Differentially Private Learning of Exponential Distributions: Simple Algorithms and Tight Bounds
This work addresses the challenge of private statistical estimation for exponential distributions, which is incremental as it builds on existing differential privacy techniques but introduces a novel parameter-free approach.
The paper tackles the problem of learning exponential distributions under differential privacy by developing a simple algorithm that privately estimates the rate parameter without dependence on its true value, achieving near-optimal sample efficiency and tight bounds. It extends the method to Pareto distributions and provides matching lower bounds, offering the first tight characterization for this task.
We study the problem of learning exponential distributions under differential privacy. Given $n$ i.i.d.\ samples from $\mathrm{Exp}(λ)$, the goal is to privately estimate $λ$ so that the learned distribution is close in total variation distance to the truth. We present a simple pure $ε$-differentially private algorithm that avoids the classical dependence on the true value of $λ$. Our method leverages a structural property of the exponential distribution: its $(1-1/e)$-quantile equals $1/λ$, 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 $(ε,δ)$-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 $λ$-free approach.