Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913380084744192 |
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| author | Jiang, Wei Yang, Sifan Yang, Wenhao Wang, Yibo Wan, Yuanyu Zhang, Lijun |
| author_facet | Jiang, Wei Yang, Sifan Yang, Wenhao Wang, Yibo Wan, Yuanyu Zhang, Lijun |
| contents | This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to match the optimal sample complexities in unconstrained settings; 2) their analysis is exclusively applicable to non-convex functions, without considering convex and strongly convex objectives. To address these issues, we introduce novel projection-free variance reduction algorithms and analyze their complexities under different criteria. For gradient mapping, our complexities improve existing results and match the optimal rates for unconstrained problems. For the widely-used Frank-Wolfe gap criterion, we provide theoretical guarantees that align with those for single-level problems. Additionally, by using a stage-wise adaptation, we further obtain complexities for convex and strongly convex functions. Finally, numerical experiments on different tasks demonstrate the effectiveness of our methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03787 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization Jiang, Wei Yang, Sifan Yang, Wenhao Wang, Yibo Wan, Yuanyu Zhang, Lijun Optimization and Control Machine Learning This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to match the optimal sample complexities in unconstrained settings; 2) their analysis is exclusively applicable to non-convex functions, without considering convex and strongly convex objectives. To address these issues, we introduce novel projection-free variance reduction algorithms and analyze their complexities under different criteria. For gradient mapping, our complexities improve existing results and match the optimal rates for unconstrained problems. For the widely-used Frank-Wolfe gap criterion, we provide theoretical guarantees that align with those for single-level problems. Additionally, by using a stage-wise adaptation, we further obtain complexities for convex and strongly convex functions. Finally, numerical experiments on different tasks demonstrate the effectiveness of our methods. |
| title | Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2406.03787 |