.. _chapter_bptt:
Backpropagation Through Time
============================
So far we repeatedly alluded to things like *exploding gradients*,
*vanishing gradients*, *truncating backprop*, and the need to *detach
the computational graph*. For instance, in the previous section we
invoked ``s.detach()`` on the sequence. None of this was really fully
explained, in the interest of being able to build a model quickly and to
see how it works. In this section we will delve a bit more deeply into
the details of backpropagation for sequence models and why (and how) the
math works. For a more detailed discussion, e.g. about randomization and
backprop also see the paper by `Tallec and Ollivier,
2017 `__.
We encountered some of the effects of gradient explosion when we first
implemented recurrent neural networks (:numref:`chapter_rnn_scratch`).
In particular, if you solved the problems in the problem set, you would
have seen that gradient clipping is vital to ensure proper convergence.
To provide a better understanding of this issue, this section will
review how gradients are computed for sequences. Note that there is
nothing conceptually new in how it works. After all, we are still merely
applying the chain rule to compute gradients. Nonetheless it is worth
while reviewing backpropagation (:numref:`chapter_backprop`) for
another time.
Forward propagation in a recurrent neural network is relatively
straightforward. Back-propagation through time is actually a specific
application of back propagation in recurrent neural networks. It
requires us to expand the recurrent neural network one time step at a
time to obtain the dependencies between model variables and parameters.
Then, based on the chain rule, we apply back propagation to compute and
store gradients. Since sequences can be rather long this means that the
dependency can be rather lengthy. E.g. for a sequence of 1000 characters
the first symbol could potentially have significant influence on the
symbol at position 1000. This is not really computationally feasible (it
takes too long and requires too much memory) and it requires over 1000
matrix-vector products before we would arrive at that very elusive
gradient. This is a process fraught with computational and statistical
uncertainty. In the following we will address what happens and how to
address this in practice.
A Simplified Recurrent Network
------------------------------
We start with a simplified model of how an RNN works. This model ignores
details about the specifics of the hidden state and how it is being
updated. These details are immaterial to the analysis and would only
serve to clutter the notation and make it look more intimidating.
.. math:: h_t = f(x_t, h_{t-1}, w) \text{ and } o_t = g(h_t, w)
Here :math:`h_t` denotes the hidden state, :math:`x_t` the input and
:math:`o_t` the output. We have a chain of values
:math:`\{\ldots (h_{t-1}, x_{t-1}, o_{t-1}), (h_{t}, x_{t}, o_t), \ldots\}`
that depend on each other via recursive computation. The forward pass is
fairly straightforward. All we need is to loop through the
:math:`(x_t, h_t, o_t)` triples one step at a time. This is then
evaluated by an objective function measuring the discrepancy between
outputs :math:`o_t` and some desired target :math:`y_t`
.. math:: L(x,y, w) = \sum_{t=1}^T l(y_t, o_t).
For backpropagation matters are a bit more tricky. Let’s compute the
gradients with regard to the parameters :math:`w` of the objective
function :math:`L`. We get that
.. math::
\begin{aligned}
\partial_{w} L & = \sum_{t=1}^T \partial_w l(y_t, o_t) \\
& = \sum_{t=1}^T \partial_{o_t} l(y_t, o_t) \left[\partial_w g(h_t, w) + \partial_{h_t} g(h_t,w) \partial_w h_t\right]
\end{aligned}
The first part of the derivative is easy to compute (this is after all
the instantaneous loss gradient at time :math:`t`). The second part is
where things get tricky, since we need to compute the effect of the
parameters on :math:`h_t`.
If we have three sequences :math:`\{a_{t}\},\{b_{t}\},\{c_{t}\}`
satisfying :math:`a_{0}=0,a_{1}=b_{1}`, and
:math:`a_{t}=b_{t}+c_{t}a_{t-1}` for :math:`t=1,2,\ldots`, then it is
easy to show
.. math::
a_{t}=b_{t}+\sum_{i=1}^{t-1}\left(\prod_{j=i+1}^{t}c_{j}\right)b_{i}
for :math:`t\geq1`, where we assume that the summation is zero when
:math:`t=1`. Therefore, the following recursion
.. math::
\partial_{w}h_{t}=\partial_{w}f(x_{t},h_{t-1},w)+\partial_{h}f(x_{t},h_{t-1},w)\partial_{w}h_{t-1}
implies
.. math::
\partial_{w}h_{t}=\partial_{w}f(x_{t},h_{t-1},w)+\sum_{i=1}^{t-1}\left(\prod_{j=i+1}^{t}\partial_{h}f(x_{j},h_{j-1},w)\right)\partial_{w}f(x_{i},h_{i-1},w).
