Error bounds for sparse classifiers in high-dimensions

TL;DR

This paper establishes an L2 recovery bound for sparse classifiers in high-dimensional settings, applicable to various loss functions and estimators, and introduces a generalized Slope estimator with adaptive sparsity features.

Contribution

It provides new theoretical bounds for sparse estimators with hinge and logistic loss, and introduces a generalized Slope estimator with an efficient algorithm.

Findings

Bound scales as (k*/n) log(p/k*) for coefficients estimation.

The generalized Slope estimator adapts to unknown sparsity.

Empirical results match the best existing bounds.

Abstract

We prove an L2 recovery bound for a family of sparse estimators defined as minimizers of some empirical loss functions -- which include hinge loss and logistic loss. More precisely, we achieve an upper-bound for coefficients estimation scaling as (k*/n)\log(p/k*): n,p is the size of the design matrix and k* the dimension of the theoretical loss minimizer. This is done under standard assumptions, for which we derive stronger versions of a cone condition and a restricted strong convexity. Our bound holds with high probability and in expectation and applies to an L1-regularized estimator and to a recently introduced Slope estimator, which we generalize for classification problems. Slope presents the advantage of adapting to unknown sparsity. Thus, we propose a tractable proximal algorithm to compute it and assess its empirical performance. Our results match the best existing bounds for classification and regression problems.