Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909172210073600 |
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| author | Puchkin, Nikita Gorbunov, Eduard Kutuzov, Nikolay Gasnikov, Alexander |
| author_facet | Puchkin, Nikita Gorbunov, Eduard Kutuzov, Nikolay Gasnikov, Alexander |
| contents | We consider stochastic optimization problems with heavy-tailed noise with structured density. For such problems, we show that it is possible to get faster rates of convergence than $\mathcal{O}(K^{-2(α- 1)/α})$, when the stochastic gradients have finite moments of order $α\in (1, 2]$. In particular, our analysis allows the noise norm to have an unbounded expectation. To achieve these results, we stabilize stochastic gradients, using smoothed medians of means. We prove that the resulting estimates have negligible bias and controllable variance. This allows us to carefully incorporate them into clipped-SGD and clipped-SSTM and derive new high-probability complexity bounds in the considered setup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04161 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems Puchkin, Nikita Gorbunov, Eduard Kutuzov, Nikolay Gasnikov, Alexander Optimization and Control Data Structures and Algorithms Machine Learning Statistics Theory We consider stochastic optimization problems with heavy-tailed noise with structured density. For such problems, we show that it is possible to get faster rates of convergence than $\mathcal{O}(K^{-2(α- 1)/α})$, when the stochastic gradients have finite moments of order $α\in (1, 2]$. In particular, our analysis allows the noise norm to have an unbounded expectation. To achieve these results, we stabilize stochastic gradients, using smoothed medians of means. We prove that the resulting estimates have negligible bias and controllable variance. This allows us to carefully incorporate them into clipped-SGD and clipped-SSTM and derive new high-probability complexity bounds in the considered setup. |
| title | Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems |
| topic | Optimization and Control Data Structures and Algorithms Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2311.04161 |