A local stochastic Lipschitz condition with application to Lasso for high dimensional generalized linear models

For regularized estimation, the upper tail behavior of the random Lipschitz coefficient associated with empirical loss functions is known to play an important role in the error bound of Lasso for high dimensional generalized linear models. The upper tail behavior is known for linear models but much less so for nonlinear models. We establish exponential type inequalities for the upper tail of the coefficient and illustrate an application of the results to Lasso likelihood estimation for high dimensional generalized linear models.