.. _sec_rnn:
Recurrent Neural Networks
=========================
In :numref:`sec_language_model` we introduced :math:`n`-gram models,
where the conditional probability of word :math:`x_t` at time step
:math:`t` only depends on the :math:`n-1` previous words. If we want to
incorporate the possible effect of words earlier than time step
:math:`t-(n-1)` on :math:`x_t`, we need to increase :math:`n`. However,
the number of model parameters would also increase exponentially with
it, as we need to store :math:`|\mathcal{V}|^n` numbers for a vocabulary
set :math:`\mathcal{V}`. Hence, rather than modeling
:math:`P(x_t \mid x_{t-1}, \ldots, x_{t-n+1})` it is preferable to use a
latent variable model:
.. math:: P(x_t \mid x_{t-1}, \ldots, x_1) \approx P(x_t \mid h_{t-1}),
where :math:`h_{t-1}` is a *hidden state* (also known as a hidden
variable) that stores the sequence information up to time step
:math:`t-1`. In general, the hidden state at any time step :math:`t`
could be computed based on both the current input :math:`x_{t}` and the
previous hidden state :math:`h_{t-1}`:
.. math:: h_t = f(x_{t}, h_{t-1}).
:label: eq_ht_xt
For a sufficiently powerful function :math:`f` in :eq:`eq_ht_xt`,
the latent variable model is not an approximation. After all,
:math:`h_t` may simply store all the data it has observed so far.
However, it could potentially make both computation and storage
expensive.
Recall that we have discussed hidden layers with hidden units in
:numref:`chap_perceptrons`. It is noteworthy that hidden layers and
hidden states refer to two very different concepts. Hidden layers are,
as explained, layers that are hidden from view on the path from input to
output. Hidden states are technically speaking *inputs* to whatever we
do at a given step, and they can only be computed by looking at data at
previous time steps.
*Recurrent neural networks* (RNNs) are neural networks with hidden
states. Before introducing the RNN model, we first revisit the MLP model
introduced in :numref:`sec_mlp`.
Neural Networks without Hidden States
-------------------------------------
Let us take a look at an MLP with a single hidden layer. Let the hidden
layer's activation function be :math:`\phi`. Given a minibatch of
examples :math:`\mathbf{X} \in \mathbb{R}^{n \times d}` with batch size
:math:`n` and :math:`d` inputs, the hidden layer's output
:math:`\mathbf{H} \in \mathbb{R}^{n \times h}` is calculated as
.. math:: \mathbf{H} = \phi(\mathbf{X} \mathbf{W}_{xh} + \mathbf{b}_h).
:label: rnn_h_without_state
In :eq:`rnn_h_without_state`, we have the weight parameter
:math:`\mathbf{W}_{xh} \in \mathbb{R}^{d \times h}`, the bias parameter
:math:`\mathbf{b}_h \in \mathbb{R}^{1 \times h}`, and the number of
hidden units :math:`h`, for the hidden layer. Thus, broadcasting (see
:numref:`subsec_broadcasting`) is applied during the summation. Next,
the hidden variable :math:`\mathbf{H}` is used as the input of the
output layer. The output layer is given by
.. math:: \mathbf{O} = \mathbf{H} \mathbf{W}_{hq} + \mathbf{b}_q,
where :math:`\mathbf{O} \in \mathbb{R}^{n \times q}` is the output
variable, :math:`\mathbf{W}_{hq} \in \mathbb{R}^{h \times q}` is the
weight parameter, and :math:`\mathbf{b}_q \in \mathbb{R}^{1 \times q}`
is the bias parameter of the output layer. If it is a classification
problem, we can use :math:`\text{softmax}(\mathbf{O})` to compute the
probability distribution of the output categories.
This is entirely analogous to the regression problem we solved
previously in :numref:`sec_sequence`, hence we omit details. Suffice
it to say that we can pick feature-label pairs at random and learn the
parameters of our network via automatic differentiation and stochastic
gradient descent.
.. _subsec_rnn_w_hidden_states:
Recurrent Neural Networks with Hidden States
--------------------------------------------
Matters are entirely different when we have hidden states. Let us look
at the structure in some more detail.
Assume that we have a minibatch of inputs
:math:`\mathbf{X}_t \in \mathbb{R}^{n \times d}` at time step :math:`t`.
