  In this paper, we study smooth stochastic multi-level composition
optimization problems, where the objective function is a nested composition of
$T$ functions. We assume access to noisy evaluations of the functions and their
gradients, through a stochastic first-order oracle. For solving this class of
problems, we propose two algorithms using moving-average stochastic estimates,
and analyze their convergence to an $\epsilon$-stationary point of the problem.
We show that the first algorithm, which is a generalization of
\cite{GhaRuswan20} to the $T$ level case, can achieve a sample complexity of
$\mathcal{O}(1/\epsilon^6)$ by using mini-batches of samples in each iteration.
By modifying this algorithm using linearized stochastic estimates of the
function values, we improve the sample complexity to
$\mathcal{O}(1/\epsilon^4)$. {\color{black}This modification not only removes
the requirement of having a mini-batch of samples in each iteration, but also
makes the algorithm parameter-free and easy to implement}. To the best of our
knowledge, this is the first time that such an online algorithm designed for
the (un)constrained multi-level setting, obtains the same sample complexity of
the smooth single-level setting, under standard assumptions (unbiasedness and
boundedness of the second moments) on the stochastic first-order oracle.
