  We present a method for solving general nonconvex-strongly-convex bilevel
optimization problems. Our method -- the \emph{Restarted Accelerated
HyperGradient Descent} (\texttt{RAHGD}) method -- finds an
$\epsilon$-first-order stationary point of the objective with
$\tilde{\mathcal{O}}(\kappa^{3.25}\epsilon^{-1.75})$ oracle complexity, where
$\kappa$ is the condition number of the lower-level objective and $\epsilon$ is
the desired accuracy. We also propose a perturbed variant of \texttt{RAHGD} for
finding an
$\big(\epsilon,\mathcal{O}(\kappa^{2.5}\sqrt{\epsilon}\,)\big)$-second-order
stationary point within the same order of oracle complexity. Our results
achieve the best-known theoretical guarantees for finding stationary points in
bilevel optimization and also improve upon the existing upper complexity bound
for finding second-order stationary points in nonconvex-strongly-concave
minimax optimization problems, setting a new state-of-the-art benchmark.
Empirical studies are conducted to validate the theoretical results in this
paper.
