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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.08766 |
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| _version_ | 1866909409474510848 |
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| author | Dagréou, Mathieu Moreau, Thomas Vaiter, Samuel Ablin, Pierre |
| author_facet | Dagréou, Mathieu Moreau, Thomas Vaiter, Samuel Ablin, Pierre |
| contents | Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ oracle calls to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, making it optimal in terms of sample complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08766 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Lower Bound and a Near-Optimal Algorithm for Bilevel Empirical Risk Minimization Dagréou, Mathieu Moreau, Thomas Vaiter, Samuel Ablin, Pierre Machine Learning Optimization and Control Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ oracle calls to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, making it optimal in terms of sample complexity. |
| title | A Lower Bound and a Near-Optimal Algorithm for Bilevel Empirical Risk Minimization |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2302.08766 |