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The Expectation-Maximization (EM) algorithm is a widely used method for maximum likelihood estimation in models with latent variables. For estimating mixtures of Gaussians, its iteration can be viewed as a soft version of the k-means clustering algorithm. Despite its wide use and applications, there are essentially no known convergence guarantees for this method. We provide global convergence guarantees for mixtures of two Gaussians with known covariance matrices. We show that the population version of EM, where the algorithm is given access to infinitely many samples from the mixture, converges geometrically to the correct mean vectors, and provide simple, closed-form expressions for the convergence rate. As a simple illustration, we show that, in one dimension, ten steps of the EM algorithm initialized at infinity result in less than 1\% error estimation of the means. In the finite sample regime, we show that, under a random initialization, Õ (d/ϵ2) samples suffice to compute the unknown vectors to within ϵ in Mahalanobis distance, where d is the dimension. In particular, the error rate of the EM based estimator is Õ (dn‾‾√) where n is the number of samples, which is optimal up to logarithmic factors.
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Learning Mixtures of Gaussians Using the DDPM Objective
Recent works have shown that diffusion models can learn essentially any distribution provided one can perform score estimation. Yet it remains poorly understood under what settings score estimation is possible, let alone when practical gradient-based algorithms for this task can provably succeed. In this work, we give the first provably efficient results along these lines for one of the most fundamental distribution families, Gaussian mixture models. We prove that gradient descent on the denoising diffusion probabilistic model (DDPM) objective can efficiently recover the ground truth parameters of the mixture model in the following two settings: 1) We show gradient descent with random initialization learns mixtures of two spherical Gaussians in d dimensions with 1/poly(d)-separated centers. 2) We show gradient descent with a warm start learns mixtures of K spherical Gaussians with Ω(log(min(K,d)))-separated centers. A key ingredient in our proofs is a new connection between score-based methods and two other approaches to distribution learning, the EM algorithm and spectral methods
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- Award ID(s):
- 1909204
- PAR ID:
- 10563576
- Publisher / Repository:
- NeurIPS
- Date Published:
- Format(s):
- Medium: X
- Location:
- New Orleans
- Sponsoring Org:
- National Science Foundation
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