We have already seen how to train linear
networks by gradient descent. In trying to do the same for multi-layer
networks we encounter a difficulty: we don't have any target values for
the hidden units. This seems to be an insurmountable problem - how could
we tell the hidden units just what to do? This unsolved question was in
fact the reason why neural networks fell out of favor after an initial
period of high popularity in the 1950s. It took 30 years before the
error backpropagation (or in short: backprop) algorithm
popularized a way to train hidden units, leading to a new wave of neural
network research and applications.
In principle, backprop provides a way to train networks with any number
of hidden units arranged in any number of layers. (There are clear
practical limits, which we will discuss later.) In fact, the network does
not have to be organized in layers - any pattern of connectivity that
permits a partial ordering of the nodes from input to output is
allowed. In other words, there must be a way to order the units such that
all connections go from "earlier" (closer to the input) to "later" ones
(closer to the output). This is equivalent to stating that their connection
pattern must not contain any cycles. Networks that respect this constraint
are called feedforward networks; their connection pattern forms a
directed acyclic graph or dag.
We want to train a multi-layer feedforward network by gradient descent to
approximate an unknown function, based on some training data consisting
of pairs (x,t). The vector x represents a pattern of input
to the network, and the vector t the corresponding target
(desired output). As we have seen before, the overall gradient with
respect to the entire training set is just the sum of the gradients for
each pattern; in what follows we will therefore describe how to compute
the gradient for just a single training pattern.
As before, we will number the units, and denote the weight from unit
j to unit i by wij.
the error signal for unit j:
the (negative) gradient for weight wij:
the set of nodes anterior to unit i:
the set of nodes posterior to unit j:
- The gradient.
As we did for linear networks before,
we expand the gradient into two factors by use of the chain rule:
The first factor is the error of unit i. The second is
Putting the two together, we get
To compute this gradient, we thus need to know the activity and the error
for all relevant nodes in the network.
- Forward activaction.
The activity of the input units
is determined by the network's external input x. For all other
units, the activity is propagated forward:
Note that before the activity of unit i can be calculated, the
activity of all its anterior nodes (forming the set Ai)
must be known. Since feedforward networks do not contain cycles,
there is an ordering of nodes from input to output that respects
- Calculating output error.
Assuming that we are using the sum-squared loss
the error for output unit o is simply
- Error backpropagation.
For hidden units, we must propagate the error back from the output nodes
(hence the name of the algorithm). Again using the chain rule, we can
expand the error of a hidden unit in terms of its posterior nodes:
Of the three factors inside the sum, the first is just the error of node
i. The second is
while the third is the derivative of node j's activation function:
For hidden units h that use the tanh activation function, we can make use
of the special identity
tanh(u)' = 1 - tanh(u)2, giving us
Putting all the pieces together we get
Note that in order to calculate the error for unit j, we must first
know the error of all its posterior nodes (forming the set Pj).
Again, as long as there are no cycles in the network, there is an ordering
of nodes from the output back to the input that respects this condition.
For example, we can simply use the reverse of the order in which activity
was propagated forward.
For layered feedforward networks that are fully connected
- that is, each node in a given layer connects to every node in
the next layer - it is often more convenient to write the backprop
algorithm in matrix notation rather than using more general graph
form given above. In this notation, the biases weights, net inputs,
activations, and error signals for all units in a layer are combined
into vectors, while all the non-bias weights from one layer to the
next form a matrix W. Layers are numbered from 0 (the input layer)
to L (the output layer). The backprop algorithm then looks as follows:
You can see that this notation is significantly more compact than the graph
form, even though it describes exactly the same sequence of operations.
- Initialize the input layer:
- Propagate activity forward: for l = 1, 2, ..., L,
where bl is the vector of bias weights.
- Calculate the error in the output layer:
- Backpropagate the error: for l = L-1, L-2, ..., 1,
where T is the matrix transposition operator.
- Update the weights and biases:
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