net.trainFcn = 'trainlm'
[net,tr] = train(net,...)
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
net.trainFcn = 'trainlm' sets the network
[net,tr] = train(net,...) trains the network
Training occurs according to
parameters, shown here with their default values:
Maximum number of epochs to train
Maximum validation failures
Minimum performance gradient
Epochs between displays (
Generate command-line output
Show training GUI
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
You can create a standard network that uses
To prepare a custom network to be trained with
to desired values.
In either case, calling
train with the resulting
network trains the network with
help feedforwardnet and
cascadeforwardnet for 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)
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:
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
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
trainlm supports training with validation
and test vectors if the network's
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
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
perf with respect to the weight and
X. Each variable is adjusted according
jj = jX * jX je = jX * E dX = -(jj+I*mu) \ je
E is all errors and
the identity matrix.
The adaptive value
mu is increased by
the change above results in a reduced performance value. The change
is then made to the network and
mu is decreased
Training stops when any of these conditions occurs:
The maximum number of
The maximum amount of
time is exceeded.
Performance is minimized to the
The performance gradient falls below
Validation performance has increased more than
since the last time it decreased (when using validation).