932 research outputs found
An Inexact Successive Quadratic Approximation Method for Convex L-1 Regularized Optimization
We study a Newton-like method for the minimization of an objective function
that is the sum of a smooth convex function and an l-1 regularization term.
This method, which is sometimes referred to in the literature as a proximal
Newton method, computes a step by minimizing a piecewise quadratic model of the
objective function. In order to make this approach efficient in practice, it is
imperative to perform this inner minimization inexactly. In this paper, we give
inexactness conditions that guarantee global convergence and that can be used
to control the local rate of convergence of the iteration. Our inexactness
conditions are based on a semi-smooth function that represents a (continuous)
measure of the optimality conditions of the problem, and that embodies the
soft-thresholding iteration. We give careful consideration to the algorithm
employed for the inner minimization, and report numerical results on two test
sets originating in machine learning
Convergence of Newton-MR under Inexact Hessian Information
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
Newton-CG beyond convexity to invex problems. Unlike the convergence analysis
of Newton-CG, which relies on spectrum preserving Hessian approximations in the
sense of L\"{o}wner partial order, our work here draws from matrix perturbation
theory to estimate the distance between the subspaces underlying the exact and
approximate Hessian matrices. Numerical experiments demonstrate a great degree
of resilience to such Hessian approximations, amounting to a highly efficient
algorithm in large-scale problems.Comment: 32 pages, 10 figure
Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
Composing Scalable Nonlinear Algebraic Solvers
Most efficient linear solvers use composable algorithmic components, with the
most common model being the combination of a Krylov accelerator and one or more
preconditioners. A similar set of concepts may be used for nonlinear algebraic
systems, where nonlinear composition of different nonlinear solvers may
significantly improve the time to solution. We describe the basic concepts of
nonlinear composition and preconditioning and present a number of solvers
applicable to nonlinear partial differential equations. We have developed a
software framework in order to easily explore the possible combinations of
solvers. We show that the performance gains from using composed solvers can be
substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table
Fully Adaptive Newton-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we develop an adaptive procedure for the numerical solution of
general, semilinear elliptic problems with possible singular perturbations. Our
approach combines both a prediction-type adaptive Newton method and an adaptive
finite element discretization (based on a robust a posteriori error analysis),
thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples
NLTGCR: A class of Nonlinear Acceleration Procedures based on Conjugate Residuals
This paper develops a new class of nonlinear acceleration algorithms based on
extending conjugate residual-type procedures from linear to nonlinear
equations. The main algorithm has strong similarities with Anderson
acceleration as well as with inexact Newton methods - depending on which
variant is implemented. We prove theoretically and verify experimentally, on a
variety of problems from simulation experiments to deep learning applications,
that our method is a powerful accelerated iterative algorithm.Comment: Under Revie
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