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Widrow-Hoff weight/bias learning function


[dW,LS] = learnwh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS)
info = learnwh('code')


learnwh is the Widrow-Hoff weight/bias learning function, and is also known as the delta or least mean squared (LMS) rule.

[dW,LS] = learnwh(W,P,Z,N,A,T,E,gW,gA,D,LP,LS) takes several inputs,


S-by-R weight matrix (or b, and S-by-1 bias vector)


R-by-Q input vectors (or ones(1,Q))


S-by-Q weighted input vectors


S-by-Q net input vectors


S-by-Q output vectors


S-by-Q layer target vectors


S-by-Q layer error vectors


S-by-R weight gradient with respect to performance


S-by-Q output gradient with respect to performance


S-by-S neuron distances


Learning parameters, none, LP = []


Learning state, initially should be = []

and returns


S-by-R weight (or bias) change matrix


New learning state

Learning occurs according to the learnwh learning parameter, shown here with its default value. — 0.01

Learning rate

info = learnwh('code') returns useful information for each code string:


Names of learning parameters


Default learning parameters


Returns 1 if this function uses gW or gA


Here you define a random input P and error E for a layer with a two-element input and three neurons. You also define the learning rate LR learning parameter.

p = rand(2,1);
e = rand(3,1); = 0.5;

Because learnwh needs only these values to calculate a weight change (see “Algorithm” below), use them to do so.

dW = learnwh([],p,[],[],[],[],e,[],[],[],lp,[])

Network Use

You can create a standard network that uses learnwh with linearlayer.

To prepare the weights and the bias of layer i of a custom network to learn with learnwh,

  1. Set net.trainFcn to 'trainb'. net.trainParam automatically becomes trainb’s default parameters.

  2. Set net.adaptFcn to 'trains'. net.adaptParam automatically becomes trains’s default parameters.

  3. Set each net.inputWeights{i,j}.learnFcn to 'learnwh'.

  4. Set each net.layerWeights{i,j}.learnFcn to 'learnwh'.

  5. Set net.biases{i}.learnFcn to 'learnwh'. Each weight and bias learning parameter property is automatically set to the learnwh default parameters.

To train the network (or enable it to adapt),

  1. Set net.trainParam (or net.adaptParam) properties to desired values.

  2. Call train (or adapt).


learnwh calculates the weight change dW for a given neuron from the neuron’s input P and error E, and the weight (or bias) learning rate LR, according to the Widrow-Hoff learning rule:

dw = lr*e*pn'


Widrow, B., and M.E. Hoff, “Adaptive switching circuits,” 1960 IRE WESCON Convention Record, New York IRE, pp. 96–104, 1960

Widrow, B., and S.D. Sterns, Adaptive Signal Processing, New York, Prentice-Hall, 1985

See Also

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Introduced before R2006a

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