DeepAI

Breaking The Dimension Dependence in Sparse Distribution Estimation under Communication Constraints

We consider the problem of estimating a d-dimensional s-sparse discrete distribution from its samples observed under a b-bit communication constraint. The best-known previous result on ℓ_2 estimation error for this problem is O( slog( d/s)/n2^b). Surprisingly, we show that when sample size n exceeds a minimum threshold n^*(s, d, b), we can achieve an ℓ_2 estimation error of O( s/n2^b). This implies that when n>n^*(s, d, b) the convergence rate does not depend on the ambient dimension d and is the same as knowing the support of the distribution beforehand. We next ask the question: “what is the minimum n^*(s, d, b) that allows dimension-free convergence?”. To upper bound n^*(s, d, b), we develop novel localization schemes to accurately and efficiently localize the unknown support. For the non-interactive setting, we show that n^*(s, d, b) = O( min( d^2log^2 d/2^b, s^4log^2 d/2^b) ). Moreover, we connect the problem with non-adaptive group testing and obtain a polynomial-time estimation scheme when n = Ω̃(s^4log^4 d/2^b). This group testing based scheme is adaptive to the sparsity parameter s, and hence can be applied without knowing it. For the interactive setting, we propose a novel tree-based estimation scheme and show that the minimum sample-size needed to achieve dimension-free convergence can be further reduced to n^*(s, d, b) = Õ( s^2log^2 d/2^b).

• 10 publications
• 34 publications
• 27 publications
10/07/2021

Pointwise Bounds for Distribution Estimation under Communication Constraints

We consider the problem of estimating a d-dimensional discrete distribut...
06/22/2022

List-Decodable Covariance Estimation

We give the first polynomial time algorithm for list-decodable covarianc...
01/10/2019

Mean Estimation from One-Bit Measurements

We consider the problem of estimating the mean of a symmetric log-concav...
09/22/2021

Sparse Uniformity Testing

In this paper we consider the uniformity testing problem for high-dimens...
02/12/2018

Some effects in adaptive robust estimation under sparsity

Adaptive estimation in the sparse mean model and in sparse regression ex...
02/23/2018

Geometric Lower Bounds for Distributed Parameter Estimation under Communication Constraints

We consider parameter estimation in distributed networks, where each nod...
01/29/2018

Temporally-Biased Sampling for Online Model Management

To maintain the accuracy of supervised learning models in the presence o...