  We propose a new first-order method for minimizing nonconvex functions with a
Lipschitz continuous gradient and Hessian. The proposed method is an
accelerated gradient descent with two restart mechanisms and finds a solution
where the gradient norm is less than $\epsilon$ in $O(\epsilon^{-7/4})$
function and gradient evaluations. Unlike existing first-order methods with
similar complexity bounds, our algorithm is parameter-free because it requires
no prior knowledge of problem-dependent parameters, e.g., the Lipschitz
constants and the target accuracy $\epsilon$. The main challenge in achieving
this advantage is estimating the Lipschitz constant of the Hessian using only
first-order information. To this end, we develop a new Hessian-free analysis
based on two technical inequalities: a Jensen-type inequality for gradients and
an error bound for the trapezoidal rule. Several numerical results illustrate
that the proposed method performs comparably to existing algorithms with
similar complexity bounds, even without parameter tuning.
