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.. _sec_backprop:
Forward Propagation, Backward Propagation, and Computational Graphs
===================================================================
So far, we have trained our models with minibatch stochastic gradient
descent. However, when we implemented the algorithm, we only worried
about the calculations involved in *forward propagation* through the
model. When it came time to calculate the gradients, we just invoked the
``backward`` function, relying on the ``GradientCollector`` module to
know what to do.
The automatic calculation of gradients profoundly simplifies the
implementation of deep learning algorithms. Before automatic
differentiation, even small changes to complicated models required
recalculating complicated derivatives by hand. Surprisingly often,
academic papers had to allocate numerous pages to deriving update rules.
While we must continue to rely on ``GradientCollector`` so we can focus
on the interesting parts, you ought to *know* how these gradients are
calculated under the hood if you want to go beyond a shallow
understanding of deep learning.
In this section, we take a deep dive into the details of backward
propagation (more commonly called *backpropagation* or *backprop*). To
convey some insight for both the techniques and their implementations,
we rely on some basic mathematics and computational graphs. To start, we
focus our exposition on a three layer (one hidden) multilayer perceptron
with weight decay (:math:`\ell_2` regularization).
Forward Propagation
-------------------
Forward propagation refers to the calculation and storage of
intermediate variables (including outputs) for the neural network in
order from the input layer to the output layer. We now work step-by-step
through the mechanics of a deep network with one hidden layer. This may
seem tedious but in the eternal words of funk virtuoso James Brown, you
must "pay the cost to be the boss".
For the sake of simplicity, letâ€™s assume that the input example is
:math:`\mathbf{x}\in \mathbb{R}^d` and that our hidden layer does not
include a bias term. Here the intermediate variable is:
.. math:: \mathbf{z}= \mathbf{W}^{(1)} \mathbf{x},
where :math:`\mathbf{W}^{(1)} \in \mathbb{R}^{h \times d}` is the weight
parameter of the hidden layer. After running the intermediate variable
:math:`\mathbf{z}\in \mathbb{R}^h` through the activation function
:math:`\phi` we obtain our hidden activation vector of length :math:`h`,
.. math:: \mathbf{h}= \phi (\mathbf{z}).
The hidden variable :math:`\mathbf{h}` is also an intermediate variable.
Assuming the parameters of the output layer only possess a weight of
:math:`\mathbf{W}^{(2)} \in \mathbb{R}^{q \times h}`, we can obtain an
output layer variable with a vector of length :math:`q`:
.. math:: \mathbf{o}= \mathbf{W}^{(2)} \mathbf{h}.
Assuming the loss function is :math:`l` and the example label is
:math:`y`, we can then calculate the loss term for a single data
example,
.. math:: L = l(\mathbf{o}, y).
According to the definition of :math:`\ell_2` regularization, given the
hyperparameter :math:`\lambda`, the regularization term is
.. math:: s = \frac{\lambda}{2} \left(\|\mathbf{W}^{(1)}\|_F^2 + \|\mathbf{W}^{(2)}\|_F^2\right),
where the Frobenius norm of the matrix is simply the :math:`L_2` norm
applied after flattening the matrix into a vector. Finally, the model's
regularized loss on a given data example is:
.. math:: J = L + s.
We refer to :math:`J` the *objective function* in the following
discussion.
Computational Graph of Forward Propagation
------------------------------------------
Plotting computational graphs helps us visualize the dependencies of
operators and variables within the calculation. :numref:`fig_forward`
contains the graph associated with the simple network described above.
The lower-left corner signifies the input and the upper right corner the
output. Notice that the direction of the arrows (which illustrate data
flow) are primarily rightward and upward.
.. _fig_forward:
.. figure:: https://raw.githubusercontent.com/d2l-ai/d2l-en/master/img/forward.svg
Computational Graph
Backpropagation
---------------
Backpropagation refers to the method of calculating the gradient of
neural network parameters. In short, the method traverses the network in
reverse order, from the output to the input layer, according to the
*chain rule* from calculus. The algorithm stores any intermediate
variables (partial derivatives) requried while calculating the gradient
with respect to some parameters. Assume that we have functions
:math:`\mathsf{Y}=f(\mathsf{X})` and
:math:`\mathsf{Z}=g(\mathsf{Y}) = g \circ f(\mathsf{X})`, in which the
input and the output :math:`\mathsf{X}, \mathsf{Y}, \mathsf{Z}` are
tensors of arbitrary shapes. By using the chain rule, we can compute the
derivative of :math:`\mathsf{Z}` wrt. :math:`\mathsf{X}` via
.. math:: \frac{\partial \mathsf{Z}}{\partial \mathsf{X}} = \text{prod}\left(\frac{\partial \mathsf{Z}}{\partial \mathsf{Y}}, \frac{\partial \mathsf{Y}}{\partial \mathsf{X}}\right).
Here we use the :math:`\text{prod}` operator to multiply its arguments
after the necessary operations, such as transposition and swapping input
positions, have been carried out. For vectors, this is straightforward:
it is simply matrix-matrix multiplication. For higher dimensional
tensors, we use the appropriate counterpart. The operator
:math:`\text{prod}` hides all the notation overhead.
The parameters of the simple network with one hidden layer are
:math:`\mathbf{W}^{(1)}` and :math:`\mathbf{W}^{(2)}`. The objective of
backpropagation is to calculate the gradients
:math:`\partial J/\partial \mathbf{W}^{(1)}` and
:math:`\partial J/\partial \mathbf{W}^{(2)}`. To accomplish this, we
apply the chain rule and calculate, in turn, the gradient of each
intermediate variable and parameter. The order of calculations are
reversed relative to those performed in forward propagation, since we
need to start with the outcome of the compute graph and work our way
towards the parameters. The first step is to calculate the gradients of
the objective function :math:`J=L+s` with respect to the loss term
:math:`L` and the regularization term :math:`s`.
