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:
$$w(n+1)=w(n)+\eta[d(n)-y(n)]x(n)$$ where
$$
d(n) = \begin{cases}
+1 &\text{if $x(n)$ belongs to class $\mathfrak{c_1}$}\\
-1 &\text{if $x(n)$ belongs to class $\mathfrak{c_2}$}
\end{cases}
$$
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 $x$
