PRIMO: Private Regression in Multiple Outcomes

We introduce a new private regression setting we call Private Regression in Multiple Outcomes (PRIMO), inspired by the common situation where a data analyst wants to perform a set of $l$ regressions while preserving privacy, where the features $X$ are shared across all $l$ regressions, and each regression $i \in [l]$ has a different vector of outcomes $y_i$. Naively applying existing private linear regression techniques $l$ times leads to a $\sqrt{l}$ multiplicative increase in error over the standard linear regression setting. We apply a variety of techniques including sufficient statistics perturbation (SSP) and geometric projection-based methods to develop scalable algorithms that outperform this baseline across a range of parameter regimes. In particular, we obtain no dependence on l in the asymptotic error when $l$ is sufficiently large. Empirically, on the task of genomic risk prediction with multiple phenotypes we find that even for values of $l$ far smaller than the theory would predict, our projection-based method improves the accuracy relative to the variant that doesn't use the projection.