Conflicting labels, a soft-margin optimum, and the meaning of its score

The feature map in [F3] realizes the stated linear kernel. At $w=0$, $b=-1$, the scores at all three observations equal $-1$. Their signed margins are respectively $1,-1,1$, so $\xi =(0,2,0)$ satisfies every soft-margin constraint, with objective $2C$.

For any feasible tuple, let $t=w+b$, the score at $x=1$. The two observations at $x=1$ require ${\xi }_2\ge \max (0,1-t)$ and ${\xi }_3\ge \max (0,1+t)$. If $-1\le t\le 1$ these lower bounds sum to $2$; if $t>1$, the second is $1+t>2$; and if $t<-1$, the first is $1-t>2$. Thus ${\xi }_2+{\xi }_3\ge 2$ for every feasible tuple, and its objective is at least $C({\xi }_2+{\xi }_3)\ge 2C$.

The candidate in [step 1.1] is therefore globally optimal. For $\alpha =(0,C,C)$ and $\mu =(C,0,0)$, ${\sum }_i{\alpha }_i{y}_i\varphi (x_i)=C-C=0=w$, ${\sum }_i{\alpha }_i{y}_i=C-C=0$, and $C-{\alpha }_i-{\mu }_i=0$ for all three indices. Each margin constraint is active, so ${\alpha }_i{g}_i^{(m)}=0$; also ${\mu }_i{\xi }_i=0$ for each $i$. These are the KKT conditions in [F2].

The dual objective is ${\sum }_i{\alpha }_i-\frac{1}{2}\mathrm{\parallel }{\sum }_i{\alpha }_i{y}_i\varphi (x_i){\mathrm{\parallel }}^2=2C$, equal to the primal value. The score at $x=1$ is $-1$, while the empirical positive-label frequency among the two observations at $x=1$ is $1\mathrm{/}2$. The logistic transform of the score is $(1+e{)}^{-1}\mathrm{\ne }1\mathrm{/}2$; the SVM optimization does not assert that this transformed value is a probability.