  This paper studies the non-asymptotic merits of the double
$\ell_1$-regularized for heterogeneous overdispersed count data via negative
binomial regressions. Under the restricted eigenvalue conditions, we prove the
oracle inequalities for Lasso estimators of two partial regression coefficients
for the first time, using concentration inequalities of empirical processes.
Furthermore, derived from the oracle inequalities, the consistency and
convergence rate for the estimators are the theoretical guarantees for further
statistical inference. Finally, both simulations and a real data analysis
demonstrate that the new methods are effective.
