  Many stochastic algorithms have been proposed to solve the bilevel
optimization problem, where the lower level function is strongly convex and the
upper level value function is nonconvex. In particular, exising Hessian
inverse-free algorithms that utilize momentum recursion or variance reduction
technqiues can reach an $\epsilon$-stationary point with a complexity of
$\tilde{O}(\epsilon^{-1.5})$ under usual smoothness conditions. However,
$\tilde{O}(\epsilon^{-1.5})$ is a complexity higher than $O(\epsilon^{-1.5})$.
How to make a Hessian inverse-free algorithm achieve the complexity of
$O(\epsilon^{-1.5})$ under usual smoothness conditions remains an unresolved
problem. In this paper, we propose a new Hessian inverse-free algorithm based
on the projected stochastic gradient descent method and variance reduction
technique of SPIDER. This algorithm can achieve a complexity of
$O(\epsilon^{-1.5})$ under usual smoothness conditions whether it runs in a
fully single loop or double loop structure. Finally, we validate our
theoretical results through synthetic experiments and demonstrate the
efficiency of our algorithm in some machine learning applications.
