Consider the problem where an experimenter needs to learn a sparse predictor
with limited budget for sampling the candidate variables. The experimenter can
either use his budget to screen all $p$  variables from $n$ i.i.d. experiments. Or
he can spend part of the budget to pre-screen on m<n experiments followed by
conducting $q>n-m$ experiments using the reduced number of pre-screened
variables.
We introduce a two-stage pre-screening and prediction approach to this problem
that using a method we call Predictive Correlation Screening (PCS) in the
first
stage followed by OLS prediction in the second stage. Our approach is well
suited to small sample sizes and high dimensions. We establish asymptotic
bounds
for the FWER of PCS and the mean square error of the two-stage predictor.