In this talk, I will present some recent progress in obtaining the optimal rate of convergence $O(\epsilon)$ in periodic homogenization of Hamilton-Jacobi equations. Our method is completely different from previous pure PDE approaches which only provides $O(\epsilon^{1/3})$. We have discovered a natural connection between the convergence rate and the underlying Hamiltonian system. This allows us to employ powerful tools from the Aubry-Mather theory and the weak KAM theory. It is a joint work with Hiroyashi Mitake and Hung V. Tran.