Simple perceptrons
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Simple Perceptrons. Or one-layer feed-forward networks. Perceptrons or Layered Feed-Forward Networks. Equation governing comp of simple perceptron. activation function, usually nonlinear, e.g. step function or sigmoid. ksi. Threshold or no threshold?. with threshold.

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Simple Perceptrons

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Simple Perceptrons

Or one-layer feed-forward networks

Perceptrons or Layered Feed-Forward Networks

Equation governing comp of simple perceptron

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


Threshold or no threshold?

with threshold

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

The General Association (Matching) Task:

Is to ask for: actual output pattern = target pattern

Threshold Units

  • 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

A Notational Simplification

  • 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.

New Form for General Association Task: geometric interpretation

Another form:

A simple learning algorithm

  • 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)

  • 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

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