In other words, for a minibatch of :math:`n` sequence examples, each row
of :math:`\mathbf{X}_t` corresponds to one example at time step
:math:`t` from the sequence. Next, denote by
:math:`\mathbf{H}_t \in \mathbb{R}^{n \times h}` the hidden variable of
time step :math:`t`. Unlike the MLP, here we save the hidden variable
:math:`\mathbf{H}_{t-1}` from the previous time step and introduce a new
weight parameter :math:`\mathbf{W}_{hh} \in \mathbb{R}^{h \times h}` to
describe how to use the hidden variable of the previous time step in the
current time step. Specifically, the calculation of the hidden variable
of the current time step is determined by the input of the current time
step together with the hidden variable of the previous time step:
.. math:: \mathbf{H}_t = \phi(\mathbf{X}_t \mathbf{W}_{xh} + \mathbf{H}_{t-1} \mathbf{W}_{hh} + \mathbf{b}_h).
:label: rnn_h_with_state
Compared with :eq:`rnn_h_without_state`,
:eq:`rnn_h_with_state` adds one more term
:math:`\mathbf{H}_{t-1} \mathbf{W}_{hh}` and thus instantiates
:eq:`eq_ht_xt`. From the relationship between hidden variables
:math:`\mathbf{H}_t` and :math:`\mathbf{H}_{t-1}` of adjacent time
steps, we know that these variables captured and retained the sequence's
historical information up to their current time step, just like the
state or memory of the neural network's current time step. Therefore,
such a hidden variable is called a *hidden state*. Since the hidden
state uses the same definition of the previous time step in the current
time step, the computation of :eq:`rnn_h_with_state` is
*recurrent*. Hence, neural networks with hidden states based on
recurrent computation are named *recurrent neural networks*. Layers that
perform the computation of :eq:`rnn_h_with_state` in RNNs are
called *recurrent layers*.
There are many different ways for constructing RNNs. RNNs with a hidden
state defined by :eq:`rnn_h_with_state` are very common. For time
step :math:`t`, the output of the output layer is similar to the
computation in the MLP:
.. math:: \mathbf{O}_t = \mathbf{H}_t \mathbf{W}_{hq} + \mathbf{b}_q.
Parameters of the RNN include the weights
:math:`\mathbf{W}_{xh} \in \mathbb{R}^{d \times h}, \mathbf{W}_{hh} \in \mathbb{R}^{h \times h}`,
and the bias :math:`\mathbf{b}_h \in \mathbb{R}^{1 \times h}` of the
hidden layer, together with the weights
:math:`\mathbf{W}_{hq} \in \mathbb{R}^{h \times q}` and the bias
:math:`\mathbf{b}_q \in \mathbb{R}^{1 \times q}` of the output layer. It
is worth mentioning that even at different time steps, RNNs always use
these model parameters. Therefore, the parameterization cost of an RNN
does not grow as the number of time steps increases.
:numref:`fig_rnn` illustrates the computational logic of an RNN at
three adjacent time steps. At any time step :math:`t`, the computation
of the hidden state can be treated as: (i) concatenating the input
:math:`\mathbf{X}_t` at the current time step :math:`t` and the hidden
state :math:`\mathbf{H}_{t-1}` at the previous time step :math:`t-1`;
(ii) feeding the concatenation result into a fully-connected layer with
the activation function :math:`\phi`. The output of such a
fully-connected layer is the hidden state :math:`\mathbf{H}_t` of the
current time step :math:`t`. In this case, the model parameters are the
concatenation of :math:`\mathbf{W}_{xh}` and :math:`\mathbf{W}_{hh}`,
and a bias of :math:`\mathbf{b}_h`, all from
:eq:`rnn_h_with_state`. The hidden state of the current time step
:math:`t`, :math:`\mathbf{H}_t`, will participate in computing the
hidden state :math:`\mathbf{H}_{t+1}` of the next time step :math:`t+1`.
What is more, :math:`\mathbf{H}_t` will also be fed into the
fully-connected output layer to compute the output :math:`\mathbf{O}_t`
of the current time step :math:`t`.
.. _fig_rnn:
.. figure:: ../img/rnn.svg
An RNN with a hidden state.
We just mentioned that the calculation of
:math:`\mathbf{X}_t \mathbf{W}_{xh} + \mathbf{H}_{t-1} \mathbf{W}_{hh}`
for the hidden state is equivalent to matrix multiplication of
concatenation of :math:`\mathbf{X}_t` and :math:`\mathbf{H}_{t-1}` and
concatenation of :math:`\mathbf{W}_{xh}` and :math:`\mathbf{W}_{hh}`.
