![]() ![]() Really enjoy the subject, because I learned many numerical methods/algorithms ffe9653768cb80dfc0da.Many years ago I studied numerical method in university. "Stochastic Newton and cubic Newton methods with simple local linear-quadratic rates". Kovalev, Dmitry Mishchenko, Konstantin Richtárik, Peter (2019). ![]() Practical Methods of Optimization (2nd ed.). Numerical optimization: Theoretical and practical aspects. Bonnans, J. Frédéric Gilbert, J. Charles Lemaréchal, Claude Sagastizábal, Claudia A.Nonlinear Programming: Analysis and Methods. One can compare with Backtracking line search method for Gradient descent, which has good theoretical guarantee under more general assumptions, and can be implemented and works well in practical large scale problems such as Deep Neural Networks. If one looks at the papers by Levenberg and Marquardt in the reference for Levenberg–Marquardt algorithm, which are the original sources for the mentioned method, one can see that there is basically no theoretical analysis in the paper by Levenberg, while the paper by Marquardt only analyses a local situation and does not prove a global convergence result. The popular modifications of Newton's method, such as quasi-Newton methods or Levenberg-Marquardt algorithm mentioned above, also have caveats:įor example, it is usually required that the cost function is (strongly) convex and the Hessian is globally bounded or Lipschitz continuous, for example this is mentioned in the section "Convergence" in this article. It can converge to a saddle point instead of to a local minimum, see the section "Geometric interpretation" in this article.See the section "Failure analysis" in Newton's method. It may not converge at all, but can enter a cycle having more than 1 point.This is clear from the very definition of Newton's method, which requires taking the inverse of the Hessian. ![]()
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