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

*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.