Surrogate Aided Unsupervised Recovery of Sparse Signals in Single Index Models for Binary Outcomes
This addresses the challenge of expensive or difficult-to-obtain true labels in high-dimensional settings like electronic medical records, offering a novel method for coefficient recovery.
The paper tackles the problem of recovering sparse regression coefficients in single index models for binary outcomes using only unlabeled data with a surrogate variable, achieving sharp finite sample performance bounds and demonstrating effectiveness through simulations and a real EMR application.
We consider the recovery of regression coefficients, denoted by $\boldsymbolβ_0$, 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 $\boldsymbolβ_0$ 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.