General nonexact oracle inequalities for classes with a subexponential envelope

TL;DR

This paper establishes nonexact oracle inequalities for empirical risk minimization with classes having subexponential envelopes, enabling fast rates without boundedness or incoherence assumptions, applicable to regularization methods like  and nuclear norms.

Contribution

It introduces nonexact oracle inequalities in an unbounded setting with subexponential envelopes, allowing fast convergence rates without RIP or incoherence conditions.

Findings

Oracle inequalities with residuals decreasing as 1/n for  and nuclear norm regularizations.

Fast rates achieved under subexponential tail assumptions without boundedness.

Applicability to convex aggregation and model selection problems.

Abstract

We show that empirical risk minimization procedures and regularized empirical risk minimization procedures satisfy nonexact oracle inequalities in an unbounded framework, under the assumption that the class has a subexponential envelope function. The main novelty, in addition to the boundedness assumption free setup, is that those inequalities can yield fast rates even in situations in which exact oracle inequalities only hold with slower rates. We apply these results to show that procedures based on $\ell_1$ and nuclear norms regularization functions satisfy oracle inequalities with a residual term that decreases like $1/n$ for every $L_q$-loss functions ($q\geq2$), while only assuming that the tail behavior of the input and output variables are well behaved. In particular, no RIP type of assumption or "incoherence condition" are needed to obtain fast residual terms in those setups. We also apply these results to the problems of convex aggregation and model selection.