Inexact restoration with subsampled trust-region methods for finite-sum minimization

Convex and nonconvex finite-sum minimization arises in many scientific computing and machine learning applications. Recently, first-order and second-order methods where objective functions, gradients and Hessians are approximated by randomly sampling components of the sum have received great attention. We propose a new trust-region method which employs suitable approximations of the objective function, gradient and Hessian built via random subsampling techniques. The choice of the sample size is deterministic and ruled by the inexact restoration approach. We discuss local and global properties for finding approximate first- and second-order optimal points and function evaluation complexity results. Numerical experience shows that the new procedure is more efficient, in terms of overall computational cost, than the standard trust-region scheme with subsampled Hessians

Economic inexact restoration for derivative-free expensive function minimization and applications

The Inexact Restoration approach has proved to be an adequate tool for handling the problem of minimizing an expensive function within an arbitrary feasible set by using different degrees of precision in the objective function. The Inexact Restoration framework allows one to obtain suitable convergence and complexity results for an approach that rationally combines low- and high-precision evaluations. In the present research, it is recognized that many problems with expensive objective functions are nonsmooth and, sometimes, even discontinuous. Having this in mind, the Inexact Restoration approach is extended to the nonsmooth or discontinuous case. Although optimization phases that rely on smoothness cannot be used in this case, basic convergence and complexity results are recovered. A derivative-free optimization phase is defined and the subproblems that arise at this phase are solved using a regularization approach that take advantage of different notions of stationarity. The new methodology is applied to the problem of reproducing a controlled experiment that mimics the failure of a dam

Minimizing Nonsmooth Convex Functions with Variable Accuracy

We consider unconstrained optimization problems with nonsmooth and convex objective function in the form of mathematical expectation. The proposed method approximates the objective function with a sample average function by using different sample size in each iteration. The sample size is chosen in an adaptive manner based on the Inexact Restoration. The method uses line search and assumes descent directions with respect to the current approximate function. We prove the almost sure convergence under the standard assumptions. The convergence rate is also considered and the worst-case complexity of $\mathcal{O} (\varepsilon^{-2})$ is proved. Numerical results for two types of problems, machine learning hinge loss and stochastic linear complementarity problems, show the efficiency of the proposed scheme

First order algorithms in variational image processing

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form ${\cal D}(Ku) + \alpha {\cal R} (u) \rightarrow \min_u$ ; where the functional ${\cal D}$ is a data fidelity term also depending on some input data $f$ and measuring the deviation of $Ku$ from such and ${\cal R}$ is a regularization functional. Moreover $K$ is a (often linear) forward operator modeling the dependence of data on an underlying image, and $\alpha$ is a positive regularization parameter. While ${\cal D}$ is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or $\ell_1$-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure

A stochastic first-order trust-region method with inexact restoration for finite-sum minimization

We propose a stochastic first-order trust-region method with inexact function and gradient evaluations for solving finite-sum minimization problems. At each iteration, the function and the gradient are approximated by sampling. The sample size in gradient approximations is smaller than the sample size in function approximations and the latter is determined using a deterministic rule inspired by the inexact restoration method, which allows the decrease of the sample size at some iterations. The trust-region step is then either accepted or rejected using a suitable merit function, which combines the function estimate with a measure of accuracy in the evaluation. We show that the proposed method eventually reaches full precision in evaluating the objective function and we provide a worst-case complexity result on the number of iterations required to achieve full precision. We validate the proposed algorithm on nonconvex binary classification problems showing good performance in terms of cost and accuracy and the important feature that a burdensome tuning of the parameters involved is not required