In Machine Learning optimization problems - a "regularization" term is often added to the optimization problem to reduce overfitting:
I have noticed that in the case of the L2-Norm regularization, this term (i.e. a function) can be considered as a Convex Term as it is basically "quadratic" in nature.
My Question: In the L2-Norm case, the optimization problem without this regularization term is likely a Non-Convex problem - but we then add a Convex Term to this problem. Do we know if doing this (i.e. adding the Convex Term to a Non-Convex Optimization Problem) automatically makes the optimization problem as Convex?
I do not think that this is the case, seeing as:
Convex Optimization Problems are generally easier to solve than Non-Convex Optimization Problems
Anecdotally, I have heard of Regularized Loss Functions (e.g. for Neural Networks) that are considered to be "very difficult" optimization problems - even though they have this Convex Term. This informally leads me to believe that in the case of L2 Regularization, the fundamental optimization problem remains Non-Convex.
However, "anecdotal and informal logic" is generally never acceptable in understanding mathematics.
Can someone please comment on this?