  We consider stochastic unconstrained bilevel optimization problems when only
the first-order gradient oracles are available. While numerous optimization
methods have been proposed for tackling bilevel problems, existing methods
either tend to require possibly expensive calculations regarding Hessians of
lower-level objectives, or lack rigorous finite-time performance guarantees. In
this work, we propose a Fully First-order Stochastic Approximation (F2SA)
method, and study its non-asymptotic convergence properties. Specifically, we
show that F2SA converges to an $\epsilon$-stationary solution of the bilevel
problem after $\epsilon^{-7/2}, \epsilon^{-5/2}$, and $\epsilon^{-3/2}$
iterations (each iteration using $O(1)$ samples) when stochastic noises are in
both level objectives, only in the upper-level objective, and not present
(deterministic settings), respectively. We further show that if we employ
momentum-assisted gradient estimators, the iteration complexities can be
improved to $\epsilon^{-5/2}, \epsilon^{-4/2}$, and $\epsilon^{-3/2}$,
respectively. We demonstrate even superior practical performance of the
proposed method over existing second-order based approaches on MNIST
data-hypercleaning experiments.
