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In this work, we develop first-order (Hessian-free) and zero-order(derivative-free) implementations of the Cubically regularized Newton methodfor solving general non-convex optimization problems. For that, we employfinite difference approximations of the derivatives. We use a special adaptivesearch procedure in our algorithms, which simultaneously fits both theregularization constant and the parameters of the finite differenceapproximations. It makes our schemes free from the need to know the actualLipschitz constants. Additionally, we equip our algorithms with the lazyHessian update that reuse a previously computed Hessian approximation matrixfor several iterations. Specifically, we prove the global complexity bound of$\mathcal{O}( n^{1/2} \epsilon^{-3/2})$ function and gradient evaluations forour new Hessian-free method, and a bound of $\mathcal{O}( n^{3/2}\epsilon^{-3/2} )$ function evaluations for the derivative-free method, where$n$ is the dimension of the problem and $\epsilon$ is the desired accuracy forthe gradient norm. These complexity bounds significantly improve the previouslyknown ones in terms of the joint dependence on $n$ and $\epsilon$, for thefirst-order and zeroth-order non-convex optimization.

First and zeroth-order implementations of the regularized Newton method with lazy approximated Hessians