Fast and Optimal Algorithms for Private Hypothesis Selection

We study the problem of private hypothesis selection: given samples from an unknown distribution drawn from a finite hypothesis class, the goal is to identify the best hypothesis under the constraint of differential privacy. Existing algorithms for this problem are either computationally expensive or achieve sub-optimal statistical rates. We propose new algorithms that achieve near-optimal rates while running in nearly linear time in the number of hypotheses. Rather than applying the exponential mechanism directly with a score function that requires pairwise comparisons between hypotheses, our approach introduces a carefully designed loss function based on a small set of strong hypotheses. This structure allows the score to be evaluated efficiently for most hypotheses, yielding significant computational savings. We further extend our algorithms to the agnostic setting, where the true distribution may not belong to the hypothesis class. As an application, we obtain faster differentially private algorithms for universal statistical estimation in low dimensional settings.