  We consider the problem of minimizing the composition of a smooth (nonconvex)
function and a smooth vector mapping, where the inner mapping is in the form of
an expectation over some random variable or a finite sum. We propose a
stochastic composite gradient method that employs an incremental
variance-reduced estimator for both the inner vector mapping and its Jacobian.
We show that this method achieves the same orders of complexity as the best
known first-order methods for minimizing expected-value and finite-sum
nonconvex functions, despite the additional outer composition which renders the
composite gradient estimator biased. This finding enables a much broader range
of applications in machine learning to benefit from the low complexity of
incremental variance-reduction methods.
