In this paper, we tackle the optimization of nonconvex functions prevalent in large-scale statistical learning tasks. We introduce the Stochastic Path-Integrated Differential EstimatoR (SPIDER), a novel method for efficiently tracking deterministic quantities with significantly lower sampling costs. Utilizing SPIDER, we develop the SPIDER-SFO algorithm, which enhances Normalized Gradient Descent (NGD) and achieves faster convergence rates for nonconvex optimization problems in both finite-sum and online cases. Our rigorous analysis shows that SPIDER-SFO achieves a gradient computational cost of $\mathcal{O}\left(\epsilon^{-3} \wedge\left(n+n^{1 / 2} \epsilon^{-2}\right)\right)$ for finding an $\epsilon$-approximate first-order stationary point. Additionally, we extend our method to find approximate second-order stationary points with the SPIDER-SFO ${ }^{+}$algorithm, which combines SPIDER-SFO with efficient Negative-CurvatureSearch techniques. This algorithm attains an $(\epsilon, \delta)$-approximate second-order stationary point at the gradient cost $\tilde{\mathcal{O}}\left(\left(\epsilon^{-3}+\delta^{-2} \epsilon^{-2}+\delta^{-5}\right) \wedge\left(n^{1 / 2} \epsilon^{-2}+n^{1 / 2} \delta^{-2} \epsilon^{-1}+\delta^{-3} \epsilon^{-1}+\delta^{-5}+n\right)\right)$. We demonstrate that our algorithms achieve the best-known optimal convergence guarantees and optimality across a wide range of $\epsilon$ and $\delta$ values, providing superior performance under specific smoothness and Hessian-Lipschitz conditions. This work sets a new benchmark in nonconvex optimization, offering robust and scalable solutions for complex stochastic optimization tasks.