Title: Surrogate Aided Unsupervised Recovery of Sparse Signals in Single Index Models for Binary Outcomes,
Abstract: We consider the recovery of regression coefficients, denoted by
$\boldsymbol{\beta}_0$, for a single index model (SIM) relating a binary
outcome $Y$ to a set of possibly high dimensional covariates $\boldsymbol{X}$,
based on a large but 'unlabeled' dataset $\mathcal{U}$, with $Y$ never
observed. On $\mathcal{U}$, we fully observe $\boldsymbol{X}$ and additionally,
a surrogate $S$ which, while not being strongly predictive of $Y$ throughout
the entirety of its support, can forecast it with high accuracy when it assumes
extreme values. Such datasets arise naturally in modern studies involving large
databases such as electronic medical records (EMR) where $Y$, unlike
$(\boldsymbol{X}, S)$, is difficult and/or expensive to obtain. In EMR studies,
an example of $Y$ and $S$ would be the true disease phenotype and the count of
the associated diagnostic codes respectively. Assuming another SIM for $S$
given $\boldsymbol{X}$, we show that under sparsity assumptions, we can recover
$\boldsymbol{\beta}_0$ proportionally by simply fitting a least squares LASSO
estimator to the subset of the observed data on $(\boldsymbol{X}, S)$
restricted to the extreme sets of $S$, with $Y$ imputed using the surrogacy of
$S$. We obtain sharp finite sample performance bounds for our estimator,
including deterministic deviation bounds and probabilistic guarantees. We
demonstrate the effectiveness of our approach through multiple simulation
studies, as well as by application to real data from an EMR study conducted at
the Partners HealthCare Systems.