Search

A stochastic second-order trust region method is proposed, which can be viewed as a second-order extension of the trust-region-ish (TRish) algorithm proposed by Curtis et al. (INFORMS J. Optim. 1(3) 200-220, 2019). In each iteration, a search direction is computed by (approximately) solving a trust region subproblem defined by stochastic gradient and Hessian estimates. The algorithm has convergence guarantees for stochastic minimization in the fully stochastic regime, meaning that guarantees hold when each stochastic gradient is required merely to be an unbiased estimate of the true gradient with bounded variance and when the stochastic Hessian estimates are bounded uniformly in norm. The algorithm is also equipped with a worst-case complexity guarantee in the nearly deterministic regime, i.e., when the stochastic gradient and Hessian estimates are very close in expectation to the true gradients and Hessians. The results of numerical experiments for training convolutional neural networks for image classification and training a recurrent neural network for time series forecasting are presented. These results show that the algorithm can outperform a stochastic gradient approach and the first-order TRish algorithm in practice.

How to select variables and identify functional forms for continuous variables is a key concern when creating a multivariable model. Ad hoc 'traditional' approaches to variable selection have been in use for at least 50 years. Similarly, methods for determining functional forms for continuous variables were first suggested many years ago. More recently, many alternative approaches to address these two challenges have been proposed, but knowledge of their properties and meaningful comparisons between them are scarce. To define a state-of-the-art and to provide evidence-supported guidance to researchers who have only a basic level of statistical knowledge many outstanding issues in multivariable modelling remain. Our main aims are to identify and illustrate such gaps in the literature and present them at a moderate technical level to the wide community of practitioners, researchers and students of statistics. We briefly discuss general issues in building descriptive regression models, strategies for variable selection, different ways of choosing functional forms for continuous variables, and methods for combining the selection of variables and functions. We discuss two examples, taken from the medical literature, to illustrate problems in the practice of modelling. Our overview revealed that there is not yet enough evidence on which to base recommendations for the selection of variables and functional forms in multivariable analysis. Such evidence may come from comparisons between alternative methods. In particular, we highlight seven important topics that require further investigation and make suggestions for the direction of further research.

A displacement aggregation strategy is proposed for the curvature pairs stored in a limited-memory BFGS (a.k.a. L-BFGS) method such that the resulting (inverse) Hessian approximations are equal to those that would be derived from a full-memory BFGS method. This means that, if a sufficiently large number of pairs are stored, then an optimization algorithm employing the limited-memory method can achieve the same theoretical convergence properties as when full-memory (inverse) Hessian approximations are stored and employed, such as a local superlinear rate of convergence under assumptions that are common for attaining such guarantees. To the best of our knowledge, this is the first work in which a local superlinear convergence rate guarantee is offered by a quasi-Newton scheme that does not either store all curvature pairs throughout the entire run of the optimization algorithm or store an explicit (inverse) Hessian approximation. Numerical results are presented to show that displacement aggregation within an adaptive L-BFGS scheme can lead to better performance than standard L-BFGS.

This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of the steepest descent method for minimizing smooth objectives. We then provide overviews of various enhancements that have been proposed to improve practical performance, as well as of several extensions that have been made in the literature, such as to solve constrained problems. The paper also includes clarification of certain technical aspects of the analysis of gradient sampling algorithms, most notably related to the assumptions one needs to make about the set of points at which the objective is continuously differentiable. Finally, we discuss possible future research directions., Comment: Submitted to: Special Methods for Nonsmooth Optimization (Springer, 2018), edited by A. Bagirov, M. Gaudioso, N. Karmitsa and M. M\"akel\"a

This paper focuses on the design of sequential quadratic optimization (commonly known as SQP) methods for solving large-scale nonlinear optimization problems. The most computationally demanding aspect of such an approach is the computation of the search direction during each iteration, for which we consider the use of matrix-free methods. In particular, we develop a method that requires an inexact solve of a single QP subproblem to establish the convergence of the overall SQP method. It is known that SQP methods can be plagued by poor behavior of the global convergence mechanism. To confront this issue, we propose the use of an exact penalty function with a dynamic penalty parameter updating strategy to be employed within the subproblem solver in such a way that the resulting search direction predicts progress toward both feasibility and optimality. We present our parameter updating strategy and prove that, under reasonable assumptions, the strategy does not modify the penalty parameter unnecessarily. We also discuss a matrix-free subproblem solver in which our updating strategy can be incorporated. We close the paper with a discussion of the results of numerical experiments that illustrate the benefits of our proposed techniques.

We expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-\L{}ojasiewicz (K-\L{}) property holds, this is strengthened to convergence to a single constrained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnegative matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems, and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions., Comment: v3: 37 pages, 7 figures v2: 32 pages, 0 figures. v1: 26 pages, 0 figures

Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity bounds. In addition, a new update strategy for the trust region radius is proposed that offers a second-order complexity bound., Comment: 10 pages

A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an objective function, an upper bound on the number of iterations (or function or derivative evaluations) required until a pth-order stationarity condition is approximately satisfied. This arguably leads to conservative characterizations based on anomalous objectives rather than on ones that are typically encountered in practice. By contrast, the strategy proposed in this paper characterizes worst-case performance separately over regions comprising a search space. These regions are defined generically based on properties of derivative values. In this manner, one can analyze the worst-case performance of an algorithm independently from any particular class of objectives. Then, once given a class of objectives, one can obtain an informative, fine-tuned complexity analysis merely by delineating the types of regions that comprise the search spaces for functions in the class. Regions defined by first- and second-order derivatives are discussed in detail and example complexity analyses are provided for a few fundamental first- and second-order algorithms when employed to minimize convex and nonconvex objectives of interest. It is also explained how the strategy can be generalized to regions defined by higher-order derivatives and for analyzing the behavior of higher-order algorithms.

An algorithm is proposed for solving stochastic and finite sum minimization problems. Based on a trust region methodology, the algorithm employs normalized steps, at least as long as the norms of the stochastic gradient estimates are within a specified interval. The complete algorithm---which dynamically chooses whether or not to employ normalized steps---is proved to have convergence guarantees that are similar to those possessed by a traditional stochastic gradient approach under various sets of conditions related to the accuracy of the stochastic gradient estimates and choice of stepsize sequence. The results of numerical experiments are presented when the method is employed to minimize convex and nonconvex machine learning test problems. These results illustrate that the method can outperform a traditional stochastic gradient approach.