He does not show the proof to why that is. I intuitively understand that we have a local linearization of the objective function and we can only use that approximation arround $\theta$, therefore using the euclidean distance between $\theta$ and $\theta'$ and limiting by $\epsilon$ we guarantee that we don't update the parameters to much to the point where the approximation has less precision.
Can someone help figure out how he gets the result for $\theta'$. I tried to construct the Lagragian an setting its' gradient to zero, but I got lost in the computation.
Thank you very much in advance.