SQ Lower Bounds for Learning Mixtures of Separated and Bounded Covariance Gaussians

Ilias Diakonikolas, Daniel M. Kane, Thanasis Pittas, Nikos Zarifis
Proceedings of Thirty Sixth Conference on Learning Theory, PMLR 195:2319-2349, 2023.

Abstract

We study the complexity of learning mixtures of separated Gaussians with common unknown bounded covariance matrix. Specifically, we focus on learning Gaussian mixture models (GMMs) on $\mathbb{R}^d$ of the form $P= \sum_{i=1}^k w_i \mathcal{N}(\vec \mu_i,\vec \Sigma_i)$, where $\vec \Sigma_i = \vec \Sigma \preceq \vec I$and $\min_{i \neq j} \|\vec \mu_i - \vec \mu_j\|_2 \geq k^\epsilon$ for some $\epsilon>0$. Known learning algorithms for this family of GMMs have complexity $(dk)^{O(1/\epsilon)}$. In this work, we prove that any Statistical Query (SQ) algorithm for this problem requires complexity at least $d^{\Omega(1/\epsilon)}$. Our SQ lower bound implies a similar lower bound for low-degree polynomial tests. Our result provides evidence that known algorithms for this problem are nearly best possible.

Cite this Paper

BibTeX
@InProceedings{pmlr-v195-diakonikolas23b, title = {SQ Lower Bounds for Learning Mixtures of Separated and Bounded Covariance Gaussians}, author = {Diakonikolas, Ilias and Kane, Daniel M. and Pittas, Thanasis and Zarifis, Nikos}, booktitle = {Proceedings of Thirty Sixth Conference on Learning Theory}, pages = {2319--2349}, year = {2023}, editor = {Neu, Gergely and Rosasco, Lorenzo}, volume = {195}, series = {Proceedings of Machine Learning Research}, month = {12--15 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v195/diakonikolas23b/diakonikolas23b.pdf}, url = {https://proceedings.mlr.press/v195/diakonikolas23b.html}, abstract = {We study the complexity of learning mixtures of separated Gaussians with common unknown bounded covariance matrix. Specifically, we focus on learning Gaussian mixture models (GMMs) on $\mathbb{R}^d$ of the form $P= \sum_{i=1}^k w_i \mathcal{N}(\vec \mu_i,\vec \Sigma_i)$, where $\vec \Sigma_i = \vec \Sigma \preceq \vec I$and $\min_{i \neq j} \|\vec \mu_i - \vec \mu_j\|_2 \geq k^\epsilon$ for some $\epsilon>0$. Known learning algorithms for this family of GMMs have complexity $(dk)^{O(1/\epsilon)}$. In this work, we prove that any Statistical Query (SQ) algorithm for this problem requires complexity at least $d^{\Omega(1/\epsilon)}$. Our SQ lower bound implies a similar lower bound for low-degree polynomial tests. Our result provides evidence that known algorithms for this problem are nearly best possible. } }
Endnote
%0 Conference Paper %T SQ Lower Bounds for Learning Mixtures of Separated and Bounded Covariance Gaussians %A Ilias Diakonikolas %A Daniel M. Kane %A Thanasis Pittas %A Nikos Zarifis %B Proceedings of Thirty Sixth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Gergely Neu %E Lorenzo Rosasco %F pmlr-v195-diakonikolas23b %I PMLR %P 2319--2349 %U https://proceedings.mlr.press/v195/diakonikolas23b.html %V 195 %X We study the complexity of learning mixtures of separated Gaussians with common unknown bounded covariance matrix. Specifically, we focus on learning Gaussian mixture models (GMMs) on $\mathbb{R}^d$ of the form $P= \sum_{i=1}^k w_i \mathcal{N}(\vec \mu_i,\vec \Sigma_i)$, where $\vec \Sigma_i = \vec \Sigma \preceq \vec I$and $\min_{i \neq j} \|\vec \mu_i - \vec \mu_j\|_2 \geq k^\epsilon$ for some $\epsilon>0$. Known learning algorithms for this family of GMMs have complexity $(dk)^{O(1/\epsilon)}$. In this work, we prove that any Statistical Query (SQ) algorithm for this problem requires complexity at least $d^{\Omega(1/\epsilon)}$. Our SQ lower bound implies a similar lower bound for low-degree polynomial tests. Our result provides evidence that known algorithms for this problem are nearly best possible.
APA
Diakonikolas, I., Kane, D.M., Pittas, T. & Zarifis, N.. (2023). SQ Lower Bounds for Learning Mixtures of Separated and Bounded Covariance Gaussians. Proceedings of Thirty Sixth Conference on Learning Theory, in Proceedings of Machine Learning Research 195:2319-2349 Available from https://proceedings.mlr.press/v195/diakonikolas23b.html.

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