This chain can get *very* long whenever :math:`t` is large. While we can
use the chain rule to compute :math:`\partial_w h_t` recursively, this
might not be ideal. Let’s discuss a number of strategies for dealing
with this problem:
**Compute the full sum.** This is very slow and gradients can blow up,
since subtle changes in the initial conditions can potentially affect
the outcome a lot. That is, we could see things similar to the butterfly
effect where minimal changes in the initial conditions lead to
disproportionate changes in the outcome. This is actually quite
undesirable in terms of the model that we want to estimate. After all,
we are looking for robust estimators that generalize well. Hence this
strategy is almost never used in practice.
**Truncate the sum after :math:`\tau` steps.** This is what we’ve been
discussing so far. This leads to an *approximation* of the true
gradient, simply by terminating the sum above at
:math:`\partial_w h_{t-\tau}`. The approximation error is thus given by
:math:`\partial_h f(x_t, h_{t-1}, w) \partial_w h_{t-1}` (multiplied by
a product of gradients involving :math:`\partial_h f`). In practice this
works quite well. It is what is commonly referred to as truncated BPTT
(backpropgation through time). One of the consequences of this is that
the model focuses primarily on short-term influence rather than
long-term consequences. This is actually *desirable*, since it biases
the estimate towards simpler and more stable models.
**Randomized Truncation.** Lastly we can replace :math:`\partial_w h_t`
by a random variable which is correct in expectation but which truncates
the sequence. This is achieved by using a sequence of :math:`\xi_t`
where :math:`\mathbf{E}[\xi_t] = 1` and :math:`\Pr(\xi_t = 0) = 1-\pi`
and furthermore :math:`\Pr(\xi_t = \pi^{-1}) = \pi`. We use this to
replace the gradient:
.. math:: z_t = \partial_w f(x_t, h_{t-1}, w) + \xi_t \partial_h f(x_t, h_{t-1}, w) \partial_w h_{t-1}
It follows from the definition of :math:`\xi_t` that
:math:`\mathbf{E}[z_t] = \partial_w h_t`. Whenever :math:`\xi_t = 0` the
expansion terminates at that point. This leads to a weighted sum of
sequences of varying lengths where long sequences are rare but
appropriately overweighted. `Tallec and Ollivier,
2017 `__ proposed this in their paper.
Unfortunately, while appealing in theory, the model does not work much
better than simple truncation, most likely due to a number of factors.
Firstly, the effect of an observation after a number of backpropagation
steps into the past is quite sufficient to capture dependencies in
practice. Secondly, the increased variance counteracts the fact that the
gradient is more accurate. Thirdly, we actually *want* models that have
only a short range of interaction. Hence BPTT has a slight regularizing
effect which can be desirable.
.. figure:: ../img/truncated-bptt.svg
From top to bottom: randomized BPTT, regularly truncated BPTT and
full BPTT
The picture above illustrates the three cases when analyzing the first
few words of *The Time Machine*: randomized truncation partitions the
text into segments of varying length. Regular truncated BPTT breaks it
into sequences of the same length, and full BPTT leads to a
computationally infeasible expression.
The Computational Graph
-----------------------
In order to visualize the dependencies between model variables and
parameters during computation in a recurrent neural network, we can draw
a computational graph for the model, as shown below. For example, the
computation of the hidden states of time step 3 :math:`\mathbf{h}_3`
depends on the model parameters :math:`\mathbf{W}_{hx}` and
:math:`\mathbf{W}_{hh}`, the hidden state of the last time step
:math:`\mathbf{h}_2`, and the input of the current time step
:math:`\mathbf{x}_3`.
.. figure:: ../img/rnn-bptt.svg
Computational dependencies for a recurrent neural network model with
three time steps. Boxes represent variables (not shaded) or
parameters (shaded) and circles represent operators.
BPTT in Detail
--------------
Now that we discussed the general principle let’s discuss BPTT in
detail, distinguishing between different sets of weight matrices
(:math:`\mathbf{W}_{hx}, \mathbf{W}_{hh}` and :math:`\mathbf{W}_{oh}`)
in a simple linear latent variable model:
.. math::
\mathbf{h}_t = \mathbf{W}_{hx} \mathbf{x}_t + \mathbf{W}_{hh} \mathbf{h}_{t-1} \text{ and }
\mathbf{o}_t = \mathbf{W}_{oh} \mathbf{h}_t
Following the discussion in :numref:`chapter_backprop` we compute
gradients :math:`\partial L/\partial \mathbf{W}_{hx}`,
:math:`\partial L/\partial \mathbf{W}_{hh}`, and
:math:`\partial L/\partial \mathbf{W}_{oh}` for
:math:`L(\mathbf{x}, \mathbf{y}, \mathbf{W}) = \sum_{t=1}^T l(\mathbf{o}_t, y_t)`.
