2. Optimal soft-margin hyperplane Let (w*, 6*, *) denote the solution to the soft-margin hyperplane quadratic program. a. (5 points) Show that if z; is misclassified by the optimal soft-margin hyperplane classifier, then > 1. Conclude that h City upper bounds the training error. Hence, the OSM objective is balancing margin maximization with minimizing a bound on the training error. b. (5 points) Show that if Ei > 0, then it is proportional to the distance from xi to the margin hypeplane associated with class yi (that is, the set {x : (w*)? x + 6* = y;}), and give the constant of proportionality. c. (Optional please do not turn in) Consider the optimization problem D min : fi 1 n i=1 st. gi(ufa + b) > 6 - Ei > 0 Vi= 1,...,n Vi= 1,...,n where D > 0, k > 0. Show that the above quadratic program yields the same classifier as the optimal soft-margin classifier for a certain parameter C, and express this C in terms of D and K. 2. Optimal soft-margin hyperplane Let (w*, 6*, *) denote the solution to the soft-margin hyperplane quadratic program. a. (5 points) Show that if z; is misclassified by the optimal soft-margin hyperplane classifier, then > 1. Conclude that h City upper bounds the training error. Hence, the OSM objective is balancing margin maximization with minimizing a bound on the training error. b. (5 points) Show that if Ei > 0, then it is proportional to the distance from xi to the margin hypeplane associated with class yi (that is, the set {x : (w*)? x + 6* = y;}), and give the constant of proportionality. c. (Optional please do not turn in) Consider the optimization problem D min : fi 1 n i=1 st. gi(ufa + b) > 6 - Ei > 0 Vi= 1,...,n Vi= 1,...,n where D > 0, k > 0. Show that the above quadratic program yields the same classifier as the optimal soft-margin classifier for a certain parameter C, and express this C in terms of D and K