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This paper studies accelerated gradient methods for nonconvex optimizationwith Lipschitz continuous gradient and Hessian. We propose two simpleaccelerated gradient methods, restarted accelerated gradient descent (AGD) andrestarted heavy ball (HB) method, and establish that our methods achieve an$\epsilon$-approximate first-order stationary point within $O(\epsilon^{-7/4})$number of gradient evaluations by elementary proofs. Theoretically, ourcomplexity does not hide any polylogarithmic factors, and thus it improves overthe best known one by the $O(\log\frac{1}{\epsilon})$ factor. Our algorithmsare simple in the sense that they only consist of Nesterov's classical AGD orPolyak's HB iterations, as well as a restart mechanism. They do not invokenegative curvature exploitation or minimization of regularized surrogatefunctions as the subroutines. In contrast with existing analysis, ourelementary proofs use less advanced techniques and do not invoke the analysisof strongly convex AGD or HB.  Code is avaliable at https://github.com/lihuanML/RestartAGD.

Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the $O(\epsilon^{-7/4})$ Complexity