  Bilevel optimization has recently attracted growing interests due to its wide
applications in modern machine learning problems. Although recent studies have
characterized the convergence rate for several such popular algorithms, it is
still unclear how much further these convergence rates can be improved. In this
paper, we address this fundamental question from two perspectives. First, we
provide the first-known lower complexity bounds of
$\widetilde{\Omega}(\frac{1}{\sqrt{\mu_x}\mu_y})$ and $\widetilde
\Omega\big(\frac{1}{\sqrt{\epsilon}}\min\{\frac{1}{\mu_y},\frac{1}{\sqrt{\epsilon^{3}}}\}\big)$
respectively for strongly-convex-strongly-convex and convex-strongly-convex
bilevel optimizations. Second, we propose an accelerated bilevel optimizer
named AccBiO, for which we provide the first-known complexity bounds without
the gradient boundedness assumption (which was made in existing analyses) under
the two aforementioned geometries. We also provide significantly tighter upper
bounds than the existing complexity when the bounded gradient assumption does
hold. We show that AccBiO achieves the optimal results (i.e., the upper and
lower bounds match up to logarithmic factors) when the inner-level problem
takes a quadratic form with a constant-level condition number. Interestingly,
our lower bounds under both geometries are larger than the corresponding
optimal complexities of minimax optimization, establishing that bilevel
optimization is provably more challenging than minimax optimization.
