We consider a discrete distribution estimation problem under the local differential privacy and the one-bit communication constraints. A fundamental privacy-utility tradeoff in this problem is formulated as the minimax squared loss. We show a tighter lower bound on the minimax squared loss, which has exactly the same form with the upper bound by the recursive Hadamard response by Chen et al. up to a constant factor of 4 for arbitrary LDP constraint and arbitrary finite data space. To derive the lower bound, we modify the van Trees inequality to involve a symmetrized Fisher information, which is invariant under the choice of the coordinate system on the probability simplex. We further characterize the maximum of the symmetrized Fisher information by considering the joint effect of the privacy and the communication constraints.

中文翻译：

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一位通信约束下局部差分私有离散分布估计的更严格逆

我们考虑局部差分隐私和一位通信约束下的离散分布估计问题。在这个问题中，一个基本的隐私效用权衡被表述为 minimax squared loss。我们在 minimax squared loss 上展示了一个更严格的下限，它与 Chen 等人的递归 Hadamard 响应的上限具有完全相同的形式。对于任意 LDP 约束和任意有限数据空间，最多为 4 的常数因子。为了推导出下界，我们修改 van Trees 不等式以涉及对称的 Fisher 信息，该信息在概率单纯形上的坐标系选择下是不变的。我们通过考虑隐私和通信约束的联合效应进一步表征了对称Fisher信息的最大值。