Analyze Gauss: Optimal Bounds for Privacy-Preserving Principal Component Analysis

We consider the problem of privately releasing a low dimensional approximation to a set of data records, represented as a matrix A in which each row corresponds to an individual and each column to an attribute. Our goal is to compute a subspace that captures the covariance of A as much as possible, classically known as principal component analysis (PCA). We assume that each row of A has l(2) norm bounded by one, and the privacy guarantee is defined with respect to addition or removal of any single row. We show that the well-known, but misnamed, randomized response algorithm, with properly tuned parameters, provides nearly optimal additive quality gap compared to the best possible singular subspace of A. We further show that when A(T)A has a large eigenvalue gap - a reason often cited for PCA - the quality improves significantly. Optimality (up to logarithmic factors) is proved using techniques inspired by the recent work of Bun, Ullman, and Vadhan on applying Tardos's fingerprinting codes to the construction of hard instances for private mechanisms for 1-way marginal queries. Along the way we define a list culling game which may be of independent interest. By combining the randomized response mechanism with the well-known following the perturbed leader algorithm of Kalai and Vempala we obtain a private online algorithm with nearly optimal regret. The regret of our algorithm even outperforms all the previously known online non-private algorithms of this type. We achieve this better bound by, satisfyingly, borrowing insights and tools from differential privacy!