Let me use the book notation to avoid confusion. You have two points
and
(which you called a1 and a2) and their target outputs (which you called assignment) are
and
.
Either point, call it just
for simplicity, is a vector that has
components
. Notice that bold
denotes a full data point, while italic
denotes a component in that data point. We add a constant 1 component to each data point
and call the component
to simplify the expression for the perceptron. If the weight vector of the perceptron is
(where
takes care of the threshold value of that perceptron), then the perceptron implements
where
returns
if its argument is positive and returns
if its argument is negative.
Example: Say the first data point
(two dimensional, so
). Add the constant component
and you have
. Therefore, the percepton's output on this point is
. If this formula returns
which is different from the target output
, the PLA adjusts the values of the weights trying to make the perceptron output agree with the target output for this point
. It uses the specific PLA update rule to achieve that.