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 ℓ2 estimation error for this problem is Oleft( fracslogleft( d/sright)n2bright). Surprisingly, we show that when sample size n exceeds a minimum threshold n^*(s, d, b), we can achieve an ℓ2 estimation error of Oleft( fracsn2bright). 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) = Oleft( min left( d2log2 d/2b, s4log2 d/2bright) right). Moreover, we connect the problem with non-adaptive group testing and obtain a polynomial-time estimation scheme when n = tildeOmegaleft(s4log4 d/2bright). 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) = tildeOleft( s2log2 d/2b right).