The renewed interest in Steepest Descent (SD) methods following the work of Barzilai and Borwein [IMA Journal of Numerical Analysis, 8 (1988)] has driven us to consider a globalization strategy based on SD, which is applicable to any line-search method. In particular, we combine Newton-type directions with scaled SD steps to have suitable descent directions. Scaling the SD directions with a suitable step length makes a significant difference with respect to similar globalization approaches, in terms of both theoretical features and computational behavior. We apply our strategy to Newton's method and the BFGS method, with computational results that appear interesting compared with the results of well-established globalization strategies devised ad hoc for those methods.
Newton method is one of the most powerful methods for finding solutions of nonlinear equations and for proving their existence. In its "pure" form it has fast convergence near the solution, but small convergence domain. On the other hand damped Newton method has slower convergence rate, but weaker conditions on the initial point. We provide new versions of Newton-like algorithms, resulting in combinations of Newton and damped Newton method with special step-size choice, and estimate its convergence domain. Under some assumptions the convergence is global. Explicit complexity results are also a…
Fast convergent, accurate, computationally efficient, parallelizable, and robust matrix inversion and parameter estimation algorithms are required in many time-critical and accuracy-critical applications such as system identification, signal and image processing, network and big data analysis, machine learning and in many others. This paper introduces new composite power series expansion with optionally chosen rates (which can be calculated simultaneously on parallel units with different computational capacities) for further convergence rate improvement of high order Newton-Schulz iteration. N…
The power flow is a non-linear problem that requires a Newton's method to be solved in dc microgrids with constant power terminals. This paper presents sufficient conditions for the quadratic convergence of the Newton's method in this type of grids. The classic Newton method as well as an approximated Newton Method are analyzed in both master-slave and island operation with droop controls. Requirements for the convergence as well as for the existence and uniqueness of the solution starting from voltages close to 1pu are presented. Computational results complement this theoretical analysis.
In a recent joint work, the author has developed a modification of Newton's method, named New Q-Newton's method, which can avoid saddle points and has quadratic rate of convergence. While good theoretical convergence guarantee has not been established for this method, experiments on small scale problems show that the method works very competitively against other well known modifications of Newton's method such as Adaptive Cubic Regularization and BFGS, as well as first order methods such as Unbounded Two-way Backtracking Gradient Descent. In this paper, we resolve the convergence guarantee iss…
Under the hypothesis that an initial point is a quasi-regular point, we use a majorant condition to present a new semi-local convergence analysis of an extension of the Gauss-Newton method for solving convex composite optimization problems. In this analysis the conditions and proof of convergence are simplified by using a simple majorant condition to define regions where a Gauss-Newton sequence is "well behaved".
In this paper, we study the Gauss-Newton method for a special class of systems of nonlinear equation. Under the hypothesis that the derivative of the function under consideration satisfies a majorant condition, semi-local convergence analysis is presented. In this analysis the conditions and proof of convergence are simplified by using a simple majorant condition to define regions where the Gauss-Newton sequence is "well behaved". Moreover, special cases of the general theory are presented as applications.
In this paper, inexact Gauss-Newton like methods for solving injective-overdetermined systems of equations are studied. We use a majorant condition, defined by a function whose derivative is not necessarily convex, to extend and improve several existing results on the local convergence of the Gauss-Newton methods.In particular, this analysis guarantees the convergence of the methods for two important new cases.
The Gauss-Newton's method for solving nonlinear least squares problems is studied in this paper. Under the hypothesis that the derivative of the function associated with the least square problem satisfies a majorant condition, a local convergence analysis is presented. This analysis allow us to obtain the optimal convergence radius, the biggest range for the uniqueness of solution, and to unify two previous and unrelated results.
In this paper, the proximal Gauss-Newton method for solving penalized nonlinear least squares problems is studied. A local convergence analysis is obtained under the assumption that the derivative of the function associated with the penalized least square problem satisfies a majorant condition. Our analysis provides a clear relationship between the majorant function and the function associated with the penalized least squares problem. The convergence for two important special cases is also derived.
In this paper, we present a local convergence analysis of inexact Gauss-Newton like methods for solving nonlinear least squares problems. Under the hypothesis that the derivative of the function associated with the least square problem satisfies a majorant condition, we obtain that the method is well-defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the least square problem. It also allows us to obtain an estimate of convergence ball for inexact Gauss-Newton like methods and some important, special cases.
