I have read in several places that R Squared is not an ideal measure when a model is fit using LASSO. However, I'm not clear on exactly why that is.
In addition, could you recommend the best alternative?
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The goal of using LASSO is obtaining a sparse representation (of a predicted quantity) in the sense of not having many covariates. Comparing models with $R^2$ tends to favor models with lots of covariates: in fact, adding covariates unrelated to the outcome will never decrease $R^2$ and almost always increases it at least a little bit. The LASSO model will identify the model with the optimal penalized log-likelihood (an unpenalized log-likelihood is monotonically related to the $R^2$). Validation statistics that are more widely used to compare LASSO models to other types of models are, for instance, the BIC or cross-validated $R^2$.