Differentially Private Learning of Exponential Distributions: Simple 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 a simple pure $\epsilon$-differentially private algorithm that avoids the classical dependence on the true value of $\lambda$. Our method leverages a structural property of the exponential distribution: its $(1-1/e)$-quantile equals $1/\lambda$, allowing us to estimate the rate parameter directly via private quantile estimation. The resulting learner is both conceptually simple and sample-efficient, achieving near-optimal guarantees. We further extend the method to Pareto distributions via a logarithmic reduction, prove nearly matching lower bounds using group privacy arguments, and show how approximate $(\epsilon,\delta)$-DP removes the need for externally supplied parameter bounds. Together, these results give the first tight characterization of exponential distribution learning under differential privacy using a simple $\lambda$-free approach.