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# srchbre

1-D interval location using Brent's method

## Syntax

[a,gX,perf,retcode,delta,tol] = srchbre(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)

## Description

srchbre is a linear search routine. It searches in a given direction to locate the minimum of the performance function in that direction. It uses a technique called Brent's technique.

[a,gX,perf,retcode,delta,tol] = srchbre(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf) takes these inputs,

 net Neural network X Vector containing current values of weights and biases Pd Delayed input vectors Tl Layer target vectors Ai Initial input delay conditions Q Batch size TS Time steps dX Search direction vector gX Gradient vector perf Performance value at current X dperf Slope of performance value at current X in direction of dX delta Initial step size tol Tolerance on search ch_perf Change in performance on previous step

and returns

 a Step size that minimizes performance gX Gradient at new minimum point perf Performance value at new minimum point retcode Return code that has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These have different meanings for different search algorithms. Some might not be used in this function. 0  Normal 1  Minimum step taken 2  Maximum step taken 3  Beta condition not met delta New initial step size, based on the current step size tol New tolerance on search

Parameters used for the Brent algorithm are

 alpha Scale factor that determines sufficient reduction in perf beta Scale factor that determines sufficiently large step size bmax Largest step size scale_tol Parameter that relates the tolerance tol to the initial step size delta, usually set to 20

The defaults for these parameters are set in the training function that calls them. See traincgf, traincgb, traincgp, trainbfg, and trainoss.

Dimensions for these variables are

 Pd No-by-Ni-by-TS cell array Each element P{i,j,ts} is a Dij-by-Q matrix. Tl Nl-by-TS cell array Each element P{i,ts} is a Vi-by-Q matrix. Ai Nl-by-LD cell array Each element Ai{i,k} is an Si-by-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)

## Examples

Here is a problem consisting of inputs p and targets t to be solved with a network.

```p = [0 1 2 3 4 5];
t = [0 0 0 1 1 1];
```

A two-layer feed-forward network is created. The network's input ranges from [0 to 10]. The first layer has two tansig neurons, and the second layer has one logsig neuron. The traincgf network training function and the srchbac search function are to be used.

### Create and Test a Network

```net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf');
a = sim(net,p)
```

### Train and Retest the Network

```net.trainParam.searchFcn = 'srchbre';
net.trainParam.epochs = 50;
net.trainParam.show = 10;
net.trainParam.goal = 0.1;
net = train(net,p,t);
a = sim(net,p)
```

## Network Use

You can create a standard network that uses srchbre with newff, newcf, or newelm. To prepare a custom network to be trained with traincgf, using the line search function srchbre,

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

2. Set net.trainParam.searchFcn to 'srchbre'.

The srchbre function can be used with any of the following training functions: traincgf, traincgb, traincgp, trainbfg, trainoss.

## Definitions

Brent's search is a linear search that is a hybrid of the golden section search and a quadratic interpolation. Function comparison methods, like the golden section search, have a first-order rate of convergence, while polynomial interpolation methods have an asymptotic rate that is faster than superlinear. On the other hand, the rate of convergence for the golden section search starts when the algorithm is initialized, whereas the asymptotic behavior for the polynomial interpolation methods can take many iterations to become apparent. Brent's search attempts to combine the best features of both approaches.

For Brent's search, you begin with the same interval of uncertainty used with the golden section search, but some additional points are computed. A quadratic function is then fitted to these points and the minimum of the quadratic function is computed. If this minimum is within the appropriate interval of uncertainty, it is used in the next stage of the search and a new quadratic approximation is performed. If the minimum falls outside the known interval of uncertainty, then a step of the golden section search is performed.

See [Bren73] for a complete description of this algorithm. This algorithm has the advantage that it does not require computation of the derivative. The derivative computation requires a backpropagation through the network, which involves more computation than a forward pass. However, the algorithm can require more performance evaluations than algorithms that use derivative information.

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### Algorithms

srchbre brackets the minimum of the performance function in the search direction dX, using Brent's algorithm, described on page 46 of Scales (see reference below). It is a hybrid algorithm based on the golden section search and the quadratic approximation.

## References

Scales, L.E., Introduction to Non-Linear Optimization, New York, Springer-Verlag, 1985