Though this can be proven in mathematics, in the following we just use a
simple code snippet to show this. To begin with, we define matrices
``X``, ``W_xh``, ``H``, and ``W_hh``, whose shapes are (3, 1), (1, 4),
(3, 4), and (4, 4), respectively. Multiplying ``X`` by ``W_xh``, and
``H`` by ``W_hh``, respectively, and then adding these two
multiplications, we obtain a matrix of shape (3, 4).
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
from mxnet import np, npx
from d2l import mxnet as d2l
npx.set_np()
X, W_xh = np.random.normal(0, 1, (3, 1)), np.random.normal(0, 1, (1, 4))
H, W_hh = np.random.normal(0, 1, (3, 4)), np.random.normal(0, 1, (4, 4))
np.dot(X, W_xh) + np.dot(H, W_hh)
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
array([[-0.21952915, 4.256434 , 4.5812645 , -5.344988 ],
[ 3.447858 , -3.0177274 , -1.6777471 , 7.535347 ],
[ 2.2390068 , 1.4199957 , 4.744728 , -8.421293 ]])
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
import torch
from d2l import torch as d2l
X, W_xh = torch.normal(0, 1, (3, 1)), torch.normal(0, 1, (1, 4))
H, W_hh = torch.normal(0, 1, (3, 4)), torch.normal(0, 1, (4, 4))
torch.matmul(X, W_xh) + torch.matmul(H, W_hh)
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tensor([[-0.5842, -0.3556, 0.6926, 1.0900],
[ 0.9317, 1.3875, -1.3222, 0.0321],
[-2.0621, -1.8653, 4.4808, -2.4656]])
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
import tensorflow as tf
from d2l import tensorflow as d2l
X, W_xh = tf.random.normal((3, 1), 0, 1), tf.random.normal((1, 4), 0, 1)
H, W_hh = tf.random.normal((3, 4), 0, 1), tf.random.normal((4, 4), 0, 1)
tf.matmul(X, W_xh) + tf.matmul(H, W_hh)
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
.. raw:: html
.. raw:: html
Now we concatenate the matrices ``X`` and ``H`` along columns (axis 1),
and the matrices ``W_xh`` and ``W_hh`` along rows (axis 0). These two
concatenations result in matrices of shape (3, 5) and of shape (5, 4),
respectively. Multiplying these two concatenated matrices, we obtain the
same output matrix of shape (3, 4) as above.
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
np.dot(np.concatenate((X, H), 1), np.concatenate((W_xh, W_hh), 0))
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
array([[-0.21952918, 4.256434 , 4.5812645 , -5.344988 ],
[ 3.4478583 , -3.0177271 , -1.677747 , 7.535347 ],
[ 2.2390068 , 1.4199957 , 4.744728 , -8.421294 ]])
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
torch.matmul(torch.cat((X, H), 1), torch.cat((W_xh, W_hh), 0))
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
tensor([[-0.5842, -0.3556, 0.6926, 1.0900],
[ 0.9317, 1.3875, -1.3222, 0.0321],
[-2.0621, -1.8653, 4.4808, -2.4656]])
.. raw:: html
.. raw:: html
.. raw:: latex
\diilbookstyleinputcell
.. code:: python
tf.matmul(tf.concat((X, H), 1), tf.concat((W_xh, W_hh), 0))
.. raw:: latex
\diilbookstyleoutputcell
.. parsed-literal::
:class: output
.. raw:: html
.. raw:: html
RNN-based Character-Level Language Models
-----------------------------------------
Recall that for language modeling in :numref:`sec_language_model`, we
aim to predict the next token based on the current and past tokens, thus
we shift the original sequence by one token as the labels. Bengio et al.
first proposed to use a neural network for language modeling
:cite:`Bengio.Ducharme.Vincent.ea.2003`. In the following we
illustrate how RNNs can be used to build a language model. Let the
minibatch size be one, and the sequence of the text be "machine". To
simplify training in subsequent sections, we tokenize text into
characters rather than words and consider a *character-level language
model*. :numref:`fig_rnn_train` demonstrates how to predict the next
character based on the current and previous characters via an RNN for
character-level language modeling.
.. _fig_rnn_train:
.. figure:: ../img/rnn-train.svg
A character-level language model based on the RNN. The input and
label sequences are "machin" and "achine", respectively.
During the training process, we run a softmax operation on the output
from the output layer for each time step, and then use the cross-entropy
loss to compute the error between the model output and the label. Due to
the recurrent computation of the hidden state in the hidden layer, the
output of time step 3 in :numref:`fig_rnn_train`,
:math:`\mathbf{O}_3`, is determined by the text sequence "m", "a", and
"c". Since the next character of the sequence in the training data is
"h", the loss of time step 3 will depend on the probability distribution
of the next character generated based on the feature sequence "m", "a",
"c" and the label "h" of this time step.
