The soft-margin SVM is regularised hinge-loss minimisation

ProofIntermediate⏱ 30 minClassical MLsvmhinge lossequivalenceregularisation

Problem

Prove that eliminating the slack variables turns the soft-margin primal into

\min_{w,b}\ \frac12\|w\|_2^2 + C\sum_i\big[1-y_i(\langle w,x_i\rangle+b)\big]_+ ,

and compare the hinge loss with the logistic and squared losses as surrogates for the zero-one loss.

Write a complete argument. Every inequality you use should be one you could state precisely.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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