Taking the derivatives with respect to :math:`W_{oh}` is fairly
straightforward and we obtain
.. math::
\partial_{\mathbf{W}_{oh}} L = \sum_{t=1}^T \mathrm{prod}
\left(\partial_{\mathbf{o}_t} l(\mathbf{o}_t, y_t), \mathbf{h}_t\right)
The dependency on :math:`\mathbf{W}_{hx}` and :math:`\mathbf{W}_{hh}` is
a bit more tricky since it involves a chain of derivatives. We begin
with
.. math::
\begin{aligned}
\partial_{\mathbf{W}_{hh}} L & = \sum_{t=1}^T \mathrm{prod}
\left(\partial_{\mathbf{o}_t} l(\mathbf{o}_t, y_t), \mathbf{W}_{oh}, \partial_{\mathbf{W}_{hh}} \mathbf{h}_t\right) \\
\partial_{\mathbf{W}_{hx}} L & = \sum_{t=1}^T \mathrm{prod}
\left(\partial_{\mathbf{o}_t} l(\mathbf{o}_t, y_t), \mathbf{W}_{oh}, \partial_{\mathbf{W}_{hx}} \mathbf{h}_t\right)
\end{aligned}
After all, hidden states depend on each other and on past inputs. The
key quantity is how past hidden states affect future hidden states.
.. math::
\partial_{\mathbf{h}_t} \mathbf{h}_{t+1} = \mathbf{W}_{hh}^\top
\text{ and thus }
\partial_{\mathbf{h}_t} \mathbf{h}_T = \left(\mathbf{W}_{hh}^\top\right)^{T-t}
Chaining terms together yields
.. math::
\begin{aligned}
\partial_{\mathbf{W}_{hh}} \mathbf{h}_t & = \sum_{j=1}^t \left(\mathbf{W}_{hh}^\top\right)^{t-j} \mathbf{h}_j \\
\partial_{\mathbf{W}_{hx}} \mathbf{h}_t & = \sum_{j=1}^t \left(\mathbf{W}_{hh}^\top\right)^{t-j} \mathbf{x}_j.
\end{aligned}
A number of things follow from this potentially very intimidating
expression. Firstly, it pays to store intermediate results, i.e. powers
of :math:`\mathbf{W}_{hh}` as we work our way through the terms of the
loss function :math:`L`. Secondly, this simple *linear* example already
exhibits some key problems of long sequence models: it involves
potentially very large powers :math:`\mathbf{W}_{hh}^j`. In it,
eigenvalues smaller than :math:`1` vanish for large :math:`j` and
eigenvalues larger than :math:`1` diverge. This is numerically unstable
and gives undue importance to potentially irrelvant past detail. One way
to address this is to truncate the sum at a computationally convenient
size. Later on in this chapter we will see how more sophisticated
sequence models such as LSTMs can alleviate this further. In code, this
truncation is effected by *detaching* the gradient after a given number
of steps.
Summary
-------
- Back-propagation through time is merely an application of backprop to
sequence models with a hidden state.
- Truncation is needed for computational convenience and numerical
stability.
- High powers of matrices can lead top divergent and vanishing
eigenvalues. This manifests itself in the form of exploding or
vanishing gradients.
- For efficient computation intermediate values are cached.
Exercises
---------
1. Assume that we have a symmetric matrix
:math:`\mathbf{M} \in \mathbb{R}^{n \times n}` with eigenvalues
:math:`\lambda_i`. Without loss of generality assume that they are
ordered in ascending order :math:`\lambda_i \leq \lambda_{i+1}`. Show
that :math:`\mathbf{M}^k` has eigenvalues :math:`\lambda_i^k`.
2. Prove that for a random vector :math:`\mathbf{x} \in \mathbb{R}^n`
with high probability :math:`\mathbf{M}^k \mathbf{x}` will by very
much aligned with the largest eigenvector :math:`\mathbf{v}_n` of
:math:`\mathbf{M}`. Formalize this statement.
3. What does the above result mean for gradients in a recurrent neural
network?
4. Besides gradient clipping, can you think of any other methods to cope
with gradient explosion in recurrent neural networks?
Scan the QR Code to `Discuss `__
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.. |image0| image:: ../img/qr_bptt.svg