In practice, each token is represented by a :math:`d`-dimensional
vector, and we use a batch size :math:`n>1`. Therefore, the input
:math:`\mathbf X_t` at time step :math:`t` will be a :math:`n\times d`
matrix, which is identical to what we discussed in
:numref:`subsec_rnn_w_hidden_states`.
.. _subsec_perplexity:
Perplexity
----------
Last, let us discuss about how to measure the language model quality,
which will be used to evaluate our RNN-based models in the subsequent
sections. One way is to check how surprising the text is. A good
language model is able to predict with high-accuracy tokens that what we
will see next. Consider the following continuations of the phrase "It is
raining", as proposed by different language models:
1. "It is raining outside"
2. "It is raining banana tree"
3. "It is raining piouw;kcj pwepoiut"
In terms of quality, example 1 is clearly the best. The words are
sensible and logically coherent. While it might not quite accurately
reflect which word follows semantically ("in San Francisco" and "in
winter" would have been perfectly reasonable extensions), the model is
able to capture which kind of word follows. Example 2 is considerably
worse by producing a nonsensical extension. Nonetheless, at least the
model has learned how to spell words and some degree of correlation
between words. Last, example 3 indicates a poorly trained model that
does not fit data properly.
We might measure the quality of the model by computing the likelihood of
the sequence. Unfortunately this is a number that is hard to understand
and difficult to compare. After all, shorter sequences are much more
likely to occur than the longer ones, hence evaluating the model on
Tolstoy's magnum opus *War and Peace* will inevitably produce a much
smaller likelihood than, say, on Saint-Exupery's novella *The Little
Prince*. What is missing is the equivalent of an average.
Information theory comes handy here. We have defined entropy, surprisal,
and cross-entropy when we introduced the softmax regression
(:numref:`subsec_info_theory_basics`) and more of information theory
is discussed in the `online appendix on information
theory `__.
If we want to compress text, we can ask about predicting the next token
given the current set of tokens. A better language model should allow us
to predict the next token more accurately. Thus, it should allow us to
spend fewer bits in compressing the sequence. So we can measure it by
the cross-entropy loss averaged over all the :math:`n` tokens of a
sequence:
.. math:: \frac{1}{n} \sum_{t=1}^n -\log P(x_t \mid x_{t-1}, \ldots, x_1),
:label: eq_avg_ce_for_lm
where :math:`P` is given by a language model and :math:`x_t` is the
actual token observed at time step :math:`t` from the sequence. This
makes the performance on documents of different lengths comparable. For
historical reasons, scientists in natural language processing prefer to
use a quantity called *perplexity*. In a nutshell, it is the exponential
of :eq:`eq_avg_ce_for_lm`:
.. math:: \exp\left(-\frac{1}{n} \sum_{t=1}^n \log P(x_t \mid x_{t-1}, \ldots, x_1)\right).
Perplexity can be best understood as the harmonic mean of the number of
real choices that we have when deciding which token to pick next. Let us
look at a number of cases:
- In the best case scenario, the model always perfectly estimates the
probability of the label token as 1. In this case the perplexity of
the model is 1.
- In the worst case scenario, the model always predicts the probability
of the label token as 0. In this situation, the perplexity is
positive infinity.
- At the baseline, the model predicts a uniform distribution over all
the available tokens of the vocabulary. In this case, the perplexity
equals the number of unique tokens of the vocabulary. In fact, if we
were to store the sequence without any compression, this would be the
best we could do to encode it. Hence, this provides a nontrivial
upper bound that any useful model must beat.
In the following sections, we will implement RNNs for character-level
language models and use perplexity to evaluate such models.
Summary
-------
- A neural network that uses recurrent computation for hidden states is
called a recurrent neural network (RNN).
- The hidden state of an RNN can capture historical information of the
sequence up to the current time step.
- The number of RNN model parameters does not grow as the number of
time steps increases.
- We can create character-level language models using an RNN.
- We can use perplexity to evaluate the quality of language models.
Exercises
---------
1. If we use an RNN to predict the next character in a text sequence,
what is the required dimension for any output?
2. Why can RNNs express the conditional probability of a token at some
time step based on all the previous tokens in the text sequence?
3. What happens to the gradient if you backpropagate through a long
sequence?
4. What are some of the problems associated with the language model
described in this section?
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`Discussions `__
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