Faster Acceleration for Steepest Descent
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918060866142208 |
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| author | Bai, Cedar Site Bullins, Brian |
| author_facet | Bai, Cedar Site Bullins, Brian |
| contents | Recent advances (Sherman, 2017; Sidford and Tian, 2018; Cohen et al., 2021) have overcome the fundamental barrier of dimension dependence in the iteration complexity of solving $\ell_\infty$ regression with first-order methods. Yet it remains unclear to what extent such acceleration can be achieved for general $\ell_p$ smooth functions. In this paper, we propose a new accelerated first-order method for convex optimization under non-Euclidean smoothness assumptions. In contrast to standard acceleration techniques, our approach uses primal-dual iterate sequences taken with respect to $\textit{differing}$ norms, which are then coupled using an $\textit{implicitly}$ determined interpolation parameter. For $\ell_p$ norm smooth problems in $d$ dimensions, our method provides an iteration complexity improvement of up to $O(d^{1-\frac{2}{p}})$ in terms of calls to a first-order oracle, thereby allowing us to circumvent long-standing barriers in accelerated non-Euclidean steepest descent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Faster Acceleration for Steepest Descent Bai, Cedar Site Bullins, Brian Optimization and Control Machine Learning Recent advances (Sherman, 2017; Sidford and Tian, 2018; Cohen et al., 2021) have overcome the fundamental barrier of dimension dependence in the iteration complexity of solving $\ell_\infty$ regression with first-order methods. Yet it remains unclear to what extent such acceleration can be achieved for general $\ell_p$ smooth functions. In this paper, we propose a new accelerated first-order method for convex optimization under non-Euclidean smoothness assumptions. In contrast to standard acceleration techniques, our approach uses primal-dual iterate sequences taken with respect to $\textit{differing}$ norms, which are then coupled using an $\textit{implicitly}$ determined interpolation parameter. For $\ell_p$ norm smooth problems in $d$ dimensions, our method provides an iteration complexity improvement of up to $O(d^{1-\frac{2}{p}})$ in terms of calls to a first-order oracle, thereby allowing us to circumvent long-standing barriers in accelerated non-Euclidean steepest descent. |
| title | Faster Acceleration for Steepest Descent |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2409.19200 |