High-dimensional and nonparametric estimation under "local" information constraints
In this talk, I will discuss estimation using interactive protocols subject to "local information constraints." Those constraints include, among others, local differential privacy (LDP), communication constraints, and a large array of restricted measurements. I will describe a general framework which lets us obtain tight minimax rate lower bounds for parameter estimation for a range of families, including density estimation of discrete probability distributions, and mean estimation for product distributions over the hypercube and high-dimensional (sparse or not) Gaussians.
I will then describe recent results on nonparametric estimation of Besov densities under one of these local constraints (communication), with a rate-optimal adaptive estimator; and explain how the lower bound framework above can be used to prove its optimality.
This talk is based on joint works with Jayadev Acharya, Aditya Vikram Singh, Ziteng Sun, and Himanshu Tyagi.