SQ Lower Bounds for Learning Bounded Covariance GMMs
This provides theoretical evidence that existing algorithms for learning bounded covariance GMMs are nearly optimal, addressing a foundational problem in machine learning theory.
The paper tackles the problem of learning Gaussian mixture models with bounded covariance and separated means, proving that any Statistical Query algorithm requires complexity at least d^{Ω(1/ε)}, which matches known upper bounds up to constants.
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 $P= \sum_{i=1}^k w_i \mathcal{N}(\boldsymbol μ_i,\mathbf Σ_i)$, where $\mathbf Σ_i = \mathbf Σ\preceq \mathbf I$ and $\min_{i \neq j} \| \boldsymbol μ_i - \boldsymbol μ_j\|_2 \geq k^ε$ for some $ε>0$. Known learning algorithms for this family of GMMs have complexity $(dk)^{O(1/ε)}$. In this work, we prove that any Statistical Query (SQ) algorithm for this problem requires complexity at least $d^{Ω(1/ε)}$. In the special case where the separation is on the order of $k^{1/2}$, we additionally obtain fine-grained SQ lower bounds with the correct exponent. Our SQ lower bounds imply similar lower bounds for low-degree polynomial tests. Conceptually, our results provide evidence that known algorithms for this problem are nearly best possible.