Private and polynomial time algorithms for learning Gaussians and beyond

• Published in: arXiv.org
• Main Authors: Ashtiani, Hassan, Liaw, Christopher
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 22.06.2022
• Subjects: Algorithms; Complexity; Information theory; Machine learning; Polynomials; Upper bounds
• ISSN: 2331-8422

We present a fairly general framework for reducing \((\varepsilon, \delta)\) differentially private (DP) statistical estimation to its non-private counterpart. As the main application of this framework, we give a polynomial time and \((\varepsilon,\delta)\)-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 \(\alpha\) is \(\widetilde{O}(d^2/\alpha^2 + d^2\sqrt{\ln(1/\delta)}/\alpha \varepsilon + d\ln(1/\delta) / \alpha \varepsilon)\) 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 \((\varepsilon, \delta)\)-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 \((\varepsilon, \delta)\)-DP algorithm for robust learning of Gaussians with sample complexity \(\widetilde{O}(d^8)\).