stem/AI/Neural Networks/SLP/Perceptron Convergence.md

Error-Correcting Perceptron Learning

• Uses a McCulloch-Pitt neuron
  • One with a hard limiter
• Unity increment
  • Learning rate of 1

If the $n$-th member of the training set, x(n), is correctly classified by the weight vector w(n) computed at the $n$-th iteration of the algorithm, no correction is made to the weight vector of the perceptron in accordance with the rule:

w(n + 1) = w(n) \text{ if  $w^Tx(n) > 0$  and  $x(n)$  belongs to class $\mathfrak{c}_1$}

w(n + 1) = w(n) \text{ if $w^Tx(n) \leq 0$ and $x(n)$ belongs to class $\mathfrak{c}_2$}

Otherwise, the weight vector of the perceptron is updated in accordance with the rule

w(n + 1) = w(n) - \eta(n)x(n) \text{ if } w^Tx(n) > 0 \text{ and } x(n) \text{ belongs to class }\mathfrak{c}_2

w(n + 1) = w(n) + \eta(n)x(n) \text{ if } w^Tx(n) \leq 0 \text{ and } x(n) \text{ belongs to class }\mathfrak{c}_1

1. Initialisation. Set w(0)=0. perform the following computations for
   time step n = 1, 2,...
2. Activation. At time step n, activate the perceptron by applying continuous-valued input vector x(n) and desired response d(n).
3. Computation of Actual Response. Compute the actual response of the perceptron:

y(n) = sgn[w^T(n)x(n)]

where sgn(\cdot) is the signum function.
4. Adaptation of Weight Vector. Update the weight vector of the perceptron:

w(n+1)=w(n)+\eta[d(n)-y(n)]x(n)

d(n) = \begin{cases}
+1 &\text{if $x(n)$ belongs to class $\mathfrak{c_1}$}\\
-1 &\text{if $x(n)$ belongs to class $\mathfrak{c_2}$}
\end{cases}

5. Continuation. Increment time step n by one and go back to step 2.

• Guarantees convergence provided
  • Patterns are linearly separable
    • Non-overlapping classes
    • Linear separation boundary
  • Learning rate not too high
• Two conflicting requirements
  1. Averaging of past inputs to provide stable weight estimates
     • Small eta
  2. Fast adaptation with respect to real changes in the underlying distribution of process responsible for x
     • Large eta