Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model

Yingyu Liang, Hui Yuan
Proceedings of Thirty Third Conference on Learning Theory, PMLR 125:2712-2737, 2020.

Abstract

In the setting of entangled single-sample distributions, the goal is to estimate some common parameter shared by a family of $n$ distributions, given one single sample from each distribution. This paper studies mean estimation for entangled single-sample Gaussians that have a common mean but different unknown variances. We propose the subset-of-signals model where an unknown subset of $m$ variances are bounded by 1 while there are no assumptions on the other variances. In this model, we analyze a simple and natural method based on iteratively averaging the truncated samples, and show that the method achieves error $O \left(\frac{\sqrt{n\ln n}}{m}\right)$ with high probability when $m=\Omega(\sqrt{n\ln n})$, slightly improving existing bounds for this range of $m$. We further prove lower bounds, showing that the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/2}\right)$ when $m$ is between $\Omega(\ln n)$ and $O(n^{1/4})$, and the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/6}\right)$ when $m$ is between $\Omega(n^{1/4})$ and $O(n^{1 - \epsilon})$ for an arbitrarily small $\epsilon>0$, improving existing lower bounds and extending to a wider range of $m$.

Cite this Paper

BibTeX
@InProceedings{pmlr-v125-liang20b, title = {Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model}, author = {Liang, Yingyu and Yuan, Hui}, booktitle = {Proceedings of Thirty Third Conference on Learning Theory}, pages = {2712--2737}, year = {2020}, editor = {Abernethy, Jacob and Agarwal, Shivani}, volume = {125}, series = {Proceedings of Machine Learning Research}, month = {09--12 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v125/liang20b/liang20b.pdf}, url = {https://proceedings.mlr.press/v125/liang20b.html}, abstract = { In the setting of entangled single-sample distributions, the goal is to estimate some common parameter shared by a family of $n$ distributions, given one single sample from each distribution. This paper studies mean estimation for entangled single-sample Gaussians that have a common mean but different unknown variances. We propose the subset-of-signals model where an unknown subset of $m$ variances are bounded by 1 while there are no assumptions on the other variances. In this model, we analyze a simple and natural method based on iteratively averaging the truncated samples, and show that the method achieves error $O \left(\frac{\sqrt{n\ln n}}{m}\right)$ with high probability when $m=\Omega(\sqrt{n\ln n})$, slightly improving existing bounds for this range of $m$. We further prove lower bounds, showing that the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/2}\right)$ when $m$ is between $\Omega(\ln n)$ and $O(n^{1/4})$, and the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/6}\right)$ when $m$ is between $\Omega(n^{1/4})$ and $O(n^{1 - \epsilon})$ for an arbitrarily small $\epsilon>0$, improving existing lower bounds and extending to a wider range of $m$.} }
Endnote
%0 Conference Paper %T Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model %A Yingyu Liang %A Hui Yuan %B Proceedings of Thirty Third Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2020 %E Jacob Abernethy %E Shivani Agarwal %F pmlr-v125-liang20b %I PMLR %P 2712--2737 %U https://proceedings.mlr.press/v125/liang20b.html %V 125 %X In the setting of entangled single-sample distributions, the goal is to estimate some common parameter shared by a family of $n$ distributions, given one single sample from each distribution. This paper studies mean estimation for entangled single-sample Gaussians that have a common mean but different unknown variances. We propose the subset-of-signals model where an unknown subset of $m$ variances are bounded by 1 while there are no assumptions on the other variances. In this model, we analyze a simple and natural method based on iteratively averaging the truncated samples, and show that the method achieves error $O \left(\frac{\sqrt{n\ln n}}{m}\right)$ with high probability when $m=\Omega(\sqrt{n\ln n})$, slightly improving existing bounds for this range of $m$. We further prove lower bounds, showing that the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/2}\right)$ when $m$ is between $\Omega(\ln n)$ and $O(n^{1/4})$, and the error is $\Omega\left(\left(\frac{n}{m^4}\right)^{1/6}\right)$ when $m$ is between $\Omega(n^{1/4})$ and $O(n^{1 - \epsilon})$ for an arbitrarily small $\epsilon>0$, improving existing lower bounds and extending to a wider range of $m$.
APA
Liang, Y. & Yuan, H.. (2020). Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model. Proceedings of Thirty Third Conference on Learning Theory, in Proceedings of Machine Learning Research 125:2712-2737 Available from https://proceedings.mlr.press/v125/liang20b.html.

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