Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization
<|reference_start|>Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization: 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.<|reference_end|>
@article{jiang2024projection-free,
    title={Projection-Free Variance Reduction Methods for Stochastic Constrained
  Multi-Level Compositional Optimization},
    author={Wei Jiang, Sifan Yang, Wenhao Yang, Yibo Wang, Yuanyu Wan, Lijun Zhang},
    journal={arXiv preprint arXiv:2406.03787},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.03787},
    primaryClass={math.OC cs.LG}
}