  Although the Lasso has been extensively studied, the relationship between its
prediction performance and the correlations of the covariates is not fully
understood. In this paper, we give new insights into this relationship in the
context of multiple linear regression. We show, in particular, that the
incorporation of a simple correlation measure into the tuning parameter can
lead to a nearly optimal prediction performance of the Lasso even for highly
correlated covariates. However, we also reveal that for moderately correlated
covariates, the prediction performance of the Lasso can be mediocre
irrespective of the choice of the tuning parameter. We finally show that our
results also lead to near-optimal rates for the least-squares estimator with
total variation penalty.
