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I am writing a scientific paper that - among other things - deals with logistic regression in the context of machine learning.
I read this article where the author states that, given a set of instance-label pairs $(\boldsymbol{x_i},y_i),\ i=1,...,l\ \ \boldsymbol(x_i) \in R^n,\ y_i \in \{-1,+1\}$, Logistic Regression solves the following optimization problem:

$$ \begin{equation}\label{eq:log} \min_{w} \frac{1}{2}\boldsymbol{w}^T\boldsymbol{w} + C\sum\limits_{i=1}^l log(1+e^{-y_i\boldsymbol{w}^T \boldsymbol{x_i}}) \end{equation}\ \ \ \ \ \ \ \ \ \ (Eq.1) $$

Is it correct if I say that: "The first summand of Equation 1 is a regularization term while the second represents the negative log-likelihood loss function"?

Thank you for your kind assistance.

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  • $\begingroup$ In here the log likelihood has a different expression: statlect.com/fundamentals-of-statistics/… $\endgroup$
    – David Masip
    Commented Apr 24, 2018 at 5:32
  • $\begingroup$ Am I right? Maybe it is due to the fact that they use y in 0, 1 and not in -1, 1 $\endgroup$
    – David Masip
    Commented Apr 24, 2018 at 5:33

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Yes the first is the L2 regularization term to keep the norm of weights parameter as small as possible.

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