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.

该论文研究了随机受限多层优化的无投影算法。介绍了新的无投影方差缩减算法并分析了它们在不同条件下的复杂性，包括梯度映射和 Frank-Wolfe 间隙准则，并通过分阶段适应进一步获得了凸函数和强凸函数的复杂性，数值实验证明了该方法的有效性。