Convergence of Gradient Descent with Small Initialization for Unregularized Matrix Completion
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914673782161408 |
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| author | Ma, Jianhao Fattahi, Salar |
| author_facet | Ma, Jianhao Fattahi, Salar |
| contents | We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $\rm{X}^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $\rm{X}^\star$ without requiring any explicit regularization. This convergence result holds true even in the over-parameterized scenario, where the true rank $r$ is unknown and conservatively over-estimated by a search rank $r'\gg r$. The existing results for this problem either require explicit regularization, a sufficiently accurate initial point, or exact knowledge of the true rank $r$.
In the over-parameterized regime where $r'\geq r$, we show that, with $\widetildeΩ(dr^9)$ observations, GD with an initial point $\|\rm{U}_0\| \leq ε$ converges near-linearly to an $ε$-neighborhood of $\rm{X}^\star$. Consequently, smaller initial points result in increasingly accurate solutions. Surprisingly, neither the convergence rate nor the final accuracy depends on the over-parameterized search rank $r'$, and they are only governed by the true rank $r$. In the exactly-parameterized regime where $r'=r$, we further enhance this result by proving that GD converges at a faster rate to achieve an arbitrarily small accuracy $ε>0$, provided the initial point satisfies $\|\rm{U}_0\| = O(1/d)$. At the crux of our method lies a novel weakly-coupled leave-one-out analysis, which allows us to establish the global convergence of GD, extending beyond what was previously possible using the classical leave-one-out analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06756 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of Gradient Descent with Small Initialization for Unregularized Matrix Completion Ma, Jianhao Fattahi, Salar Machine Learning Optimization and Control We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $\rm{X}^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $\rm{X}^\star$ without requiring any explicit regularization. This convergence result holds true even in the over-parameterized scenario, where the true rank $r$ is unknown and conservatively over-estimated by a search rank $r'\gg r$. The existing results for this problem either require explicit regularization, a sufficiently accurate initial point, or exact knowledge of the true rank $r$. In the over-parameterized regime where $r'\geq r$, we show that, with $\widetildeΩ(dr^9)$ observations, GD with an initial point $\|\rm{U}_0\| \leq ε$ converges near-linearly to an $ε$-neighborhood of $\rm{X}^\star$. Consequently, smaller initial points result in increasingly accurate solutions. Surprisingly, neither the convergence rate nor the final accuracy depends on the over-parameterized search rank $r'$, and they are only governed by the true rank $r$. In the exactly-parameterized regime where $r'=r$, we further enhance this result by proving that GD converges at a faster rate to achieve an arbitrarily small accuracy $ε>0$, provided the initial point satisfies $\|\rm{U}_0\| = O(1/d)$. At the crux of our method lies a novel weakly-coupled leave-one-out analysis, which allows us to establish the global convergence of GD, extending beyond what was previously possible using the classical leave-one-out analysis. |
| title | Convergence of Gradient Descent with Small Initialization for Unregularized Matrix Completion |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2402.06756 |