It is generally agreed that in penalized regression models, such as ridge regression, the lasso, and the elastic net, one should standardize the predictors (such as dividing each by its SD) so that the penalty applies equally to the different predictors, instead of applying to different degrees based on the scale of the predictors. But should one do this even for binary predictors, which are coded as 0 and 1?
To be more specific, suppose my goal is maximizing the predictive accuracy (i.e., minimizing the test error) of my model, and I have little a priori knowledge about the relative importance of the predictors. Which is likely to get better predictive accuracy, rescaling my binary predictors along with my continuous predictors, or leaving them alone?
A frequently mentioned paper in the context of predictor standardization and binary predictors is the following, but it doesn't seem to consider the case of penalized regression.
Gelman, A. (2008). Scaling regression inputs by dividing by two standard deviations. Statistics in Medicine, 27, 2865–2873. doi:10.1002/sim.3107. http://www.stat.columbia.edu/~gelman/research/published/standardizing7.pdf
A similar but more general question is: Are categorical variables standardized differently in penalized regression?
Edit: a more closely related question is: whether to rescale indicator / binary / dummy predictors for LASSO