        d2l.matmul(H, self.W_hz) + self.b_z)
        R = d2l.sigmoid(d2l.matmul(X, self.W_xr) + 
                        d2l.matmul(H, self.W_hr) + self.b_r)
        H_tilde = d2l.tanh(d2l.matmul(X, self.W_xh) + 
                           d2l.matmul(R * H, self.W_hh) + self.b_h)
        H = Z * H + (1 - Z) * H_tilde
        outputs.append(H)
    return outputs, H
@d2l.add_to_class(GRUScratch)
def forward(self, inputs, H=None):
    # Use lax.scan primitive instead of looping over the
    # inputs, since scan saves time in jit compilation
    def scan_fn(H, X):
        Z = d2l.sigmoid(d2l.matmul(X, self.W_xz) + d2l.matmul(H, self.W_hz) +
                        self.b_z)
        R = d2l.sigmoid(d2l.matmul(X, self.W_xr) +
                        d2l.matmul(H, self.W_hr) + self.b_r)
        H_tilde = d2l.tanh(d2l.matmul(X, self.W_xh) +
                           d2l.matmul(R * H, self.W_hh) + self.b_h)
        H = Z * H + (1 - Z) * H_tilde
        return H, H  # return carry, y

    if H is None:
        batch_size = inputs.shape[1]
        carry = jnp.zeros((batch_size, self.num_hiddens))
    else:
        carry = H

    # scan takes the scan_fn, initial carry state, xs with leading axis to be scanned
    carry, outputs = jax.lax.scan(scan_fn, carry, inputs)
    return outputs, carry
@d2l.add_to_class(GRUScratch)
def forward(self, inputs, H=None):
    if H is None:
        # Initial state with shape: (batch_size, num_hiddens)
        H = d2l.zeros((inputs.shape[1], self.num_hiddens),
                      ctx=inputs.ctx)
    outputs = []
    for X in inputs:
        Z = d2l.sigmoid(d2l.matmul(X, self.W_xz) +
                        d2l.matmul(H, self.W_hz) + self.b_z)
        R = d2l.sigmoid(d2l.matmul(X, self.W_xr) + 
                        d2l.matmul(H, self.W_hr) + self.b_r)
        H_tilde = d2l.tanh(d2l.matmul(X, self.W_xh) + 
                           d2l.matmul(R * H, self.W_hh) + self.b_h)
        H = Z * H + (1 - Z) * H_tilde
        outputs.append(H)
    return outputs, H

10.2.4.3 Training

Training a language model on The Time Machine dataset works in exactly the same manner as in Section 9.5.

data = d2l.TimeMachine(batch_size=1024, num_steps=32)
gru = GRUScratch(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLMScratch(gru, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1, num_gpus=1)
trainer.fit(model, data)

gru-c3-pytorch
data = d2l.TimeMachine(batch_size=1024, num_steps=32)
with d2l.try_gpu():
    gru = GRUScratch(num_inputs=len(data.vocab), num_hiddens=32)
    model = d2l.RNNLMScratch(gru, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1)
trainer.fit(model, data)

gru-c3-tensorflow
data = d2l.TimeMachine(batch_size=1024, num_steps=32)
gru = GRUScratch(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLMScratch(gru, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1, num_gpus=1)
trainer.fit(model, data)

gru-c3-jax
data = d2l.TimeMachine(batch_size=1024, num_steps=32)
gru = GRUScratch(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLMScratch(gru, vocab_size=len(data.vocab), lr=4)
trainer = d2l.Trainer(max_epochs=50, gradient_clip_val=1, num_gpus=1)
trainer.fit(model, data)

gru-c3-mxnet

10.2.5 Concise Implementation

In high-level APIs, we can directly instantiate a GRU model. This encapsulates all the configuration detail that we made explicit above.

class GRU(d2l.RNN):
    def __init__(self, num_inputs, num_hiddens):
        d2l.Module.__init__(self)
        self.save_hyperparameters()
        self.rnn = nn.GRU(num_inputs, num_hiddens)
class GRU(d2l.RNN):
    def __init__(self, num_inputs, num_hiddens):
        d2l.Module.__init__(self)
        self.save_hyperparameters()
        self.rnn = tf.keras.layers.GRU(num_hiddens, return_sequences=True, 
                                       return_state=True)
class GRU(d2l.RNN):
    num_hiddens: int

    @nn.compact
    def __call__(self, inputs, H=None, training=False):
        if H is None:
            batch_size = inputs.shape[1]
            H = nn.GRUCell(features=self.num_hiddens).initialize_carry(
                jax.random.PRNGKey(0), (batch_size, self.num_hiddens))

        GRU = nn.scan(nn.GRUCell, variable_broadcast="params",
                      in_axes=0, out_axes=0, split_rngs={"params": False})

        H, outputs = GRU(features=self.num_hiddens)(H, inputs)
        return outputs, H
class GRU(d2l.RNN):
    def __init__(self, num_inputs, num_hiddens):
        d2l.Module.__init__(self)
        self.save_hyperparameters()
        self.rnn = rnn.GRU(num_hiddens)

The code is significantly faster in training as it uses compiled operators rather than Python.

gru = GRU(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLM(gru, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

gru-c5-pytorch
gru = GRU(num_inputs=len(data.vocab), num_hiddens=32)
with d2l.try_gpu():
    model = d2l.RNNLM(gru, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

gru-c5-tensorflow
gru = GRU(num_hiddens=32)
model = d2l.RNNLM(gru, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

gru-c5-jax
gru = GRU(num_inputs=len(data.vocab), num_hiddens=32)
model = d2l.RNNLM(gru, vocab_size=len(data.vocab), lr=4)
trainer.fit(model, data)

gru-c5-mxnet

After training, we print out the perplexity on the training set and the predicted sequence following the provided prefix.

model.predict('it has', 20, data.vocab, d2l.try_gpu())
'it has a cond the the pron'
model.predict('it has', 20, data.vocab)
'it has in the time travell'
model.predict('it has', 20, data.vocab, trainer.state.params)
'it has in and and and and '
model.predict('it has', 20, data.vocab, d2l.try_gpu())
'it has surd the time trave'

10.2.6 Summary

Compared with LSTMs, GRUs achieve similar performance but tend to be lighter computationally. Generally, compared with simple RNNs, gated RNNS, just like LSTMs and GRUs, can better capture dependencies for sequences with large time step distances. GRUs contain basic RNNs as their extreme case whenever the reset gate is switched on. They can also skip subsequences by turning on the update gate.

10.2.7 Exercises

  1. Assume that we only want to use the input at time step \(t'\) to predict the output at time step \(t > t'\). What are the best values for the reset and update gates for each time step?
  2. Adjust the hyperparameters and analyze their influence on running time, perplexity, and the output sequence.
  3. Compare runtime, perplexity, and the output strings for rnn.RNN and rnn.GRU implementations with each other.
  4. What happens if you implement only parts of a GRU, e.g., with only a reset gate or only an update gate?