Say we have a known model $M$ with unknown parameters and more specifically, $M$ is a parametric model.
Parameter estimation on $M$ is applying an appropriate method for estimating the parameters.
My question is: If we know the form/shape of the model $M$ up to unknown parameters, and applying a machine learning algorithm such as gradient descent, does this fall under machine/statistical learning?
Or is it still only parameter estimation using a borrowed method since the model form is known?
It seems the former question is yes since we are "learning" the parameters rather than just doing some algebraic/calculus manipulation to get suitable estimators directly.
My confusion comes from that what reading I have done on machine learning/statistical learning is that the goal is to estimate the model itself or rather the function $f$ which gives the model. Of course, parameter estimation is something utilized in machine/statistical learning but seems the situation I described is a instance of machine/statistical learning, perhaps this instance would be more specifically referred to as parametric machine/statistical learning
Yes. It is both. You can use either language.
"Machine learning" is not a term with a formal, precise mathematical definition. What you describe can be considered a form of machine learning.
In practice, when doing model parameter estimation, it's common that what we really care about is the model and the parameters themselves, e.g., because they will help us make inferences about the causal structure of the real world, or because we care about the model itself. In contrast, much use of machine learning focuses more on prediction rather than inference: we care more about predicting future outputs from the model and the parameters are only a means to an end.
Ultimately, to misquote Shakespeare, a rose by any other name is still a rose.