Model selection consistency of U-statistics with convex loss and weighted lasso penalty

In the paper we consider minimisation of U-statistics with the weighted Lasso penalty and investigate their asymptotic properties in model selection and estimation. We prove that the use of appropriate weights in the penalty leads to the procedure that behaves like the oracle that knows the true model in advance, i.e. it is model selection consistent and estimates nonzero parameters with the standard rate. For the unweighted Lasso penalty, we obtain sufficient and necessary conditions for model selection consistency of estimators. The obtained results strongly based on the convexity of the loss function that is the main assumption of the paper. Our theorems can be applied to the ranking problem as well as generalised regression models. Thus, using U-statistics we can study more complex models (better describing real problems) than usually investigated linear or generalised linear models.