Why is Scikit's Support Vector Classifier returning support vectors with decision scores outside [-1,1]? Is this a mistake?

Question

I'm currently playing around with support vector machines in Scikit Learn and I've come across some unusual behaviour. For a basic simulated dataset, I've trained an SVC estimator (with linear kernel), found the support vectors and computed the decision function scores.

It's those scores that are confusing me; according to Scikit, some of the scores are greater than 1 or less than -1; isn't this odd? The margins in a linear support vector classifier should be $x^{T}w+b = \pm 1$, so any examples with score outside $[-1,1]$ shouldn't be a support vector. I've attached my code below - what's going wrong?

from sklearn.svm import SVC

clf = SVC(C = 1.0, kernel = 'linear', class_weight = 'balanced',tol=1e-10)

clf.fit(X,Y)

w = clf.coef_
b = clf.intercept_

sv = clf.support_vectors_
print(w,b)
print('Number of support vectors:', sv.shape[0])
print(clf.decision_function(sv))

Here is the output:

[[1.28546328 1.08848812]] [-0.35044999]
Number of support vectors: 44
array([-0.57011282, -0.59506839, -0.53074738, -0.36055916, -0.69388818,
         0.28055111,  1.39559261,  0.26911971, -0.90759079, -0.93444484,
         2.37910343,  0.13621966, -0.86860512, -0.93672977,  0.85133397,
        -0.85497661, -0.78276458,  0.3908212 , -0.35721162, -1.00000001,
         2.07987489, -0.89293624,  0.37559497,  0.3603185 ,  0.35699434,
         0.23356079,  0.4274206 ,  0.77953069,  0.50193484,  0.99999995,
         0.7034846 ,  0.68110044, -2.00881614,  0.17607163,  0.99999989,
         0.40207295,  0.81320519, -1.31492698,  0.84101353,  0.3534314 ,
         0.39314117, -1.62172733, -2.41029845, -0.56251937])

Answer 1

Misclassified points will always be support vectors, even when they are "badly misclassified" and lie beyond the margin, with decision function scores outside of $[-1, 1]$.

Answer 2

SVC.decision_function returns values in $\mathbb{R}$ and is defined as: $$ f(x) = \sum_{i \in SV} y_i \alpha_i \langle x, x_i \rangle + b $$ You expect to get $1$ or $-1$ when predicting classes, not scores, with SVC.predict which is $\text{sign}(f(x))$. So, I believe everything is OK.