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1-D minimization using golden section search
[a,gX,perf,retcode,delta,tol] = srchgol(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)
srchgol 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 the golden section search.
[a,gX,perf,retcode,delta,tol] = srchgol(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 golden section algorithm are
alpha | Scale factor that determines sufficient reduction in perf |
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) |
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 srchgol search function are to be used.
net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p)
net.trainParam.searchFcn = 'srchgol'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p)
You can create a standard network that uses srchgol with newff, newcf, or newelm.
To prepare a custom network to be trained with traincgf, using the line search function srchgol,
The srchgol function can be used with any of the following training functions: traincgf, traincgb, traincgp, trainbfg, trainoss.
The golden section search srchgol is a linear search that does not require the calculation of the slope. This routine begins by locating an interval in which the minimum of the performance function occurs. This is accomplished by evaluating the performance at a sequence of points, starting at a distance of delta and doubling in distance each step, along the search direction. When the performance increases between two successive iterations, a minimum has been bracketed. The next step is to reduce the size of the interval containing the minimum. Two new points are located within the initial interval. The values of the performance at these two points determine a section of the interval that can be discarded, and a new interior point is placed within the new interval. This procedure is continued until the interval of uncertainty is reduced to a width of tol, which is equal to delta/scale_tol.
See [HDB96], starting on page 12-16, for a complete description of the golden section search. Try the Neural Network Design demonstration nnd12sd1 [HDB96] for an illustration of the performance of the golden section search in combination with a conjugate gradient algorithm.