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

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