Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions

We obtain estimation error rates and sharp oracle inequalities for regularization procedures of the form f̂ ∈ argmin f∈F ( 1 N N ∑ i=1 `(f(Xi), Yi) + λ ‖f‖ ) when ‖·‖ is any norm, F is a convex class of functions and ` is a Lipschitz loss function satisfying a Bernstein condition over F . We explore both the bounded and subgaussian stochastic frameworks for the distribution of the f(Xi)’s, with no assumption on the distribution of the Yi’s. The general results rely on two main objects: a complexity function, and a sparsity equation, that depend on the specific setting in hand (loss ` and norm ‖·‖). As a proof of concept, we obtain minimax rates of convergence in the following problems: 1) matrix completion with any Lipschitz loss function, including the hinge and logistic loss for the so-called 1-bit matrix completion instance of the problem, and quantile losses for the general case, which enables to estimate any quantile on the entries of the matrix; 2) logistic LASSO and variants such as the logistic SLOPE; 3) kernel methods, where the loss is the hinge loss, and the regularization function is the RKHS norm.