Differentially Private Ordinary Least Squares

More specifically, we use Theorem B.1 from (Sheffet, 2015) that states that given a matrix A whose all of its singular values at greater than T ( , δ) where T ( , δ) = 2B (√ 2r ln(4/δ) + 2 ln(4/δ) ) , publishing RA is ( , δ)differentially private for a r-row matrix R whose entries sampled are i.i.d normal Gaussians. Since we have that all of the singular values of A′ are greater than w (as specified in Algorithm 1), outputtingRA′ is ( /2, δ/2)-differentially private. The rest of the proof boils down to showing that (i) the if-else-condition is ( /2, 0)-differentially private and that (ii) w.p. ≤ δ/2 any matrix A whose smallest singular value is smaller than w passes the if-condition (step 3). If both these facts hold, then knowing whether we pass the if-condition or not is ( /2)-differentially private and the output of the algorithm is ( /2, δ)-differentially private, hence basic composition gives the overall bound of ( , δ)differential privacy.