Journal ArticleDOI
A trust region algorithm with a worst-case iteration complexity of $$\mathcal{O}(\epsilon ^{-3/2})$$O(∈-3/2) for nonconvex optimization
TLDR
It is proved that the trust region algorithm, entitled trace, follows a trust region framework, but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst-case global complexity bound.Abstract:
We propose a trust region algorithm for solving nonconvex smooth optimization problems. For any $$\overline{\epsilon }\in (0,\infty )$$∈¯?(0,?), the algorithm requires at most $$\mathcal{O}(\epsilon ^{-3/2})$$O(∈-3/2) iterations, function evaluations, and derivative evaluations to drive the norm of the gradient of the objective function below any $$\epsilon \in (0,\overline{\epsilon }]$$∈?(0,∈¯]. This improves upon the $$\mathcal{O}(\epsilon ^{-2})$$O(∈-2) bound known to hold for some other trust region algorithms and matches the $$\mathcal{O}(\epsilon ^{-3/2})$$O(∈-3/2) bound for the recently proposed Adaptive Regularisation framework using Cubics, also known as the arc algorithm. Our algorithm, entitled trace, follows a trust region framework, but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst-case global complexity bound. Importantly, we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms. We also prove a worst-case upper bound on the number of iterations, function evaluations, and derivative evaluations that the algorithm requires to obtain an approximate second-order stationary point.read more
Citations
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Journal ArticleDOI
Theoretical Insights Into the Optimization Landscape of Over-Parameterized Shallow Neural Networks
TL;DR: In this paper, the problem of learning a shallow neural network that best fits a training data set was studied in the over-parameterized regime, where the numbers of observations are fewer than the number of parameters in the model.
Journal ArticleDOI
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
Yuejie Chi,Yue Lu,Yuxin Chen +2 more
TL;DR: This tutorial-style overview highlights the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees and reviews two contrasting approaches: two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and global landscape analysis and initialization-free algorithms.
Proceedings Article
How to escape saddle points efficiently
TL;DR: In this article, the authors show that perturbed gradient descent can escape saddle points almost for free, in a number of iterations which depends only poly-logarithmically on dimension.
Journal ArticleDOI
Global rates of convergence for nonconvex optimization on manifolds
TL;DR: The first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds apply in particular for optimization constrained to compact submanifolds of $\mathbb{R}^n$, under simpler assumptions.
Journal ArticleDOI
Newton-type methods for non-convex optimization under inexact Hessian information
TL;DR: In this article, the authors consider variants of trust-region and adaptive cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated, and provide iteration complexity to achieve $$\varepsilon $$ -approximate second-order optimality which have been shown to be tight.
References
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Book
Numerical Optimization
Jorge Nocedal,Stephen J. Wright +1 more
TL;DR: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization, responding to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.
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TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
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Nonlinear Programming: Theory and Algorithms
TL;DR: The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.
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Numerical methods for unconstrained optimization and nonlinear equations
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
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