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.