  Predictor screening rules, which discard predictors from the design matrix
before fitting a model, have had sizable impacts on the speed with which
$\ell_1$-regularized regression problems, such as the lasso, can be solved.
Current state-of-the-art screening rules, however, have difficulties in dealing
with highly-correlated predictors, often becoming too conservative. In this
paper, we present a new screening rule to deal with this issue: the Hessian
Screening Rule. The rule uses second-order information from the model in order
to provide more accurate screening as well as higher-quality warm starts. In
our experiments on $\ell_1$-regularized least-squares (the lasso) and logistic
regression, we show that the rule outperforms all other alternatives in
simulated experiments with high correlation, as well as in the majority of real
datasets that we study.
