Bayesian regularization backpropagation
net.trainFcn = 'trainbr'
[net,tr] = train(net,...)
trainbr is a network training function that updates the weight and bias
values according to Levenberg-Marquardt optimization. It minimizes a combination of squared
errors and weights, and then determines the correct combination so as to produce a network that
generalizes well. The process is called Bayesian regularization.
net.trainFcn = 'trainbr' sets the network
[net,tr] = train(net,...) trains the network with
Training occurs according to
trainbr training parameters, shown here
with their default values:
Maximum number of epochs to train
Marquardt adjustment parameter
Decrease factor for
Increase factor for
Maximum value for
Maximum validation failures
Minimum performance gradient
Epochs between displays (
Generate command-line output
Show training GUI
Maximum time to train in seconds
Validation stops are disabled by default (
max_fail = 0) so that training
can continue until an optimal combination of errors and weights is found. However, some
weight/bias minimization can still be achieved with shorter training times if validation is
enabled by setting
max_fail to 6 or some other strictly positive value.
You can create a standard network that uses
cascadeforwardnet. To prepare a custom
network to be trained with
NET.trainParam properties to desired
In either case, calling
train with the resulting network trains the
cascadeforwardnet for examples.
Here is a problem consisting of inputs
p and targets
t to be solved with a network. It involves fitting a noisy sine wave.
p = [-1:.05:1]; t = sin(2*pi*p)+0.1*randn(size(p));
A feed-forward network is created with a hidden layer of 2 neurons.
net = feedforwardnet(2,'trainbr');
Here the network is trained and tested.
net = train(net,p,t); a = net(p)
This function uses the Jacobian for calculations, which assumes that performance is a mean
or sum of squared errors. Therefore networks trained with this function must use either the
sse performance function.
trainbr can train any network as long as its weight, net input, and
transfer functions have derivative functions.
Bayesian regularization minimizes a linear combination of squared errors and weights. It also modifies the linear combination so that at the end of training the resulting network has good generalization qualities. See MacKay (Neural Computation, Vol. 4, No. 3, 1992, pp. 415 to 447) and Foresee and Hagan (Proceedings of the International Joint Conference on Neural Networks, June, 1997) for more detailed discussions of Bayesian regularization.
This Bayesian regularization takes place within the Levenberg-Marquardt algorithm.
Backpropagation is used to calculate the Jacobian
jX of performance
perf with respect to the weight and bias variables
Each variable is adjusted according to Levenberg-Marquardt,
jj = jX * jX je = jX * E dX = -(jj+I*mu) \ je
E is all errors and
I is the identity
The adaptive value
mu is increased by
the change shown above results in a reduced performance value. The change is then made to the
mu is decreased by
Training stops when any of these conditions occurs:
The maximum number of
epochs (repetitions) is reached.
The maximum amount of
time is exceeded.
Performance is minimized to the
The performance gradient falls below
 MacKay, David J. C. "Bayesian interpolation." Neural computation. Vol. 4, No. 3, 1992, pp. 415–447.
 Foresee, F. Dan, and Martin T. Hagan. "Gauss-Newton approximation to Bayesian learning." Proceedings of the International Joint Conference on Neural Networks, June, 1997.