Nearly-Optimal Private Selection via Gaussian Mechanism
Ethan Leeman, Pasin Manurangsi
TL;DR
This work addresses private selection under differential privacy using the Gaussian mechanism with a budget $\rho$. By combining subsampling, a gap-focused recursive strategy, and a final BinTree step, the authors achieve an error that scales as $\tilde{O}\!\left(\dfrac{\log |\mathcal{Y}|}{\sqrt{\rho}}\right)$ up to polylog factors, nearly matching the Exponential Mechanism and improving upon Steinke's previous $O(\log^{3/2} |\mathcal{Y}|/\sqrt{\rho})$ bound. The analysis introduces a careful construction of good events and a recursive inductive proof to bound probabilities and errors, with a high-probability guarantee that translates into an expected-error bound via a combining step. The results demonstrate that nearly-optimal private selection is achievable with adaptive Gaussian queries on low-sensitivity losses, and they discuss future directions such as removing lower-order polylog terms, extending to Laplace noise, and handling infinite candidate sets. The work advances practical DP tools for selection tasks and highlights a pathway to close the gap with the Exponential Mechanism in finite settings.
Abstract
Steinke (2025) recently asked the following intriguing open question: Can we solve the differentially private selection problem with nearly-optimal error by only (adaptively) invoking Gaussian mechanism on low-sensitivity queries? We resolve this question positively. In particular, for a candidate set $\mathcal{Y}$, we achieve error guarantee of $\tilde{O}(\log |\mathcal{Y}|)$, which is within a factor of $(\log \log |\mathcal{Y}|)^{O(1)}$ of the exponential mechanism (McSherry and Talwar, 2007). This improves on Steinke's mechanism which achieves an error of $O(\log^{3/2} |\mathcal{Y}|)$.