Damped Newton Method with Near-Optimal Global $O(k⁻³ )$ Convergence Rate

This paper investigates the global convergence of stepsized Newton methods for convex functions. We propose several simple stepsize schedules with fast global convergence guarantees, up to O (k⁻³), nearly matching lower complexity bounds Ω (k-3.5) of second-order methods. For cases with multiple plausible smoothness parameterizations or an unknown smoothness constant, we introduce a stepsize backtracking procedure that ensures convergence as if the optimal smoothness parameters were known.