Similarity of perceptron criterion and SVM

In the book "Neural Networks and Deep Learning" by Aggarwal there is an exercise 2.10.1:

Consider the following loss function for training pair $$(\overline{X},y)$$: $$L=max(0, a -y(\overline{W} \cdot \overline{X}))$$ The test instances are predicted as $$\hat{y}=sign(\overline{W} \cdot \overline{X})$$. A value of $$a=0$$ corresponds to the perceptron criterion and a value of $$a=1$$ corresponds to the SVM. Show that any value of $$a>0$$ leads to the SVM with an unchanged optimal solution when no regularization is used.

I am trying to solve this exercise and from the previous chapters of the book I drew a conclusion that the stochastic update for such a loss function will be $$\overline{W} \Leftarrow \overline{W} + \alpha y \overline{X} [I(y \hat{y} < a)]$$ so it's different from original SVM for cases where $$y \hat{y} \in (a,1)$$. For this range, the weights in original SVM will be updated, but for "modified" SVM the wegiths will not be updated. So I don't see a reason why the "modified" SVM (with $$a$$ instead of $$1$$) should be equal to the original SVM. How to solve this exercise?

• Just a tip: you might get more and/or better answers for this kind of theoretical questions on stats.stackexchange.com Jul 29, 2019 at 14:56
• @Erwan thanks for a tip
– DeeM
Jul 30, 2019 at 14:19