The Privacy Price of Tail-Risk Learning: Effective Tail Sample Size in Differentially Private CVaR Optimization

Differential privacy changes the effective sample size governing CVaR learning. For tail mass $τ$, the privacy-relevant sample size is not $n$, but $nτ$; equivalently, the effective private tail sample size is $εnτ$. Private CVaR excess risk decomposes into ordinary tail-risk statistical error and a privacy price. This decomposition is complete for scalar estimation and finite classes: scalar estimation has rate $Θ(B \min\{1,(nτ)^{-1/2}+(εnτ)^{-1}\})$, and finite classes of size $M$ have rate $Θ(B \min\{1,\sqrt{\log(2M)/(nτ)}+\log(2M)/(εnτ)\})$. These complete rates hold under pure DP, and their lower bounds extend to approximate DP in the stated small-$δ$ regimes. For convex Lipschitz learning, modular upper and lower reductions show that the CVaR-specific privacy term necessarily scales as $1/(εnτ)$, with dimension dependence inherited from private stochastic convex optimization. Together, these results identify ordinary private learning on $Θ(nτ)$ informative tail records as the canonical hard subproblem inside private CVaR learning.