  We propose a projection-free conditional gradient-type algorithm for smooth
stochastic multi-level composition optimization, where the objective function
is a nested composition of $T$ functions and the constraint set is a closed
convex set. Our algorithm assumes access to noisy evaluations of the functions
and their gradients, through a stochastic first-order oracle satisfying certain
standard unbiasedness and second moment assumptions. We show that the number of
calls to the stochastic first-order oracle and the linear-minimization oracle
required by the proposed algorithm, to obtain an $\epsilon$-stationary
solution, are of order $\mathcal{O}_T(\epsilon^{-2})$ and
$\mathcal{O}_T(\epsilon^{-3})$ respectively, where $\mathcal{O}_T$ hides
constants in $T$. Notably, the dependence of these complexity bounds on
$\epsilon$ and $T$ are separate in the sense that changing one does not impact
the dependence of the bounds on the other. Moreover, our algorithm is
parameter-free and does not require any (increasing) order of mini-batches to
converge unlike the common practice in the analysis of stochastic conditional
gradient-type algorithms.
