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Bayesian regularization backpropagation


net.trainFcn = 'trainbr' sets the network trainFcn property.


[trainedNet,tr] = train(net,...) trains the network with trainbr.

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.

Training occurs according to trainbr training parameters, shown here with their default values:

  • net.trainParam.epochs — Maximum number of epochs to train. The default value is 1000.

  • net.trainParam.goal — Performance goal. The default value is 0.

  • — Marquardt adjustment parameter. The default value is 0.005.

  • net.trainParam.mu_dec — Decrease factor for mu. The default value is 0.1.

  • net.trainParam.mu_inc — Increase factor for mu. The default value is 10.

  • net.trainParam.mu_max — Maximum value for mu. The default value is 1e10.

  • net.trainParam.max_fail — Maximum validation failures. The default value is 0.

  • net.trainParam.min_grad — Minimum performance gradient. The default value is 1e-7.

  • — Epochs between displays (NaN for no displays). The default value is 25.

  • net.trainParam.showCommandLine — Generate command-line output. The default value is false.

  • net.trainParam.showWindow — Show training GUI. The default value is true.

  • net.trainParam.time — Maximum time to train in seconds. The default value is inf.

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.


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This example shows how to solve a problem consisting of inputs p and targets t by using 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)

Input Arguments

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Input network, specified as a network object. To create a network object, use for example, feedforwardnet or narxnet.

Output Arguments

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Trained network, returned as a network object.

Training record (epoch and perf), returned as a structure whose fields depend on the network training function (net.NET.trainFcn). It can include fields such as:

  • Training, data division, and performance functions and parameters

  • Data division indices for training, validation and test sets

  • Data division masks for training validation and test sets

  • Number of epochs (num_epochs) and the best epoch (best_epoch).

  • A list of training state names (states).

  • Fields for each state name recording its value throughout training

  • Performances of the best network (best_perf, best_vperf, best_tperf)


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 mse or sse performance function.

More About

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Network Use

You can create a standard network that uses trainbr with feedforwardnet or cascadeforwardnet. To prepare a custom network to be trained with trainbr,

  1. Set NET.trainFcn to 'trainbr'. This sets NET.trainParam to trainbr’s default parameters.

  2. Set NET.trainParam properties to desired values.

In either case, calling train with the resulting network trains the network with trainbr. See feedforwardnet and cascadeforwardnet for examples.


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 X. Each variable is adjusted according to Levenberg-Marquardt,

jj = jX * jX
je = jX * E
dX = -(jj+I*mu) \ je

where E is all errors and I is the identity matrix.

The adaptive value mu is increased by mu_inc until the change shown above results in a reduced performance value. The change is then made to the network, and mu is decreased by mu_dec.

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 goal.

  • The performance gradient falls below min_grad.

  • mu exceeds mu_max.


[1] MacKay, David J. C. "Bayesian interpolation." Neural computation. Vol. 4, No. 3, 1992, pp. 415–447.

[2] Foresee, F. Dan, and Martin T. Hagan. "Gauss-Newton approximation to Bayesian learning." Proceedings of the International Joint Conference on Neural Networks, June, 1997.

Version History

Introduced before R2006a