差分隐私让尾部风险学习的有效样本量从n降至nτ，带来额外隐私代价。
The Privacy Price of Tail-Risk Learning: Effective Tail Sample Size in Differentially Private CVaR Optimization
- 用有效尾部样本量nτ替代n，揭示隐私对尾部风险学习的影响
- 在纯差分隐私下，尾部风险误差率达到Θ(1/(εnτ))
- 适用于关注隐私保护的金融风控、极端事件建模的研究者
差分隐私会改变CVaR学习的有效样本量。对于尾部质量τ，隐私相关的样本量不是n，而是nτ；等价地，有效私有尾部样本量为εnτ。私有CVaR超额风险可分解为普通尾部风险统计误差与隐私代价。该分解在标量估计和有限类情况下是完整的：标量估计的误差率为Θ(B min{1, (nτ)^{-1/2} + (εnτ)^{-1}})，有限类规模为M时的误差率为Θ(B min{1, √(log(2M)/(nτ)) + log(2M)/(εnτ)})。这些完整速率在纯差分隐私下成立，其下界在近似差分隐私的小δ范围内也成立。对于凸Lipschitz学习，模块化上下界归约表明，专用于CVaR的隐私项必然以1/(εnτ)缩放，维度依赖继承自私有随机凸优化。这些结果将私有CVaR学习中的核心难题识别为在Θ(nτ)个有信息的尾部记录上进行私有学习。
原文摘要 · Abstract (English)
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.
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