If a primary interest is to conduct automatic variable selection, only LASSO can do that. A general guideline is that if it is thought that only a few variables are strongly related to the response, LASSO would be the better choice, since it would eliminate the unimportant features. On the other hand, if many predictors are thought to have at least a small degree of explanatory power, Ridge might be a better choice since it would allow all features to retain some influence in the model. However, it is usually advantageous to train both models and compare their performance, as depending on the dataset, one might perform better than the other.
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