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trainlm

Levenberg-Marquardt backpropagation

Syntax

net.trainFcn = 'trainlm'
[net,tr] = train(net,...)

Description

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

trainlm is often the fastest backpropagation algorithm in the toolbox, and is highly recommended as a first-choice supervised algorithm, although it does require more memory than other algorithms.

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

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

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

net.trainParam.epochs1000

Maximum number of epochs to train

net.trainParam.goal0

Performance goal

net.trainParam.max_fail6

Maximum validation failures

net.trainParam.min_grad1e-7

Minimum performance gradient

net.trainParam.mu0.001

Initial mu

net.trainParam.mu_dec0.1

mu decrease factor

net.trainParam.mu_inc10

mu increase factor

net.trainParam.mu_max1e10

Maximum mu

net.trainParam.show25

Epochs between displays (NaN for no displays)

net.trainParam.showCommandLine0

Generate command-line output

net.trainParam.showWindow1

Show training GUI

net.trainParam.timeinf

Maximum time to train in seconds

Validation vectors are 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. Test vectors are used as a further check that the network is generalizing well, but do not have any effect on training.

trainlm is the default training function for several network creation functions including newcf, newdtdnn, newff, and newnarx.

Network Use

You can create a standard network that uses trainlm with feedforwardnet or cascadeforwardnet.

To prepare a custom network to be trained with trainlm,

  1. Set net.trainFcn to 'trainlm'. This sets net.trainParam to trainlm's default parameters.

  2. Set net.trainParam properties to desired values.

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

See help feedforwardnet and help cascadeforwardnet for examples.

Examples

Here a neural network is trained to predict median house prices.

[x,t] = house_dataset;
net = feedforwardnet(10,'trainlm');
net = train(net,x,t);
y = net(x)

Definitions

Like the quasi-Newton methods, the Levenberg-Marquardt algorithm was designed to approach second-order training speed without having to compute the Hessian matrix. When the performance function has the form of a sum of squares (as is typical in training feedforward networks), then the Hessian matrix can be approximated as

H = JTJ

and the gradient can be computed as

g = JTe

where J is the Jacobian matrix that contains first derivatives of the network errors with respect to the weights and biases, and e is a vector of network errors. The Jacobian matrix can be computed through a standard backpropagation technique (see [HaMe94]) that is much less complex than computing the Hessian matrix.

The Levenberg-Marquardt algorithm uses this approximation to the Hessian matrix in the following Newton-like update:

xk+1=xk[JTJ+μI]1JTe

When the scalar µ is zero, this is just Newton's method, using the approximate Hessian matrix. When µ is large, this becomes gradient descent with a small step size. Newton's method is faster and more accurate near an error minimum, so the aim is to shift toward Newton's method as quickly as possible. Thus, µ is decreased after each successful step (reduction in performance function) and is increased only when a tentative step would increase the performance function. In this way, the performance function is always reduced at each iteration of the algorithm.

The original description of the Levenberg-Marquardt algorithm is given in [Marq63]. The application of Levenberg-Marquardt to neural network training is described in [HaMe94] and starting on page 12-19 of [HDB96]. This algorithm appears to be the fastest method for training moderate-sized feedforward neural networks (up to several hundred weights). It also has an efficient implementation in MATLAB® software, because the solution of the matrix equation is a built-in function, so its attributes become even more pronounced in a MATLAB environment.

Try the Neural Network Design demonstration nnd12m [HDB96] for an illustration of the performance of the batch Levenberg-Marquardt algorithm.

Limitations

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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Algorithms

trainlm supports training with validation and test vectors if the network's NET.divideFcn property is set to a data division function. Validation vectors are 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. Test vectors are used as a further check that the network is generalizing well, but do not have any effect on training.

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 states continue to decrease the amount of memory needed and increase training times.

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.

  • Validation performance has increased more than max_fail times since the last time it decreased (when using validation).

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