.. math:: \frac{\partial J}{\partial L} = 1 \; \text{and} \; \frac{\partial J}{\partial s} = 1.
Next, we compute the gradient of the objective function with respect to
variable of the output layer :math:`\mathbf{o}` according to the chain
rule.
.. math::
\frac{\partial J}{\partial \mathbf{o}}
= \text{prod}\left(\frac{\partial J}{\partial L}, \frac{\partial L}{\partial \mathbf{o}}\right)
= \frac{\partial L}{\partial \mathbf{o}}
\in \mathbb{R}^q.
Next, we calculate the gradients of the regularization term with respect
to both parameters.
.. math::
\frac{\partial s}{\partial \mathbf{W}^{(1)}} = \lambda \mathbf{W}^{(1)}
\; \text{and} \;
\frac{\partial s}{\partial \mathbf{W}^{(2)}} = \lambda \mathbf{W}^{(2)}.
Now we are able calculate the gradient
:math:`\partial J/\partial \mathbf{W}^{(2)} \in \mathbb{R}^{q \times h}`
of the model parameters closest to the output layer. Using the chain
rule yields:
.. math::
\frac{\partial J}{\partial \mathbf{W}^{(2)}}
= \text{prod}\left(\frac{\partial J}{\partial \mathbf{o}}, \frac{\partial \mathbf{o}}{\partial \mathbf{W}^{(2)}}\right) + \text{prod}\left(\frac{\partial J}{\partial s}, \frac{\partial s}{\partial \mathbf{W}^{(2)}}\right)
= \frac{\partial J}{\partial \mathbf{o}} \mathbf{h}^\top + \lambda \mathbf{W}^{(2)}.
To obtain the gradient with respect to :math:`\mathbf{W}^{(1)}` we need
to continue backpropagation along the output layer to the hidden layer.
The gradient with respect to the hidden layer's outputs
:math:`\partial J/\partial \mathbf{h} \in \mathbb{R}^h` is given by
.. math::
\frac{\partial J}{\partial \mathbf{h}}
= \text{prod}\left(\frac{\partial J}{\partial \mathbf{o}}, \frac{\partial \mathbf{o}}{\partial \mathbf{h}}\right)
= {\mathbf{W}^{(2)}}^\top \frac{\partial J}{\partial \mathbf{o}}.
Since the activation function :math:`\phi` applies elementwise,
calculating the gradient
:math:`\partial J/\partial \mathbf{z} \in \mathbb{R}^h` of the
intermediate variable :math:`\mathbf{z}` requires that we use the
elementwise multiplication operator, which we denote by :math:`\odot`.
.. math::
\frac{\partial J}{\partial \mathbf{z}}
= \text{prod}\left(\frac{\partial J}{\partial \mathbf{h}}, \frac{\partial \mathbf{h}}{\partial \mathbf{z}}\right)
= \frac{\partial J}{\partial \mathbf{h}} \odot \phi'\left(\mathbf{z}\right).
Finally, we can obtain the gradient
:math:`\partial J/\partial \mathbf{W}^{(1)} \in \mathbb{R}^{h \times d}`
of the model parameters closest to the input layer. According to the
chain rule, we get
.. math::
\frac{\partial J}{\partial \mathbf{W}^{(1)}}
= \text{prod}\left(\frac{\partial J}{\partial \mathbf{z}}, \frac{\partial \mathbf{z}}{\partial \mathbf{W}^{(1)}}\right) + \text{prod}\left(\frac{\partial J}{\partial s}, \frac{\partial s}{\partial \mathbf{W}^{(1)}}\right)
= \frac{\partial J}{\partial \mathbf{z}} \mathbf{x}^\top + \lambda \mathbf{W}^{(1)}.
Training a Model
----------------
When training networks, forward and backward propagation depend on each
other. In particular, for forward propagation, we traverse the compute
graph in the direction of dependencies and compute all the variables on
its path. These are then used for backpropagation where the compute
order on the graph is reversed. One of the consequences is that we need
to retain the intermediate values until backpropagation is complete.
This is also one of the reasons why backpropagation requires
significantly more memory than plain prediction. We compute tensors as
gradients and need to retain all the intermediate variables to invoke
the chain rule. Another reason is that we typically train with
minibatches containing more than one variable, thus more intermediate
activations need to be stored.
Summary
-------
- Forward propagation sequentially calculates and stores intermediate
variables within the compute graph defined by the neural network. It
proceeds from input to output layer.
- Back propagation sequentially calculates and stores the gradients of
intermediate variables and parameters within the neural network in
the reversed order.
- When training deep learning models, forward propagation and back
propagation are interdependent.
- Training requires significantly more memory and storage.
Exercises
---------
1. Assume that the inputs :math:`\mathbf{x}` to some scalar function
:math:`f` are :math:`n \times m` matrices. What is the dimensionality
of the gradient of :math:`f` with respect to :math:`\mathbf{x}`?
2. Add a bias to the hidden layer of the model described in this
section.
- Draw the corresponding compute graph.
- Derive the forward and backward propagation equations.
3. Compute the memory footprint for training and inference in model
described in the current chapter.
4. Assume that you want to compute *second* derivatives. What happens to
the compute graph? How long do you expect the calculation to take?
5. Assume that the compute graph is too large for your GPU.
- Can you partition it over more than one GPU?
- What are the advantages and disadvantages over training on a
smaller minibatch?