If $ y = 1$, we want $\theta^Tx \geq 1$ (not just $\geq 0$)
If $ y = 0$, we want $\theta^Tx \leq -1$ (not just $< 0$)
My question is the following: What if we have an observation in which $-1<\theta^Tx<1$? How do we classify the observation in that case? Is it $y = 0$? Or is it $y = 1$? Or is it neither?