BFGS quasi-Newton backpropagation
net.trainFcn = 'trainbfg'
[net,tr] = train(net,...)
trainbfg is a network training function that updates weight and bias
values according to the BFGS quasi-Newton method.
net.trainFcn = 'trainbfg' sets the network
[net,tr] = train(net,...) trains the network with
Training occurs according to
trainbfg training parameters, shown here
with their default values:
Maximum number of epochs to train
Show training window
|Epochs between displays (|
Generate command-line output
Maximum time to train in seconds
Minimum performance gradient
Maximum validation failures
Name of line search routine to use
Parameters related to line search methods (not all used for all methods):
Scale factor that determines sufficient reduction in
Scale factor that determines sufficiently large step size
Initial step size in interval location step
Parameter to avoid small reductions in performance, usually set to 0.1 (see
Lower limit on change in step size
Upper limit on change in step size
Maximum step length
Minimum step length
Maximum step size
In case of multiple batches, they are considered independent. Any nonzero value implies a fragmented batch, so the final layer’s conditions of a previous trained epoch are used as initial conditions for the next epoch.
You can create a standard network that uses
cascadeforwardnet. To prepare a custom
network to be trained with
NET.trainParam properties to desired
In either case, calling
train with the resulting network trains the
This example shows how to train a neural network using the
trainbfg train function.
Here a neural network is trained to predict body fat percentages.
[x, t] = bodyfat_dataset; net = feedforwardnet(10, 'trainbfg'); net = train(net, x, t); y = net(x);
Newton’s method is an alternative to the conjugate gradient methods for fast optimization. The basic step of Newton’s method is
where is the Hessian matrix (second derivatives) of the performance index at the
current values of the weights and biases. Newton’s method often converges faster than conjugate
gradient methods. Unfortunately, it is complex and expensive to compute the Hessian matrix for
feedforward neural networks. There is a class of algorithms that is based on Newton’s method,
but which does not require calculation of second derivatives. These are called quasi-Newton (or
secant) methods. They update an approximate Hessian matrix at each iteration of the algorithm.
The update is computed as a function of the gradient. The quasi-Newton method that has been
most successful in published studies is the Broyden, Fletcher, Goldfarb, and Shanno (BFGS)
update. This algorithm is implemented in the
The BFGS algorithm is described in [DeSc83]. This algorithm requires
more computation in each iteration and more storage than the conjugate gradient methods,
although it generally converges in fewer iterations. The approximate Hessian must be stored,
and its dimension is n
n, where n is equal to the number of weights and biases
in the network. For very large networks it might be better to use Rprop or one of the conjugate
gradient algorithms. For smaller networks, however,
trainbfg can be an
efficient training function.
trainbfg can train any network as long as its weight, net input, and
transfer functions have derivative functions.
Backpropagation is used to calculate derivatives of performance
with respect to the weight and bias variables
X. Each variable is adjusted
according to the following:
X = X + a*dX;
dX is the search direction. The parameter
selected to minimize the performance along the search direction. The line search function
searchFcn is used to locate the minimum point. The first search direction is
the negative of the gradient of performance. In succeeding iterations the search direction is
computed according to the following formula:
dX = -H\gX;
gX is the gradient and
H is a approximate
Hessian matrix. See page 119 of Gill, Murray, and Wright (Practical
Optimization, 1981) for a more detailed discussion of the BFGS quasi-Newton
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
The performance gradient falls below
Validation performance has increased more than
max_fail times since
the last time it decreased (when using validation).
Gill, Murray, & Wright, Practical Optimization, 1981