Efficient and Differentially Private Statistical Estimation via a Sum-of-Squares Exponential Mechanism

Abstract

As machine learning is applied to more privacy-sensitive data, it is becoming increasingly crucial to develop algorithms that maintain privacy. However, even the most basic high-dimensional statistical estimation tasks were not fully understood under differential privacy, specifically, there were no known efficient algorithms for mean estimation using the optimal number of samples under pure differential privacy.

We propose a new method for designing efficient and information-theoretically optimal algorithms for statistical estimation tasks that preserve privacy, using a combination of the Sum-of-Squares hierarchy and the exponential mechanism. The Sum-of-Squares hierarchy, a convex programming method, has been used to design efficient algorithms in robust statistics. The exponential mechanism, which has been widely used in differential privacy, is often used to design information-theoretically optimal algorithms, but can be inefficient. By combining these two approaches, we are able to create efficient algorithms that are also information-theoretically optimal. We apply this approach to mean estimation for heavy-tailed distributions and learning Gaussian distributions and achieve optimal results. We also show that this approach can be applied to other problems captured by the Sum-of-Squares hierarchy through a meta-theorem. Additionally, our algorithms highlight the strong connection between robustness and privacy.

We establish information-theoretical lower bounds to show the statistical optimality of our approaches. Technically we use packing lower bounds; however, the novelty of our lower bounds is in capturing the high probability setting.