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The results we are going to prove
hold regardless of the particular kind of cost function used (as long
as its continuous in the output) and
regardless of the particular algorithm which is employed to compute the gradient.
Here we shortly explain how gradients are computed by the standard
BPTT algorithm (e.g., [27], see also
crossreference Chapter 14 for more details) because its analytical
form is better suited to the forthcoming analyses.
The error at time is denoted by .
Considering only the error at time , output unit 's error signal is

and some non-output unit 's backpropagated error signal
at time
is

where

is unit 's current net input,

is the activation of
a non-input unit
with differentiable transfer function ,
and is the weight on the connection from unit to .
The corresponding contribution
to 's total weight update is
, where is the
learning rate, and stands for an arbitrary unit
connected to unit .

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Juergen Schmidhuber
2003-02-19