  Bilevel optimization is a popular two-level hierarchical optimization, which
has been widely applied to many machine learning tasks such as hyperparameter
learning, meta learning and continual learning. Although many bilevel
optimization methods recently have been developed, the bilevel methods are not
well studied when the lower-level problem is nonconvex. To fill this gap, in
the paper, we study a class of nonconvex bilevel optimization problems, where
both upper-level and lower-level problems are nonconvex, and the lower-level
problem satisfies Polyak-{\L}ojasiewicz (PL) condition. We propose an efficient
momentum-based gradient bilevel method (MGBiO) to solve these deterministic
problems. Meanwhile, we propose a class of efficient momentum-based stochastic
gradient bilevel methods (MSGBiO and VR-MSGBiO) to solve these stochastic
problems. Moreover, we provide a useful convergence analysis framework for our
methods. Specifically, under some mild conditions, we prove that our MGBiO
method has a sample (or gradient) complexity of $O(\epsilon^{-2})$ for finding
an $\epsilon$-stationary solution of the deterministic bilevel problems (i.e.,
$\|\nabla F(x)\|\leq \epsilon$), which improves the existing best results by a
factor of $O(\epsilon^{-1})$. Meanwhile, we prove that our MSGBiO and VR-MSGBiO
methods have sample complexities of $\tilde{O}(\epsilon^{-4})$ and
$\tilde{O}(\epsilon^{-3})$, respectively, in finding an $\epsilon$-stationary
solution of the stochastic bilevel problems (i.e., $\mathbb{E}\|\nabla
F(x)\|\leq \epsilon$), which improves the existing best results by a factor of
$\tilde{O}(\epsilon^{-3})$. Extensive experimental results on bilevel PL game
and hyper-representation learning demonstrate the efficiency of our algorithms.
This paper commemorates the mathematician Boris Polyak (1935 -2023).
