Two-Stage Sampling, Prediction and Adaptive Regression via Correlation Screening

This paper proposes a general adaptive procedure for budget-limited predictor design in high dimensions called two-stage Sampling, Prediction and Adaptive Regression via Correlation Screening (SPARCS). The SPARCS can be applied to high-dimensional prediction problems in experimental science, medicine, finance, and engineering, as illustrated by the following. Suppose that one wishes to run a sequence of experiments to learn a sparse multivariate predictor of a dependent variable Y (disease prognosis for instance) based on a p dimensional set of independent variables X = [X
₁
,..., X
_p
]
^T
(assayed biomarkers). Assume that the cost of acquiring the full set of variables X increases linearly in its dimension. The SPARCS breaks the data collection into two stages in order to achieve an optimal tradeoff between sampling cost and predictor performance. In the first stage, we collect a few (n) expensive samples {y
_i
, s
_i
}
_i=1
ⁿ
, at the full dimension p ≫ n of X, winnowing the number of variables down to a smaller dimension l <; p using a type of cross correlation or regression coefficient screening. In the second stage, we collect a larger number (t - n) of cheaper samples of the l variables that passed the screening of the first stage. At the second stage, a low-dimensional predictor is constructed by solving the standard regression problem using all t samples of the selected variables. The SPARCS is an adaptive online algorithm that implements false positive control on the selected variables, is well suited to small sample sizes, and is scalable to high dimensions. We establish asymptotic bounds for the familywise error rate, specify high dimensional convergence rates for support recovery, and establish optimal sample allocation rules to the first and second stages.