Errors and Error Surfaces

Revisiting the AND function, for certain values of $w_1,w_2$, the perceptron may not give us the correct output for a fixed bias $w_0$. For example, if we have two inputs $x_1,x_2$, such that $w_1=-1$ and $w_2=-1$ with $w_0$ fixed at $-1$, the point $(1,1)$ will be in the negative half-space, since $w_0+w_1\ast x_1+w_2\ast x_2<0$ for this input.

Notice that the positive half-space this time is to the left of the separator. Similarly, for various other combinations of $w_1,w_2$, some points may be classified incorrectly. Hence, we are interested in finding a combination of weights such that the error is minimum.