  Recently emerging large-scale biomedical data pose exciting opportunities for
scientific discoveries. However, the ultrahigh dimensionality and
non-negligible measurement errors in the data may create difficulties in
estimation. There are limited methods for high-dimensional covariates with
measurement error, that usually require knowledge of the noise distribution and
focus on linear or generalized linear models. In this work, we develop
high-dimensional measurement error models for a class of Lipschitz loss
functions that encompasses logistic regression, hinge loss and quantile
regression, among others. Our estimator is designed to minimize the $L_1$ norm
among all estimators belonging to suitable feasible sets, without requiring any
knowledge of the noise distribution. Subsequently, we generalize these
estimators to a Lasso analog version that is computationally scalable to higher
dimensions. We derive theoretical guarantees in terms of finite sample
statistical error bounds and sign consistency, even when the dimensionality
increases exponentially with the sample size. Extensive simulation studies
demonstrate superior performance compared to existing methods in classification
and quantile regression problems. An application to a gender classification
task based on brain functional connectivity in the Human Connectome Project
data illustrates improved accuracy under our approach, and the ability to
reliably identify significant brain connections that drive gender differences.
