Toward Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixture Models
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909630692589568 |
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| author | Xu, Weihang Fazel, Maryam Du, Simon S. |
| author_facet | Xu, Weihang Fazel, Maryam Du, Simon S. |
| contents | We study the gradient Expectation-Maximization (EM) algorithm for Gaussian Mixture Models (GMM) in the over-parameterized setting, where a general GMM with $n>1$ components learns from data that are generated by a single ground truth Gaussian distribution. While results for the special case of 2-Gaussian mixtures are well-known, a general global convergence analysis for arbitrary $n$ remains unresolved and faces several new technical barriers since the convergence becomes sub-linear and non-monotonic. To address these challenges, we construct a novel likelihood-based convergence analysis framework and rigorously prove that gradient EM converges globally with a sublinear rate $O(1/\sqrt{t})$. This is the first global convergence result for Gaussian mixtures with more than $2$ components. The sublinear convergence rate is due to the algorithmic nature of learning over-parameterized GMM with gradient EM. We also identify a new emerging technical challenge for learning general over-parameterized GMM: the existence of bad local regions that can trap gradient EM for an exponential number of steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toward Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixture Models Xu, Weihang Fazel, Maryam Du, Simon S. Machine Learning Optimization and Control We study the gradient Expectation-Maximization (EM) algorithm for Gaussian Mixture Models (GMM) in the over-parameterized setting, where a general GMM with $n>1$ components learns from data that are generated by a single ground truth Gaussian distribution. While results for the special case of 2-Gaussian mixtures are well-known, a general global convergence analysis for arbitrary $n$ remains unresolved and faces several new technical barriers since the convergence becomes sub-linear and non-monotonic. To address these challenges, we construct a novel likelihood-based convergence analysis framework and rigorously prove that gradient EM converges globally with a sublinear rate $O(1/\sqrt{t})$. This is the first global convergence result for Gaussian mixtures with more than $2$ components. The sublinear convergence rate is due to the algorithmic nature of learning over-parameterized GMM with gradient EM. We also identify a new emerging technical challenge for learning general over-parameterized GMM: the existence of bad local regions that can trap gradient EM for an exponential number of steps. |
| title | Toward Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixture Models |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2407.00490 |