One Release for All: Private Simple Linear Regression Via Improved Simplex Mapping

We present a single-release extension of DP-RSS for private simple linear regression when both input and output are bounded in $[0,1]$. Rather than privately releasing several related sufficient-statistic estimates, we encode the required information in one unit-sensitivity vector and apply the Laplace mechanism once at the full privacy budget. The required least-squares sufficient statistics are then recovered by deterministic post-processing and used directly in an ordinary least-squares solve. We obtain variance reductions of $1.67$--$2.67\times$ relative to DP-RSS for several statistics. Experiments show that the benefit is most pronounced in the high-privacy regime, while the methods converge as privacy noise becomes negligible.