.. _chapter_plain_rnn:
Recurrent Neural Networks
=========================
In :numref:`chapter_language_model` we introduced :math:`n`-gram
models, where the conditional probability of word :math:`x_t` at
position :math:`t` only depends on the :math:`n-1` previous words. If we
want to check the possible effect of words earlier than :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:`|V|^n` numbers for a vocabulary :math:`V`. Hence, rather
than modeling :math:`p(x_t|x_{t-1}, \ldots x_{t-n+1})` it is preferable
to use a latent variable model in which we have
.. math:: p(x_t|x_{t-1}, \ldots x_1) \approx p(x_t|x_{t-1}, h_{t}).
Here :math:`h_t` is a *latent variable* that stores the sequence
information. A latent variable is also called as *hidden variable*,
*hidden state* or *hidden state variable*. The hidden state at time
:math:`t` could be computed based on both input :math:`x_{t-1}` and
hidden state :math:`h_{t-1}` at time :math:`t-1`, that is
.. math:: h_t = f(x_{t-1}, h_{t-1}).
For a sufficiently powerful function :math:`f`, the latent variable
model is not an approximation. After all, :math:`h_t` could simply store
all the data it observed so far. We discussed this in
:numref:`chapter_sequence`. But it could potentially makes both
computation and storage expensive.
Note that we also use :math:`h` to denote by the number of hidden units
of a hidden layer. 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.
Instead, they can only be computed by looking at data at previous
iterations. In this sense they have much in common with latent variable
models in statistics, such as clustering or topic models where the
clusters affect the output but cannot be directly observed.
Recurrent neural networks are neural networks with hidden states. Before
introducing this model, let’s first revisit the multi-layer perceptron
introduced in :numref:`chapter_mlp`.
Recurrent Networks Without Hidden States
----------------------------------------
Let take a look at a multilayer perceptron with a single hidden layer.
Given a mini-batch of instances
:math:`\mathbf{X} \in \mathbb{R}^{n \times d}` with sample size
:math:`n` and :math:`d` inputs. Let the hidden layer’s activation
function be :math:`\phi`. Hence 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
Here, we have the weight parameter
:math:`\mathbf{W}_{xh} \in \mathbb{R}^{d \times h}`, bias parameter
:math:`\mathbf{b}_h \in \mathbb{R}^{1 \times h}`, and the number of
hidden units :math:`h`, for the hidden layer.
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.
Here, :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 category.
This is entirely analogous to the regression problem we solved
previously in :numref:`chapter_sequence`, hence we omit details.
Suffice it to say that we can pick :math:`(x_t, x_{t-1})` pairs at
random and estimate the parameters :math:`\mathbf{W}` and
:math:`\mathbf{b}` of our network via autograd and stochastic gradient
descent.
Recurrent Networks with Hidden States
-------------------------------------
Matters are entirely different when we have hidden states. Let’s look at
the structure in some more detail. Remember that we often call iteration
:math:`t` as time :math:`t` in an optimization algorithm, time in a
recurrent neural network refers to steps within an iteration. Assume we
have :math:`\mathbf{X}_t \in \mathbb{R}^{n \times d}`,
:math:`t=1,\ldots,T`, in an iteration. And
:math:`\mathbf{H}_t \in \mathbb{R}^{n \times h}` is the hidden variable
of time step :math:`t` from the sequence. Unlike the multilayer
perceptron, 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).
Compared with :eq:`rnn_h_without_state`, we added one more
:math:`\mathbf{H}_{t-1} \mathbf{W}_{hh}` here. From the relationship
between hidden variables :math:`\mathbf{H}_t` and
:math:`\mathbf{H}_{t-1}` of adjacent time steps, we know that those
variables captured and retained the sequence’s historical information up
to the 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 the
equation above is recurrent, hence the name recurrent neural network
(RNN).
There are many different RNN construction methods. RNNs with a hidden
state defined by the equation above are very common. For time step
:math:`t`, the output of the output layer is similar to the computation
in the multilayer perceptron:
.. math:: \mathbf{O}_t = \mathbf{H}_t \mathbf{W}_{hq} + \mathbf{b}_q
RNN parameters include the weight
:math:`\mathbf{W}_{xh} \in \mathbb{R}^{d \times h}, \mathbf{W}_{hh} \in \mathbb{R}^{h \times h}`
of the hidden layer with the bias
:math:`\mathbf{b}_h \in \mathbb{R}^{1 \times h}`, and the weight
:math:`\mathbf{W}_{hq} \in \mathbb{R}^{h \times q}` of the output layer
with the bias :math:`\mathbf{b}_q \in \mathbb{R}^{1 \times q}`. It is
worth mentioning that RNNs always use these model parameters, even for
different time steps. Therefore, the number of RNN model parameters does
not grow as the number of time steps increases.
