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The obvious
approach is to write down an error function
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and try to show
that each step of the learning procedure
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reduces the
error.
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For
stochastic online learning we would like to show
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that
each step reduces the expected error, where the
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expectation
is across the choice of training cases.
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It
cannot be a squared error because the size of the
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update
does not depend on the size of the mistake.
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The textbook
tries to use the sum of the distances on the
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wrong side of
the decision surface as an error measure.
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Its
conclusion is that the perceptron convergence
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procedure
is not guaranteed to reduce the total error
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at
each step.
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This is
true for that error function even if there is a set of
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weights
that gets the right answer for every training case.
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