A local convergence analysis of the Gauss-Newton method for solving injective-overdetermined systems of nonlinear equations under a majorant condition is provided. The convergence as well as results on its rate are established without a convexity hypothesis on the derivative of the majorant function. The optimal convergence radius, the biggest range for uniqueness of the solution along with some other special cases are also obtained.range for uniqueness of the solution along with some other special cases.
In this paper we study the Newton's method for finding a singularity of a differentiable vector field defined on a Riemannian manifold. Under the assumption of invertibility of covariant derivative of the vector field at its singularity, we establish the well definition of the method in a suitable neighborhood of this singularity. Moreover, we also show that the generated sequence by Newton method converges for the solution with superlinear rate.
When combining the numerical concept of variational discretization and semi-smooth Newton methods for the numerical solution of pde constrained optimization with control constraints, special emphasis has to be taken on the implementation, convergence and globalization of the numerical algorithm. In the present work we address all these issues. In particular we prove fast local convergence of the algorithm and propose two different globalization strategies which are applicable in many practically relevant mathematical settings. We illustrate our analytical and algorithmical findings by numerica…
We present a new theoretical analysis of local superlinear convergence of classical quasi-Newton methods from the convex Broyden class. As a result, we obtain a significant improvement in the currently known estimates of the convergence rates for these methods. In particular, we show that the corresponding rate of the Broyden-Fletcher-Goldfarb-Shanno method depends only on the product of the dimensionality of the problem and the logarithm of its condition number.
The University of Electro-Communications（电气通信大学）；Tokyo University of Science（东京理科大学）
Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions. The present paper proposes a Riemannian stochastic quasi-Newton algorithm with variance reduction (R-SQN-VR). The key challenges of averaging, adding, and subtracting multiple gradients are addressed with notions of retraction and vector transport. We present convergence analyses of R-SQN-VR on both non-convex and retraction-convex functions under retraction and vector transport operators. The proposed algorithm is evaluated on the Karcher mean compu…
In this paper, we present the first stepsize schedule for Newton method resulting in fast global and local convergence guarantees. In particular, a) we prove an $O\left( \frac 1 {k^2} \right)$ global rate, which matches the state-of-the-art global rate of cubically regularized Newton method of Polyak and Nesterov (2006) and of regularized Newton method of Mishchenko (2021) and Doikov and Nesterov (2021), b) we prove a local quadratic rate, which matches the best-known local rate of second-order methods, and c) our stepsize formula is simple, explicit, and does not require solving any subproble…
In this paper we propose a Newton method for shape functions defined on an image set generated by the (Micheletti) metric group. We review basic properties of the metric group and a quotient associated with the metric group and a fixed domain. Taking into account the special structure of the second shape derivative and its symmetric part allows us to distinguish between two Hessians, the domain shape Hessian and the boundary shape Hessian. Using the domain Hessian we define a Newton method on the metric group by discretising the tangent space of the quotient via approximate normal functions us…
In this paper, we study greedy variants of quasi-Newton methods. They are based on the updating formulas from a certain subclass of the Broyden family. In particular, this subclass includes the well-known DFP, BFGS and SR1 updates. However, in contrast to the classical quasi-Newton methods, which use the difference of successive iterates for updating the Hessian approximations, our methods apply basis vectors, greedily selected so as to maximize a certain measure of progress. For greedy quasi-Newton methods, we establish an explicit non-asymptotic bound on their rate of local superlinear conve…
Optimization is important in machine learning problems, and quasi-Newton methods have a reputation as the most efficient numerical schemes for smooth unconstrained optimization. In this paper, we consider the explicit superlinear convergence rates of quasi-Newton methods and address two open problems mentioned by Rodomanov and Nesterov. First, we extend Rodomanov and Nesterov's results to random quasi-Newton methods, which include common DFP, BFGS, SR1 methods. Such random methods adopt a random direction for updating the approximate Hessian matrix in each iteration. Second, we focus on the sp…
The use of network Newton methods for the decentralized optimization of a sum cost distributed through agents of a network is considered. Network Newton methods reinterpret distributed gradient descent as a penalty method, observe that the corresponding Hessian is sparse, and approximate the Newton step by truncating a Taylor expansion of the inverse Hessian. Truncating the series at $K$ terms yields the NN-$K$ that requires aggregating information from $K$ hops away. Network Newton is introduced and shown to converge to the solution of the penalized objective function at a rate that is at lea…
