trainbr (Neural Network Toolbox)

Neural Network Toolbox

trainbr

Bayesian regularization backpropagation

Syntax

[net,TR,Ac,El] = trainbr(net,Pd,Tl,Ai,Q,TS,VV,TV)

info = trainbr(code)

Description

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.

trainbr(net,Pd,Tl,Ai,Q,TS,VV,TV) takes these inputs,

net -- Neural network

Pd -- Delayed input vectors

Tl -- Layer target vectors

Ai -- Initial input delay conditions

Q -- Batch size

TS -- Time steps

VV -- Either empty matrix [] or structure of validation vectors

TV -- Either empty matrix [] or structure of test vectors

and returns,

net -- Trained network

TR -- Training record of various values over each epoch:
- TR.epoch -- Epoch number
  TR.perf -- Training performance
  TR.vperf -- Validation performance
  TR.tperf -- Test performance
  TR.mu -- Adaptive mu value
Ac -- Collective layer outputs for last epoch.

El -- Layer errors for last epoch

Training occurs according to the trainlm's training parameters, shown here with their default values:

net.trainParam.epochs 100 Maximum number of epochs to train

net.trainParam.goal 0 Performance goal

net.trainParam.mu 0.005 Marquardt adjustment parameter

net.trainParam.mu_dec 0.1 Decrease factor for mu

net.trainParam.mu_inc 10 Increase factor for mu

net.trainParam.mu_max 1e-10 Maximum value for mu

net.trainParam.max_fail 5 Maximum validation failures

net.trainParam.mem_reduc 1

Factor to use for memory/speed trade-off

net.trainParam.min_grad 1e-10 Minimum performance gradient

net.trainParam.show 25 Epochs between showing progress

net.trainParam.time inf Maximum time to train in seconds

Dimensions for these variables are:

Pd -- No x Ni x TS cell array, each element P{i,j,ts} is a Dij x Q matrix

Tl -- Nl x TS cell array, each element P{i,ts} is a Vi x Q matrix

Ai -- Nl x LD cell array, each element Ai{i,k} is an Si x Q matrix

where

Ni = net.numInputs

Nl = net.numLayers

LD = net.numLayerDelays

Ri = net.inputs{i}.size

Si = net.layers{i}.size

Vi = net.targets{i}.size

Dij = Ri * length(net.inputWeights{i,j}.delays)

If VV is not [], it must be a structure of validation vectors,

VV.PD -- Validation delayed inputs

VV.Tl -- Validation layer targets

VV.Ai -- Validation initial input conditions

VV.Q -- Validation batch size

VV.TS -- Validation time steps

which is normally used to stop training early if the network performance on the validation vectors fails to improve or remains the same for max_fail epochs in a row.

If TV is not [], it must be a structure of validation vectors,

    TV.PD -- Validation delayed inputs

    TV.Tl -- Validation layer targets

    TV.Ai -- Validation initial input conditions

   TV.Q -- Validation batch size

   TV.TS -- Validation time steps

which is used to test the generalization capability of the trained network.

trainbr(code) returns useful information for each code string:

'pnames' -- Names of training parameters

'pdefaults' -- Default training parameters

Examples

Here is a problem consisting of inputs p and targets t that we would like to solve with a network. It involves fitting a noisy sine wave.

p = [-1:.05:1];
t = sin(2*pi*p)+0.1*randn(size(p));

Here a two-layer feed-forward network is created. The network's input ranges from [-1 to 1]. The first layer has 20 tansig neurons, the second layer has one purelin neuron. The trainbr network training function is to be used. The plot of the resulting network output should show a smooth response, without overfitting.

Create a Network

net=newff([-1 1],[20,1],{'tansig','purelin'},'trainbr');

Train and Test the Network
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net = train(net,p,t);
a = sim(net,p)
plot(p,a,p,t,'+')

Network Use

You can create a standard network that uses trainbr with newff, newcf, or newelm.

To prepare a custom network to be trained with trainbr

Set net.trainFcn to 'trainlm'. This will set net.trainParam to trainbr's default parameters.
Set net.trainParam properties to desired values.

In either case, calling train with the resulting network will train the network with trainbr.

See newff, newcf, and newelm for examples.

Algorithm

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) and Foresee and Hagan (Proceedings of the International Joint Conference on Neural Networks) 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.

The parameter mem_reduc indicates how to use memory and speed to calculate the Jacobian jX. If mem_reduc is 1, then trainlm runs the fastest, but can require a lot of memory. Increasing mem_reduc to 2 cuts some of the memory required by a factor of two, but slows trainlm somewhat. Higher values continue to decrease the amount of memory needed and increase the training times.

Training stops when any one of these conditions occurs:

The maximum number of epochs (repetitions) is reached.
The maximum amount of time has been exceeded.
Performance has been minimized to the goal.
The performance gradient falls below mingrad.
mu exceeds mu_max.
Validation performance has increased more than max_fail times since the last time it decreased (when using validation).

See Also

newff, newcf, traingdm, traingda, traingdx, trainlm, trainrp, traincgf, traincgb, trainscg, traincgp, trainoss

References

Foresee, F. D., and M. T. Hagan, "Gauss-Newton approximation to Bayesian regularization," Proceedings of the 1997 International Joint Conference on Neural Networks, 1997.

MacKay, D. J. C., "Bayesian interpolation," Neural Computation, vol. 4, no. 3, pp. 415-447, 1992.

trainbfg trainc