Private and polynomial time algorithms for learning Gaussians and beyond
This addresses the challenge of efficient and private statistical estimation for machine learning practitioners, offering incremental improvements in sample complexity over existing works.
The paper tackles the problem of learning Gaussian distributions under differential privacy constraints, presenting a framework that yields polynomial-time algorithms with sample complexities matching or improving upon prior non-efficient or less efficient methods, such as achieving $\widetilde{O}(d^2/α^2 + d^2\sqrt{\ln(1/δ)}/α\varepsilon + d\ln(1/δ) / α\varepsilon)$ for standard learning and $\widetilde{O}(d^{3.5})$ for robust learning.
We present a fairly general framework for reducing $(\varepsilon, δ)$ differentially private (DP) statistical estimation to its non-private counterpart. As the main application of this framework, we give a polynomial time and $(\varepsilon,δ)$-DP algorithm for learning (unrestricted) Gaussian distributions in $\mathbb{R}^d$. The sample complexity of our approach for learning the Gaussian up to total variation distance $α$ is $\widetilde{O}(d^2/α^2 + d^2\sqrt{\ln(1/δ)}/α\varepsilon + d\ln(1/δ) / α\varepsilon)$ matching (up to logarithmic factors) the best known information-theoretic (non-efficient) sample complexity upper bound due to Aden-Ali, Ashtiani, and Kamath (ALT'21). In an independent work, Kamath, Mouzakis, Singhal, Steinke, and Ullman (arXiv:2111.04609) proved a similar result using a different approach and with $O(d^{5/2})$ sample complexity dependence on $d$. As another application of our framework, we provide the first polynomial time $(\varepsilon, δ)$-DP algorithm for robust learning of (unrestricted) Gaussians with sample complexity $\widetilde{O}(d^{3.5})$. In another independent work, Kothari, Manurangsi, and Velingker (arXiv:2112.03548) also provided a polynomial time $(\varepsilon, δ)$-DP algorithm for robust learning of Gaussians with sample complexity $\widetilde{O}(d^8)$.