We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form $(\frac{n L^2}{μ^2 k})^{k/2…
Non-asymptotic analysis of quasi-Newton methods have gained traction recently. In particular, several works have established a non-asymptotic superlinear rate of $\mathcal{O}((1/\sqrt{t})^t)$ for the (classic) BFGS method by exploiting the fact that its error of Newton direction approximation approaches zero. Moreover, a greedy variant of BFGS was recently proposed which accelerates its convergence by directly approximating the Hessian, instead of the Newton direction, and achieves a fast local quadratic convergence rate. Alas, the local quadratic convergence of Greedy-BFGS requires way more u…
In this paper, we study and prove the non-asymptotic superlinear convergence rate of the Broyden class of quasi-Newton algorithms which includes the Davidon--Fletcher--Powell (DFP) method and the Broyden--Fletcher--Goldfarb--Shanno (BFGS) method. The asymptotic superlinear convergence rate of these quasi-Newton methods has been extensively studied in the literature, but their explicit finite-time local convergence rate is not fully investigated. In this paper, we provide a finite-time (non-asymptotic) convergence analysis for Broyden quasi-Newton algorithms under the assumptions that the objec…
In this paper, we propose and analyze some practical Newton methods for electronic structure calculations. We show the convergence and the local quadratic convergence rate for the Newton method when the Newton search directions are well-obtained. In particular, we investigate some basic implementation issues in determining the search directions and step sizes which ensures the convergence of the subproblem at each iteration and accelerates the algorithm, respectively. It is shown by our numerical experiments that our Newton methods perform better than the existing conjugate gradient method, an…
Algorithms based on the hard thresholding principle have been well studied with sounding theoretical guarantees in the compressed sensing and more general sparsity-constrained optimization. It is widely observed in existing empirical studies that when a restricted Newton step was used (as the debiasing step), the hard-thresholding algorithms tend to meet halting conditions in a significantly low number of iterations and are very efficient. Hence, the thus obtained Newton hard-thresholding algorithms call for stronger theoretical guarantees than for their simple hard-thresholding counterparts. …
We consider the use of a curvature-adaptive step size in gradient-based iterative methods, including quasi-Newton methods, for minimizing self-concordant functions, extending an approach first proposed for Newton's method by Nesterov. This step size has a simple expression that can be computed analytically; hence, line searches are not needed. We show that using this step size in the BFGS method (and quasi-Newton methods in the Broyden convex class other than the DFP method) results in superlinear convergence for strongly convex self-concordant functions. We present numerical experiments compa…
We show that Newton's method converges globally at a linear rate for objective functions whose Hessians are stable. This class of problems includes many functions which are not strongly convex, such as logistic regression. Our linear convergence result is (i) affine-invariant, and holds even if an (ii) approximate Hessian is used, and if the subproblems are (iii) only solved approximately. Thus we theoretically demonstrate the superiority of Newton's method over first-order methods, which would only achieve a sublinear $O(1/t^2)$ rate under similar conditions.
The problem of minimizing an objective that can be written as the sum of a set of $n$ smooth and strongly convex functions is considered. The Incremental Quasi-Newton (IQN) method proposed here belongs to the family of stochastic and incremental methods that have a cost per iteration independent of $n$. IQN iterations are a stochastic version of BFGS iterations that use memory to reduce the variance of stochastic approximations. The convergence properties of IQN bridge a gap between deterministic and stochastic quasi-Newton methods. Deterministic quasi-Newton methods exploit the possibility of…
Recently, there has been a surge of interest in designing variants of the classical Newton-CG in which the Hessian of a (strongly) convex function is replaced by suitable approximations. This is mainly motivated by large-scale finite-sum minimization problems that arise in many machine learning applications. Going beyond convexity, inexact Hessian information has also been recently considered in the context of algorithms such as trust-region or (adaptive) cubic regularization for general non-convex problems. Here, we do that for Newton-MR, which extends the application range of the classical N…
This work presents a novel version of recently developed Gauss-Newton method for solving systems of nonlinear equations, based on upper bound of solution residual and quadratic regularization ideas. We obtained for such method global convergence bounds and under natural non-degeneracy assumptions we present local quadratic convergence results. We developed stochastic optimization algorithms for presented Gauss-Newton method and justified sub-linear and linear convergence rates for these algorithms using weak growth condition (WGC) and Polyak-Lojasiewicz (PL) inequality. We show that Gauss-Newt…
