Differentially Private Learning of Exponential Distributions: Adaptive Algorithms and Tight Bounds

We study the problem of learning exponential distributions under differential privacy. Given n i.i.d. samples from \mathrm{Exp}(\lambda), the goal is to privately estimate \lambda so that the learned distribution is close in total variation distance to the truth. We present two complementary pure DP algorithms: one adapts the classical maximum likelihood estimator via clipping and Laplace noise, while the other leverages the fact that the (1-1/e)-quantile equals 1/\lambda. Each method excels in a different regime, and we combine them into an adaptive best-of-both algorithm achieving near-optimal sample complexity for all \lambda. We further extend our approach to Pareto distributions via a logarithmic reduction, prove nearly matching lower bounds using packing and group privacy \cite{Karwa2017FiniteSD}, and show how approximate (\epsilon,\delta)-DP removes the need for externally supplied bounds. Together, these results give the first tight characterization of exponential distribution learning under DP and illustrate the power of adaptive strategies for heavy-tailed laws.

Recommended citation: Mahpud, Bar; Sheffet, Or. (2025). "Differentially Private Learning of Exponential Distributions: Adaptive Algorithms and Tight Bounds." (under review).
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