混洗模型中的私有向量均值估计：最佳速率需要许多消息

我们研究了隐私混洗模型中的隐私向量均值估计问题，其中 nnn 个用户各自在 ddd 维度中都有一个单位向量。我们提出了一种新的多消息协议，该协议使用 O~(min⁡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)O~(min(nε2,d)) 条消息/用户实现最优误差。此外，我们表明，任何实现最优误差的（无偏）协议都要求每个用户发送 Ω(min⁡(nε2,d)/log⁡(n))\Omega(\min(n\varepsilon^2,d)/\log(n))Ω(min(nε2,d)/log(n)) 条消息，证明了我们的消息复杂度在对数因子范围内的最优性。

nnn nn $nn$ nn n n n n n n ddd dd $dd$ dd div> d d d d d d O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)O~(min(nε2,d)) O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) div> O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(minâ¡(nε2,d)) O~ O ~ (minâ¡(nε2,d)) ( min â¡ ( n ε2 ε 2 , d ) ) \tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(min(nε2,d)) O~(min(nε2,d)) O~ O~ O~ O~ O O ~ ~ ~ (min(n ε2,d)) ( ( min ( n ε2 ε 2 2 2 2 2 2 2 , d ) ) ) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\ min(n\varepsilon^2,d)/\log(n))Ω(min(nε2,d)/log(n)) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) > Ω(minâ¡(nε2,d)/logâ¡(n)) Ω ( min â¡ ( n ε2 ε 2 , d ) / log â¡ ( n ) ) \Omega(\min(n\varepsilon^2,d) )/\log(n)) Ω(min(nε2,d)/log(n)) Ω(min(nε2,d)/log(n)) Ω ( min ( n ε2 ε 2 2 2 2 2 2 2 , d ) div> / log g ( n )) O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})O(dnd/(d+2)Δ4/(d+2)) O(dnd/(d+2)Δ4/(d+2))\ma thcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)εâ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)Δ4/(d+2)) O ( d > nd/(d+2) d d d d d d O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)O~(min(nε2,d)) O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) div> O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(minâ¡(nε2,d)) O~ O ~ (minâ¡(nε2,d)) ( min â¡ ( n ε2 ε 2 , d ) ) \tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(min(nε2,d)) O~(min(nε2,d)) O~ O~ O~ O~ O O ~ ~ ~ (min(n ε2,d)) ( ( min ( n ε2 ε 2 2 2 2 2 2 2 , d ) ) ) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\ min(n\varepsilon^2,d)/\log(n))Ω(min(nε2,d)/log(n)) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) > Ω(minâ¡(nε2,d)/logâ¡(n)) Ω ( min â¡ ( n ε2 ε 2 , d ) / log â¡ ( n ) ) \Omega(\min(n\varepsilon^2,d) )/\log(n)) Ω(min(nε2,d)/log(n)) Ω(min(nε2,d)/log(n)) Ω ( min ( n ε2 ε 2 2 2 2 2 2 2 , d ) div> / log g ( n )) O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})O(dnd/(d+2)Δ4/(d+2)) O(dnd/(d+2)Δ4/(d+2))\ma thcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)εâ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)Δ4/(d+2)) O ( d > nd/(d+2) $O~(minâ¡(nÎ\mu 2,d))\backslash tilde\{\backslash mathcal\{O\}\}\backslash left(\backslash min(n\backslash varepsilon^2,d)\backslash right)$ O~(minâ¡(nε2,d))\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(minâ¡(nε2,d)) O~ O ~ (minâ¡(nε2,d)) ( min â¡ ( n ε2 ε 2 , d ) ) \tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right) O~(min(nε2,d)) O~(min(nε2,d)) O~ O~ O~ O~ O O ~ ~ ~ (min(n ε2,d)) ( ( min ( n ε2 ε 2 2 2 2 2 2 2 , d ) ) ) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\ min(n\varepsilon^2,d)/\log(n))Ω(min(nε2,d)/log(n)) Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) $Î\copyright (minâ¡(nÎ\mu 2,d)/logâ¡(n))\backslash Omega(\backslash min(n\backslash varepsilon^2,d)/\backslash log(n))$ Ω(minâ¡(nε2,d)/logâ¡(n))\Omega(\min(n\varepsilon^2,d)/\log(n)) > Ω(minâ¡(nε2,d)/logâ¡(n)) Ω ( min â¡ ( n ε2 ε 2 , d ) / log â¡ ( n ) ) \Omega(\min(n\varepsilon^2,d) )/\log(n)) Ω(min(nε2,d)/log(n)) Ω(min(nε2,d)/log(n)) Ω ( min ( n ε2 ε 2 2 2 2 2 2 2 , d ) div> / log g ( n )) O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})O(dnd/(d+2)Δ4/(d+2)) O(dnd/(d+2)Δ4/(d+2))\ma thcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)εâ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)Δ4/(d+2)) O ( d > nd/(d+2) / log g ( n )) O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})O(dnd/(d+2)Δ4/(d+2)) O(dnd/(d+2)Δ4/(d+2))\ma thcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) $O(dnd/(d+2)Î\mu â4/(d+2))\backslash mathcal\{O\}(dn^\{d/(d+2)\}\backslash varepsilon^\{-4/(d+2)\})$ O(dnd/(d+2)Δ4/(d+2))\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)}) O(dnd/(d+2)Δ4/(d+2)) O ( d nd/(d+2) n d/(d+2) d / ( d + 2 ) 4/(d+2) ε 4/(d+2) 4 / ( d + 2 ) ) 哦 ( d n d / ( d +