Regularized Newton Method with Global \({\boldsymbol{\mathcal{O}(1/{k}^2)}}\) Convergence
提出一种牛顿型方法，融合三次正则化与自适应Levenberg-Marquardt惩罚，对任意Lipschitz Hessian凸目标函数从任意初始化快速收敛，全局收敛率达O(1/k^2)，局部超线性收敛，并设计了无需先验参数H的线搜索过程。
.We present a Newton-type method that converges fast from any initialization and for arbitrary convex objectives with Lipschitz Hessians. We achieve this by merging the ideas of cubic regularization with a certain adaptive Levenberg–Marquardt penalty. In particular, we show that the iterates given by \(x^{k+1}=x^k - (\nabla^2 f(x^k) + \sqrt{H\|\nabla f(x^k)\|} \mathbf{I} )^{-1}\nabla f(x^k)\) , where \(H\gt 0\) is a constant, converge globally with a \(\mathcal{O}(\frac{1}{k^2})\) rate. Our method is the first variant of Newton's method that has both cheap iterations and provably fast global convergence. Moreover, we prove that locally our method converges superlinearly when the objective is strongly convex. To boost the method's performance, we present a line search procedure that does not need prior knowledge of \(H\) and is provably efficient.Keywordssecond-order optimizationNewton's methodLevenberg–Marquardt algorithmglobal convergenceMSC codes65K05