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Simple Perceptrons

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Simple Perceptrons

Or one-layer feed-forward networks

activation function, usually nonlinear, e.g. step function or sigmoid

ksi

with threshold

without threshold; threshold simulated with connections to an input terminal permanently tied to -1

Is to ask for: actual output pattern = target pattern

- Start with simplest threshold unit, practical for 1-level perceptrons

- Also assume the targets have plus/minus 1 values and no values in between those extremes, that is,

- Then all that matter is that for each input pattern, the net input (weighted sum) h to each output unit has the same sign as the target zeta

- To simplify notation, note that the output units are independent
- [In a multilayer nn, however, the hidden (non-output) layers aren’t independent]
- So let’s consider only one output at a time
- Drop the i subscripts

Weights and each input pattern live in the same space.

Advantage: can geometrically represent these two vectors together.

Another form:

- Also called the Perceptron Rule
- Go through the input patterns one by one
- For each pattern go through the output units one by one, asking whether output is the desired one.
- If so, leave the weight into that unit alone
- Else in the spirit of Hebb add to each connection something proportional to product of the input and desired output

Simplified Simple Learning Algorithm(for one neuron case)

- Start with w = 0 (not necessary)
- Cycle through the learning patterns
- For each pattern ksi
- If the output (O) != desired output (zeta), add product of the desired output and the input to w. (i.e., w = w + z*x)

- For each pattern ksi
- Keep cycling through the patterns until done.
- Convergence is guaranteed provided the two classes of input points are linearly separable.
- Perceptron convergence theorem guarantees this

Weight Update Formula,“Hebbian” from blue book, too complicated