Related papers: Root-finding: from Newton to Halley and beyond
Many machine learning models depend on solving a large scale optimization problem. Recently, sub-sampled Newton methods have emerged to attract much attention for optimization due to their efficiency at each iteration, rectified a weakness…
Newton-type methods enjoy fast local convergence and strong empirical performance, but achieving global guarantees comparable to first-order methods remains challenging. Even for simple strongly convex problems, no straightforward variant…
Newton's method is the most widespread high-order method, demanding the gradient and the Hessian of the objective function. However, one of the main disadvantages of Newtons method is its lack of global convergence and high iteration cost.…
We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton's method for analytic functions. The complex formulation of the method allows an analysis in a complex variables…
We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear…
We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…
We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\mathcal{S}_d$ of $3.33d\log^2 d(1 + o(1))$ points with…
In this paper, we will extend the falling and rising factorial transforms \cite{ref. 1} which in this case every arbitrary function can be applied. Then, the properties of these transforms will be investigated and some corollaries will be…
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
The purpose of this work is to investigate root finding problems defined on (quasi-)metric spaces, and ranging in Euclidean spaces. The motivation for this line of inquiry stems from recent models in biology and phylogenetics, where…
A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author.…
We extend Robust Optimization to fractional programming, where both the objective and the constraints contain uncertain parameters. Earlier work did not consider uncertainty in both the objective and the constraints, or did not use Robust…
Newton's method has been thoroughly studied for the class of self-concordant functions. However, a local analysis specific to strongly self-concordant functions (a subclass of the former) is missing from the literature. The local quadratic…
We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with…
We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…
Boosting algorithms are frequently used in applied data science and in research. To date, the distinction between boosting with either gradient descent or second-order Newton updates is often not made in both applied and methodological…
We demonstrate that the most popular variants of all common algebraic multidimensional rootfinding algorithms are unstable by analyzing the conditioning of subproblems that are constructed at intermediate steps. In particular, we give…
{A defining characteristic of Newton's method is local superlinear convergence within a neighbourhood of a strict local minimum. However, outside this neighborhood Newton's method can converge slowly or even diverge. A common approach to…
A new variant of Newton's method for empirical risk minimization is studied, where at each iteration of the optimization algorithm, the gradient and Hessian of the objective function are replaced by robust estimators taken from existing…