On Momentum-Based Gradient Methods for Bilevel Optimization with Nonconvex Lower-Level

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, which both upper-level and lower-level problems are nonconvex, and the lower-level problem satisfies Polyak-Lojasiewicz (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 $O(\epsilon^{-3})$. This manuscript commemorates the mathematician Boris Polyak (1935 -2023).