:numref:`fig_rnn` shows the computational logic of an RNN at three
adjacent time steps. In time step :math:`t`, the computation of the
hidden state can be treated as an entry of a fully connected layer with
the activation function :math:`\phi` after concatenating the input
:math:`\mathbf{X}_t` with the hidden state :math:`\mathbf{H}_{t-1}` of
the previous time step. The output of the fully connected layer is the
hidden state of the current time step :math:`\mathbf{H}_t`. Its model
parameter is the concatenation of :math:`\mathbf{W}_{xh}` and
:math:`\mathbf{W}_{hh}`, with a bias of :math:`\mathbf{b}_h`. 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`, the result of which will become the
input for the fully connected output layer of the current time step.
.. _fig_rnn:
.. figure:: ../img/rnn.svg
An RNN with a hidden state.
Steps in a Language Model
-------------------------
Now we illustrate how RNNs can be used to build a language model. For
simplicity of illustration we use words rather than characters, since
the former are easier to comprehend. Let the number of mini-batch
examples be 1, and the sequence of the text be the beginning of our
dataset, i.e. “the time machine by h. g. wells”. The figure below
illustrates how to estimate the next word based on the present and
previous words. 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 function to compute the error between the result and
the label. Due to the recurrent computation of the hidden state in the
hidden layer, the output of time step 3, :math:`\mathbf{O}_3`, is
determined by the text sequence “the”, “time”, “machine”. Since the next
word of the sequence in the training data is “by”, the loss of time step
3 will depend on the probability distribution of the next word generated
based on the sequence “the”, “time”, “machine” and the label “by” of
this time step.
.. figure:: ../img/rnn-train.svg
Word-level RNN language model. The input and label sequences are
``The Time Machine by H.`` and ``Time Machine by H. G.``
respectively.
In practice, each word is presented 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 before.
Perplexity
----------
Last, let’s discuss about how to measure the sequence model quality. One
way is to check how surprising the text is. A good language model is
able to predict with high accuracy 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 so accurately
reflect which word follows (``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 and borderline dysgrammatical extension. Nonetheless, at
least the model has learned how to spell words and some degree of
correlation between words. Lastly, example 3 indicates a poorly trained
model that doesn’t fit data.
We might measure the quality of the model by computing :math:`p(w)`,
i.e. 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 than long 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. If we want to compress text we can
ask about estimating the next symbol given the current set of symbols. A
lower bound on the number of bits is given by
:math:`-\log_2 p(x_t|x_{t-1}, \ldots x_1)`. A good language model should
allow us to predict the next word quite accurately and thus it should
allow us to spend very few bits on compressing the sequence. So we can
measure it by the average number of bits that we need to spend.
.. math:: \frac{1}{n} \sum_{t=1}^n -\log p(x_t|x_{t-1}, \ldots x_1)
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 rather than bitrate. In a nutshell
it is the exponential of the above:
.. math:: \mathrm{PPL} := \exp\left(-\frac{1}{n} \sum_{t=1}^n \log p(x_t|x_{t-1}, \ldots x_1)\right)
It can be best understood as the harmonic mean of the number of real
choices that we have when deciding which word to pick next. Note that
perplexity naturally generalizes the notion of the cross-entropy loss
defined when we introduced the softmax regression
(:numref:`chapter_softmax`). That is, for a single symbol both
definitions are identical bar the fact that one is the exponential of
the other. Let’s look at a number of cases:
- In the best case scenario, the model always estimates the probability
of the next symbol as :math:`1`. In this case the perplexity of the
model is :math:`1`.
- In the worst case scenario, the model always predicts the probability
of the label category as 0. In this situation, the perplexity is
infinite.
- At the baseline, the model predicts a uniform distribution over all
tokens. In this case the perplexity equals the size of the dictionary
``len(vocab)``. 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 model must satisfy.
Summary
-------
- A network that uses recurrent computation is called a recurrent
neural network (RNN).
- The hidden state of the 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 language models using a character-level RNN.
Exercises
---------
1. If we use an RNN to predict the next character in a text sequence,
how many output dimensions do we need?
2. Can you design a mapping for which an RNN with hidden states is
exact? Hint - what about a finite number of words?
3. What happens to the gradient if you backpropagate through a long
sequence?
4. What are some of the problems associated with the simple sequence
model described above?
Scan the QR Code to `Discuss `__
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|image0|
.. |image0| image:: ../img/qr_rnn.svg