# 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).

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