Weighted Lasso Estimates for Sparse Logistic Regression: Non-asymptotic Properties with Measurement Error

TL;DR

This paper introduces weighted Lasso estimates for high-dimensional sparse logistic regression, providing non-asymptotic theoretical guarantees and demonstrating improved performance through simulations and real data applications.

Contribution

The paper proposes two new weighted Lasso methods based on covariates using McDiarmid inequality, with non-asymptotic bounds and practical evaluation.

Findings

Non-asymptotic oracle inequalities established for the proposed methods.

Improved estimation and prediction errors compared to previous weighted estimates.

Validated effectiveness through simulations and real data analysis.

Abstract

When we are interested in high-dimensional system and focus on classification performance, the $\ell_{1}$-penalized logistic regression is becoming important and popular. However, the Lasso estimates could be problematic when penalties of different coefficients are all the same and not related to the data. We proposed two types of weighted Lasso estimates depending on covariates by the McDiarmid inequality. Given sample size $n$ and dimension of covariates $p$, the finite sample behavior of our proposed methods with a diverging number of predictors is illustrated by non-asymptotic oracle inequalities such as $\ell_{1}$-estimation error and squared prediction error of the unknown parameters. We compare the performance of our methods with former weighted estimates on simulated data, then apply these methods to do real data analysis.