Devising mechanisms with good beyond-worst-case input-dependent performance has been an important focus of differential privacy, with techniques such as smooth sensitivity, propose-test-release, or inverse sensitivity mechanism being developed to achieve this goal. This makes it very natural to use the notion of universal optimality in differential privacy. Universal optimality is a strong instance-specific optimality guarantee for problems on weighted graphs, which roughly states that for any fixed underlying (unweighted) graph, the algorithm is optimal in the worst-case sense, with respect to the possible setting of the edge weights. In this paper, we give the first such result in differential privacy. Namely, we prove that a simple differentially private mechanism for approximately releasing the minimum spanning tree is near-optimal in the sense of universal optimality for the $\ell_1$ neighbor relation. Previously, it was only known that this mechanism is nearly optimal in the worst case. We then focus on the $\ell_\infty$ neighbor relation, for which the described mechanism is not optimal. We show that one may implement the exponential mechanism for MST in polynomial time, and that this results in universal near-optimality for both the $\ell_1$ and the $\ell_\infty$ neighbor relations.

翻译：开发具有良好超越最坏情况输入依赖性能的机制一直是差分隐私的重要关注点，为此发展了平滑敏感度、提出-测试-发布或逆敏感度机制等技术来实现这一目标。这使得在差分隐私中引入通用最优性的概念变得非常自然。通用最优性是对加权图问题的一种强实例特定最优性保证，其大致含义是：对于任意固定的底层（未加权）图，该算法在边权重的可能设置下是最坏情况意义上的最优。在本文中，我们给出了差分隐私领域的首个此类结果。具体而言，我们证明了一种用于近似发布最小生成树的简单差分隐私机制，在$\ell_1$邻接关系下满足通用最优性意义上的近似最优性。此前，仅已知该机制在最坏情况下是近似最优的。然后，我们聚焦于$\ell_0$邻接关系，对于该关系，所描述的机制并非最优。我们表明，可以在多项式时间内实现用于最小生成树的指数机制，并且这能在$\ell_1$和$\ell_0$邻接关系下都实现通用近似最优性。