componentwise boosting dates back at least to Bühlman and Yu (2003), where in each boosting iteration a set of base-learners (e.g. simple linear models) depending on a subset of the covariates are used. This implies some variable selection.
Let us consider the case, in which we use base-learners with only one covariate. If we have $P$ covariates, we then have $P$ base learners in each boosting iteration. At the end of each boosting iteration, we select the best base-learner among this $P$ base-learners for the update. Since only one covariate is in cluded in each base-learner, we update only the parameters for one covariate.
In the literature one can find componentwise boosting based on functional gradiend descent on the one hand and componentwise boosting based on one-step Fisher scoring (e.g. Tutz and Binder 2006; Tutz and Binder 2007 ...) on the other hand. My question refers to the latter, in which a penalized maximum likelihood estimation (ridge penalty) is used to update the parameters by one-step Fisher scoring.
Here comes my question: If we have only metric covariates and use componentwise boosting with only one single predictor in each step, do we then need penalized maximum likelihood estimation?
My guess is no, because in my view penalization is only needed if we use componentwise boosting, where we want to update a set of predictors in each step (and not only one single predictor). Another reason where I see the need for penalization is the case, where we have a single multicategorial covariate (with $K$ categories) that has to be updated. Each multicategorial covariate has $K-1$ associated parameters and thus updating this multicategorial covariate means updating all $K-1$ associated parameters. Here, one can do penalization to update only a subset of the $K-1$ associated parameters, namely the 'important' ones, while the 'unimportant' ones are shrinked towards zero due to penalization.
Since we want to update one single metric predictor in each boosting step, we don't need penalization, right?