  In high dimensional settings where a small number of regressors are expected
to be important, the Lasso estimator can be used to obtain a sparse solution
vector with the expectation that most of the non-zero coefficients are
associated with true signals. While several approaches have been developed to
control the inclusion of false predictors with the Lasso, these approaches are
limited by relying on asymptotic theory, having to empirically estimate terms
based on theoretical quantities, assuming a continuous response class with
Gaussian noise and design matrices, or high computation costs. In this paper we
show how: (1) an existing model (the SQRT-Lasso) can be recast as a method of
controlling the number of expected false positives, (2) how a similar estimator
can used for all other generalized linear model classes, and (3) this approach
can be fit with existing fast Lasso optimization solvers. Our justification for
false positive control using randomly weighted self-normalized sum theory is to
our knowledge novel. Moreover, our estimator's properties hold in finite
samples up to some approximation error which we find in practical settings to
be negligible under a strict mutual incoherence condition.
