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

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.