Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization

• URL: http://arxiv.org/abs/2406.03787v1
• Date: Thu, 6 Jun 2024 06:56:56 GMT
• Title: Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization
• Authors: Wei Jiang, Sifan Yang, Wenhao Yang, Yibo Wang, Yuanyu Wan, Lijun Zhang,
• Abstract summary: This paper investigates projection-free algorithms for constrained multi-level optimization functions. By using a stage-wise adaptation, we obtain complexities for convex and strongly convex functions.
• Score: 34.628967272528044
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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