  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.
