【转】Principles of training multi-layer neural network using backpropagation

Principles of training multi-layer neural network using backpropagation

The project describes teaching process of multi-layer neural network employing backpropagation algorithm. To illustrate this process the three layer neural network with two inputs and one output,which is shown in the picture below, is used:

Each neuron is composed of two units. First unit adds products of
weights coefficients and input signals. The second unit realise
nonlinear
function, called neuron activation function. Signal e is adder output signal, and y = f(e) is output signal of nonlinear
element. Signal y is also output signal of neuron.

To teach the neural network we need training data set. The training data set consists of input signals (x₁ and
x₂ ) assigned with corresponding target (desired output) z.
The network training is an iterative process. In each
iteration weights coefficients of nodes are modified using new data from
training data set. Modification is calculated using algorithm
described below:
Each teaching step starts with forcing both input signals from training
set. After this stage we can determine output signals values for
each neuron in each network layer. Pictures below illustrate how signal
is propagating through the network, Symbols w_(xm)n
represent weights of connections between network input x_m and neuron n in input layer. Symbols y_n
represents output signal of neuron n.

Propagation of signals through the hidden layer. Symbols w_mn represent weights of connections between output of neuron
m and input of neuron n in the next layer.

Propagation of signals through the output layer.

In the next algorithm step the output signal of the network y is compared with the desired output value (the target), which is found
in training data set. The difference is called error signal d of
output layer neuron.

It is impossible to compute error signal for internal neurons directly,
because output values of these neurons are unknown. For many years
the effective method for training multiplayer networks has been
unknown. Only in the middle eighties the backpropagation algorithm has
been
worked out. The idea is to propagate error signal d (computed in
single teaching step) back to all neurons, which output signals were input for discussed neuron.

The weights‘ coefficients w_mn used to propagate errors
back are equal to this used during computing output value. Only the
direction of data flow is changed (signals are propagated from output to
inputs one after the other). This technique is used for all network
layers. If propagated errors came from few neurons they are added. The
illustration is below:

When the error signal for each neuron is computed, the weights
coefficients of each neuron input node may be modified. In formulas
below
df(e)/de represents derivative of neuron activation function (which weights are modified).

Coefficient h
affects network teaching speed. There are a few
techniques to select this parameter. The first method is to start
teaching process with large value of the parameter. While weights
coefficients are being established the parameter is being decreased
gradually. The second, more complicated, method starts teaching with
small parameter value. During the teaching process the parameter is
being increased when the teaching is advanced and then decreased again
in
the final stage. Starting teaching process with low parameter value
enables to determine weights coefficients signs.

References
Ryszard Tadeusiewcz "Sieci neuronowe", Kraków 1992