PRIMO: Private Regression in Multiple Outcomes
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We introduce a new differentially private regression setting we call Private Regression in Multiple Outcomes (PRIMO), inspired the common situation where a data analyst wants to perform a set of l regressions while preserving privacy, where the covariates X are shared across all l regressions, and each regression i ∈ [l] has a different vector of outcomes y_i. While naively applying private linear regression techniques l times leads to a √(l) multiplicative increase in error over the standard linear regression setting, in Subsection 4.1 we modify techniques based on sufficient statistics perturbation (SSP) to yield greatly improved dependence on l. In Subsection 4.2 we prove an equivalence to the problem of privately releasing the answers to a special class of low-sensitivity queries we call inner product queries. Via this equivalence, we adapt the geometric projection-based methods from prior work on private query release to the PRIMO setting. Under the assumption the labels Y are public, the projection gives improved results over the Gaussian mechanism when n < l√(d), with no asymptotic dependence on l in the error. In Subsection 4.3 we study the complexity of our projection algorithm, and analyze a faster sub-sampling based variant in Subsection 4.4. Finally in Section 5 we apply our algorithms to the task of private genomic risk prediction for multiple phenotypes using data from the 1000 Genomes project. We find that for moderately large values of l our techniques drastically improve the accuracy relative to both the naive baseline that uses existing private regression methods and our modified SSP algorithm that doesn't use the projection.
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