Isotonic regression meets LASSO

This paper studies a two step procedure for monotone increasing additive single index models with Gaussian designs. The proposed procedure is simple, easy to implement with existing software, and consists of consecutively applying LASSO and isotonic regression. Aside from formalizing this procedure, we provide theoretical guarantees regarding its performance: 1) we show that our procedure controls the in-sample squared error; 2) we demonstrate that one can use the procedure for predicting new observations, by showing that the absolute prediction error can be controlled with high-probability. Our bounds show a tradeoff of two rates: the mini-max rate for estimating high dimensional quadratic loss, and the minimax nonparametric rate for estimating a monotone increasing function.