trainlm (Neural Network Toolbox)

Levenberg-Marquardt backpropagation

[net,tr] = trainlm(net,Pd,Tl,Ai,Q,TS,VV)
info = trainlm(code)

Description

trainlm is a network training function that updates weight and bias values according to Levenberg-Marquardt optimization.

trainlm(net,Pd,Tl,Ai,Q,TS,VV) 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.

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.

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

net.trainParam.epochs 10 Maximum number of epochs to train

net.trainParam.goal 0 Performance goal

net.trainParam.lr 0.01 Learning rate

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 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.

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

'pnames' - Names of training parameters.

'pdefaults' - Default training parameters.

Network Use

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

To prepare a custom network to be trained with trainlm:

1.: Set net.trainFcn to 'trainlm'. This will set net.trainParam to trainlm's default parameters.

2.: Set net.trainParam properties to desired values.

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

See newff, newcf, and newelm for examples.

Algorithm

trainlm can train any network as long as its weight, net input, and transfer functions have derivative functions.

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 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 training times.

Training stops when any of these conditions occur:

1.: The maximum number of epochs (repetitions) is reached.

2.: The maximum amount of time has been exceeded.

3.: Performance has been minimized to the goal.

4.: The performance gradient falls below mingrad.

5.: mu exceeds mu_max.

6.: Validation performance has increase more than max_fail times since the last time it decreased (when using validation).