AI 中文总结
研究通过随机机制一致且差分隐私地释放计数查询输出的问题，推导最小错误概率闭式表达式及最优机制，扩展框架到与随机变换级联设置，应用于AWGN信道上的PSK传输，证明高隐私下无编码传输最优。
AI 中文摘要
我们研究了通过一种既一致又具有\((\epsilon,\delta)\)-差分隐私的随机机制来释放计数查询输出的问题。一致性要求释放的值在查询的可行范围内，而效用通过最坏情况错误概率来衡量。我们首先推导了最小可实现错误概率的闭式表达式并获得了显式最优机制。通过利用该机制满足的主动差分隐私约束，我们通过传播论证刻画了整个最优机制类，识别了所有优化器共有的结构特性。接着我们将框架扩展到隐私机制与任意固定随机变换级联的设置，该变换代表源与目的地之间通信介质的预定部分。我们首先建立了介质的这种部分固定不会导致效用损失的充要条件。然后基于凸混合和谱扰动推导了最优可实现性能的上下界。最后，我们将该理论应用于加性高斯白噪声（AWGN）信道上的（M）-ary相移键控（PSK）传输，并表明在高隐私 regime 中无编码传输实际上是最优的。
英文摘要
We study the problem of releasing counting-query outputs through a stochastic mechanism that is both consistent and \((ε,δ)\)-differentially private. Consistency requires the released value to lie within the feasible range of the query, while utility is measured by the worst-case probability of error. We first derive a closed-form expression for the minimum achievable error probability and obtain an explicit optimal mechanism. By exploiting the active differential privacy constraints satisfied by this mechanism, we then characterize the entire class of optimal mechanisms via a propagation argument, identifying the structural properties shared by all optimizers. We next extend the framework to the setting in which the privacy mechanism is cascaded with an arbitrary fixed stochastic transformation representing a predetermined portion of the communication medium between the source and the destination. We first establish necessary and sufficient conditions under which this partial fixation of the medium incurs no loss in utility. We then derive upper and lower bounds on the optimal achievable performance based on convex mixing and spectral perturbation. Finally, we apply the theory to (M)-ary phase-shift keying (PSK) transmission over an additive white Gaussian noise (AWGN) channel and show that uncoded transmission is effectively optimal in the high-privacy regime.
