#### Classification

In this case, a typical choice is to project on the half-space, Section 8.6.2, formed by the hyperplane

${y}_{n}〈{f}_{n-1},\kappa \left(\cdot ,{\mathbit{x}}_{n}\right)〉=\rho ,$

and the corresponding projection operator becomes

${P}_{k}\left({f}_{n-1}\right)={f}_{n-1}+{\beta }_{k}\kappa \left(\cdot ,{\mathbit{x}}_{k}\right),$

where

$\begin{array}{l}\hfill {\beta }_{k}=\left\{\begin{array}{ll}\frac{\rho -{y}_{k}〈{f}_{n-1},\kappa \left(\cdot ,{\mathbit{x}}_{k}\right)〉}{\kappa \left({\mathbit{x}}_{k},{\mathbit{x}}_{k}\right)},& \text{if}\rho -{y}_{k}〈{f}_{n-1},\kappa \left(\cdot ,{\mathbit{x}}_{k}\right)〉>0,\\ 0,& \text{otherwise}.\end{array}\right\\end{array}$

(11.110)

Recall that the corresponding half-space is the 0-level set of the hinge loss function, ρ(yn,f(xn)) defined in (11.59).

Thus, for both cases, regression ...

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