  In variable or graph selection problems, finding a right-sized model or
controlling the number of false positives is notoriously difficult. Recently, a
meta-algorithm called Stability Selection was proposed that can provide
reliable finite-sample control of the number of false positives. Its benefits
were demonstrated when used in conjunction with the lasso and orthogonal
matching pursuit algorithms.
  In this paper, we investigate the applicability of stability selection to
structured selection algorithms: the group lasso and the structured
input-output lasso. We find that using stability selection often increases the
power of both algorithms, but that the presence of complex structure reduces
the reliability of error control under stability selection. We give strategies
for setting tuning parameters to obtain a good model size under stability
selection, and highlight its strengths and weaknesses compared to competing
methods screen and clean and cross-validation. We give guidelines about when to
use which error control method.
