3 / 8 Perceptron & SVM

Hinge and Squeeze

The soft-margin SVM is often stated as: minimize (1/2)∥w∥2+C⋅∑ξi(1/2)|w|^2 + C \cdot \sum \xi_i subject to yi(w⋅xi+b)≥1−ξiy_i(w \cdot x_i+b) \geq 1 - \xi_i and ξi≥0\xi_i \geq 0. Which unconstrained objective is exactly equivalent to this formulation?

A

min⁡w,bC∥w∥2+∑imax⁡(0,1−yi(w⋅xi+b))2\min_{w,b} C|w|^2 + \sum_i \max(0, 1 - y_i(w \cdot x_i+b))^2

B

min⁡w,b∥w∥1+C⋅∑iξi2\min_{w,b} |w|_1 + C \cdot \sum_i \xi_i^2

C

min⁡w,b(1/2)∥w∥2+C⋅∑imax⁡(0,1−yi(w⋅xi+b))\min_{w,b} (1/2)|w|^2 + C \cdot \sum_i \max(0, 1 - y_i(w \cdot x_i+b))

D

min⁡w,b(1/2)∥w∥2+C⋅∑imax⁡(0,yi(w⋅xi+b)−1)\min_{w,b} (1/2)|w|^2 + C \cdot \sum_i \max(0, y_i(w \cdot x_i+b) - 1)

E

min⁡w,b(1/2)∥w∥2−C⋅∑iyi(w⋅xi+b)\min_{w,b} (1/2)|w|^2 - C \cdot \sum_i y_i(w \cdot x_i+b)