We consider high-dimensional regression with a count response modeled by Poisson or negative binomial generalized linear model (GLM). We propose a penalized maximum likelihood estimator with a properly chosen complexity penalty and establish its adaptive minimaxity across models of various sparsity. To make the procedure computationally feasible for high-dimensional data we consider its LASSO and SLOPE convex surrogates. Their performance is illustrated through simulated and real-data examples.