全支持高斯机制下的固定组成混洗渐近
Fixed-Composition Shuffle Asymptotics in the Full-Support Gaussian Regime
AI总结：
针对二进制输入、固定有限输出字母表且全支持的本地随机化器，研究混洗隐私放大，推导内部固定组成下的协方差和Fisher常数，并证明精确似然比恒等式和回归分解，得到Jensen-Shannon散度和平滑散度的渐近公式。
摘要：
我们研究了针对二进制输入、具有固定有限输出字母表且全支持的本地随机化器的混洗隐私放大。对于包含恰好k个1（共n个用户）的数据集，令T_{n,k}表示混洗直方图分布。在内部固定组成机制下，我们识别了控制相邻对(T_{n,k},T_{n,k+1})的协方差和Fisher常数。正确的协方差为Sigma_pi=(1-pi)Sigma_0+pi Sigma_1，而非混合的多项式协方差。令v=W_1-W_0，所得常数为I_pi=v^T Sigma_pi^+ v。我们证明了精确的似然比恒等式和回归分解，其中残差矩E[R^2]=O(n^{-2})和E[R^4]=O(n^{-4})，对内部组成一致成立。因此，JSD(T_{n,k}||T_{n,k+1})=I_pi/(8n)+O(n^{-2})，且相同的常数控制平滑散度渐近。对于mu_n=(I_pi/n)^{1/2}，在epsilon=t mu_n处的定向曲棍球棒隐私曲线等于mu_n{phi(t)-t Phi(-t)}+O(n^{-1})，对紧集上的t一致成立。还提供了精确的有限n计算公式和固定消息解绑特例。
英文摘要：
We study privacy amplification by shuffling for binary-input local randomizers with a fixed finite output alphabet and full support. For a dataset containing exactly k ones among n users, let T_{n,k} denote the shuffled histogram law. For fixed-composition neighboring shuffled histogram laws in the interior regime, we identify the covariance and Fisher constant governing the neighboring pair (T_{n,k},T_{n,k+1}). For a composition parameter pi in [0,1], the correct covariance is Sigma_pi=(1-pi)Sigma_0+pi Sigma_1 rather than the multinomial covariance of the mixture. With v=W_1-W_0, the resulting constant is I_pi=v^T Sigma_pi^+ v. We prove exact likelihood-ratio identities and a regression decomposition with residual moments E[R^2]=O(n^{-2}) and E[R^4]=O(n^{-4}), uniformly over interior compositions. Consequently, JSD(T_{n,k}||T_{n,k+1})=I_{k/n}/(8n)+O(n^{-2}), and the same constant governs smooth divergence asymptotics. For mu_n=(I_{k/n}/n)^{1/2}, both directed hockey-stick privacy curves at epsilon=t mu_n equal mu_n{phi(t)-t Phi(-t)}+O(n^{-1}) uniformly for t in compact sets. Exact finite-n accounting formulas and fixed-message unbundled specializations are also provided.
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