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BFGS quasi-Newton backpropagation



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

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

trainbfg is a network training function that updates weight and bias values according to the BFGS quasi-Newton method.

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

  • net.trainParam.epochs — Maximum number of epochs to train. The default value is 1000.

  • net.trainParam.showWindow — Show training GUI. The default value is true.

  • — Epochs between displays (NaN for no displays). The default value is 25.

  • net.trainParam.showCommandLine — Generate command-line output. The default value is false.

  • net.trainParam.goal — Performance goal. The default value is 0.

  • net.trainParam.time — Maximum time to train in seconds. The default value is inf.

  • net.trainParam.min_grad — Minimum performance gradient. The default value is 1e-6.

  • net.trainParam.max_fail — Maximum validation failures. The default value is 6.

  • net.trainParam.searchFcn — Name of line search routine to use. The default value is 'srchbac'.

Parameters related to line search methods (not all used for all methods):

  • net.trainParam.scal_tol — Divide into delta to determine tolerance for linear search. The default value is 20.

  • net.trainParam.alpha — Scale factor that determines sufficient reduction in perf. The default value is 0.001.

  • net.trainParam.beta — Scale factor that determines sufficiently large step size. The default value is 0.1.

  • — Initial step size in interval location step. The default value is 0.01.

  • net.trainParam.gamma — Parameter to avoid small reductions in performance, usually set to 0.1 (see srch_cha). The default value is 0.1.

  • net.trainParam.low_lim — Lower limit on change in step size. The default value is 0.1.

  • net.trainParam.up_lim — Upper limit on change in step size. The default value is 0.5.

  • net.trainParam.maxstep — Maximum step length. The default value is 100.

  • net.trainParam.minstep — Minimum step length. The default value is 1.0e-6.

  • net.trainParam.bmax — Maximum step size. The default value is 26.

  • net.trainParam.batch_frag — 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. The default value is 0.


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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);

Input Arguments

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Input network, specified as a network object. To create a network object, use for example, feedforwardnet or narxnet.

Output Arguments

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Trained network, returned as a network object..

Training record (epoch and perf), returned as a structure whose fields depend on the network training function (net.NET.trainFcn). It can include fields such as:

  • Training, data division, and performance functions and parameters

  • Data division indices for training, validation and test sets

  • Data division masks for training validation and test sets

  • Number of epochs (num_epochs) and the best epoch (best_epoch).

  • A list of training state names (states).

  • Fields for each state name recording its value throughout training

  • Performances of the best network (best_perf, best_vperf, best_tperf)

More About

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Network Use

You can create a standard network that uses trainbfg with feedfowardnet or cascadeforwardnet. To prepare a custom network to be trained with trainbfg:

  1. Set NET.trainFcn to 'trainbfg'. This sets NET.trainParam to trainbfg’s default parameters.

  2. Set NET.trainParam properties to desired values.

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

BFGS Quasi-Newton Backpropagation

Newton’s method is an alternative to the conjugate gradient methods for fast optimization. The basic step of Newton’s method is


where Ak1 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 trainbfg routine.

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 x 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 perf with respect to the weight and bias variables X. Each variable is adjusted according to the following:

X = X + a*dX;

where dX is the search direction. The parameter a is 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;

where 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 method.

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.

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


[1] Gill, Murray, & Wright, Practical Optimization, 1981

Version History

Introduced before R2006a