# What does the number of support vectors tell us?

I've trained a SVM regression model, and noticed a large number of support vectors required. I have found the following discussions where it seems people are concluding that a larger number of support vectors means overfitting:
Is a high number of support vectors a sign of overfitting?
What is the relation between the number of Support Vectors and training data and classifiers performance?

In my case, there are 688 observations in the training set, out of which 589 are support vectors. However, my performance metrics on the test set are great when compared to the training set, the $$R^{2}$$ for train/test is 0.914/0.883 and $$MSE$$ is 0.0021/0.0026. To choose the hyperparameters I did use cross validation, and the optimal parameters are $$C=220$$, $$\epsilon=0.01$$ and $$\gamma=3\times10^{-4}$$.
If you are having doubts about the test set, I separated it immediately upon loading the data, and all the scaling is fitted on the training set. So everything indicates that the model makes good predictions, and that there is no overfitting. So what's the actual deal with the number of support vectors? Are they really a sign of overfitting, or just of a complex system? I'm wondering about these questions regardless of my specific case.

EDIT: Sorry forgot to mention. I'm using RBF function, which as far as I understand uses the dual form, which needs to learn $$N$$ parameters, equal to the number of observations, instead of number of features. In my case I have 40 features.

## 1 Answer

The more SVs you have, the more flexible is the decision function. It's easy to see how: if you had only 2 SVs, then the decision function is rather simple.

Having more SVs is not, by itself, an indicator of overfitting. Rather, it's reverse inference, because we expect an overfitted model to have a large number of SVs.

In all cases, the only thing that matters is generalizability to held out data, so, if your model works, then it works.

• I think "decision function" is not right here, since we're talking regression, not classification. Many SVs likely imply a highly non-linear fitted function - if, as @Sjotroll says, the model is not overfitted. Commented Feb 23, 2023 at 14:53