Recently proposed adaptive Sketch & Project (SP) methods connect several well-known projection methods such as Randomized Kaczmarz (RK), Randomized Block Kaczmarz (RBK), Motzkin Relaxation (MR), Randomized Coordinate Descent (RCD), Capped Coordinate Descent (CCD), etc. into one framework for solving linear systems. In this work, we first propose a Stochastic Steepest Descent (SSD) framework that connects SP methods with the well-known Steepest Descent (SD) method for solving positive-definite linear system of equations. We then introduce two greedy sampling strategies in the SSD framework that allow us to obtain algorithms such as Sampling Kaczmarz Motzkin (SKM), Sampling Block Kaczmarz (SBK), Sampling Coordinate Descent (SCD), etc. In doing so, we generalize the existing sampling rules into one framework and develop an efficient version of SP methods. Furthermore, we incorporated the Polyak momentum technique into the SSD method to accelerate the resulting algorithms. We provide global convergence results for both the SSD method and the momentum induced SSD method. Moreover, we prove $\mathcal{O}(\frac{1}{k})$ convergence rate for the Cesaro average of iterates generated by both methods. By varying parameters in the SSD method, we obtain classical convergence results of the SD method as well as the SP methods as special cases. We design computational experiments to demonstrate the performance of the proposed greedy sampling methods as well as the momentum methods. The proposed greedy methods significantly outperform the existing methods for a wide variety of datasets such as random test instances as well as real-world datasets (LIBSVM, sparse datasets from matrix market collection). Finally, the momentum algorithms designed in this work accelerate the algorithmic performance of the SSD methods.
We present some extensions to the limited memory steepest descent method based on spectral properties and cyclic iterations. Our aim is to show that it is possible to combine sweep and delayed strategies for improving the performance of gradient methods. Numerical results are reported which indicate that our new methods are better than the original version. Some remarks on the stability and parallel implementation are shown in the end.
Gradient-type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD-id) is one of such methods. The convergence behavior of the PSD-id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non-asymptotic estimates indicate a superlinear convergence of the PSD-id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis…
Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N-GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, while the second employs a predefined small step. A simple global convergence proof is provided for the N-GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioni…
Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；University of British Columbia（不列颠哥伦比亚大学）
Much recent attention has been devoted to gradient descent algorithms where the steepest descent step size is replaced by a similar one from a previous iteration or gets updated only once every second step, thus forming a {\em faster gradient descent method}. For unconstrained convex quadratic optimization these methods can converge much faster than steepest descent. But the context of interest here is application to certain ill-posed inverse problems, where the steepest descent method is known to have a smoothing, regularizing effect, and where a strict optimization solution is not necessary.…
The self regulation problem as an inexact steepest descent method for multicriteria optimization
IME-Universidade Federal de Goiás（戈亚斯联邦大学数学与工程学院）；DM, Universidade Federal do Piauí（皮奥伊联邦大学数学系）；COPPE/Sistemas-Universidade Federal do Rio de Janeiro（里约热内卢联邦大学科佩系统工程研究所）；Aix-Marseille University（艾克斯-马赛大学）
In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link was first observed in a previous work which sought minimisers of a shape derivative over the space of Lipschitz functions which may be closely related to the $\infty$-Laplacian. We provide some results of shape optimisation using this novel Lipschitz approach, highlighting the difference between the Lipschitz and $W^{1,\infty}$ semi-norms. After this, we provide an overview of the necessary results from Optimal trans…
A convergence analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints
Purdue University（普渡大学）；University of Vienna（维也纳大学）
The coordinate descent method is an effective iterative method for solving large linear least-squares problems. In this paper, for the highly coherent columns case, we construct an effective coordinate descent method which iteratively projects the estimate onto a solution space formed by two greedily selected hyperplanes via Gram-Schmidt orthogonalization. Our methods may be regarded as a simple block version of coordinate descent method which involves two active columns. The convergence analysis of this method is provided and numerical simulations also confirm the effectiveness for matrices w…
A q-Levenberg-Marquardt method is an iterative procedure that blends a q-steepest descent and q-Gauss-Newton methods. When the current solution is far from the correct one the algorithm acts as the q-steepest descent method. Otherwise the algorithm acts as the q-Gauss-Newton method. A damping parameter is used to interpolate between these two methods. The q-parameter is used to escape from local minima and to speed up the search process near the optimal solution.
We present a novel greedy Gauss-Seidel method for solving large linear least squares problem. This method improves the greedy randomized coordinate descent (GRCD) method proposed recently by Bai and Wu [Bai ZZ, and Wu WT. On greedy randomized coordinate descent methods for solving large linear least-squares problems. Numer Linear Algebra Appl. 2019;26(4):1--15], which in turn improves the popular randomized Gauss-Seidel method. Convergence analysis of the new method is provided. Numerical experiments show that, for the same accuracy, our method outperforms the GRCD method in term of the comput…
We develop two greedy sampling rules for the Sketch & Project method for solving linear feasibility problems. The proposed greedy sampling rules generalize the existing max-distance sampling rule and uniform sampling rule and generate faster variants of Sketch & Project methods. We also introduce greedy capped sampling rules that improve the existing capped sampling rules. Moreover, we incorporate the so-called heavy ball momentum technique to the proposed greedy Sketch & Project method. By varying the parameters such as sampling rules, sketching vectors; we recover several well-known algorith…
