Close the consistency-rate gap

Close the gap between the n^{-3/16} rate sufficient for consistency of the Laplace-convolution NPMLE and the n^{-1/2} rate necessary for consistency.

Background

The paper proves that the NPMLE consistently recovers a bounded latent distribution in 1-Wasserstein distance when the Laplace noise scale grows sufficiently slowly, corresponding to a privacy-loss budget that decays more slowly than approximately n^{-3/16} up to logarithmic factors. It also establishes that uniformly consistent recovery by any estimator is impossible when the noise scale is at least of order sqrt(n), corresponding to a privacy-loss budget of order n^{-1/2} or smaller.

These results leave an unresolved range between the sufficient consistency threshold and the impossibility threshold. The stated problem is to determine the sharp transition, or otherwise narrow the gap between the two rates.