Download or read book Topics in Non convex Optimization and Learning written by Hongyi Zhang (Ph. D.) and published by . This book was released on 2019 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: Non-convex optimization and learning play an important role in data science and machine learning, yet so far they still elude our understanding in many aspects. In this thesis, I study two important aspects of non-convex optimization and learning: Riemannian optimization and deep neural networks. In the first part, I develop iteration complexity analysis for Riemannian optimization, i.e., optimization problems defined on Riemannian manifolds. Through bounding the distortion introduced by the metric curvature, iteration complexity of Riemannian (stochastic) gradient descent methods is derived. I also show that some fast first-order methods in Euclidean space, such as Nesterov's accelerated gradient descent (AGD) and stochastic variance reduced gradient (SVRG), have Riemannian counterparts that are also fast under certain conditions. In the second part, I challenge two common practices in deep learning, namely empirical risk minimization (ERM) and normalization. Specifically, I show (1) training on convex combinations of samples improves model robustness and generalization, and (2) a good initialization is sufficient for training deep residual networks without normalization. The method in (1), called mixup, is motivated by a data-dependent Lipschitzness regularization of the network. The method in (2), called Zerolnit, makes the network update scale invariant to its depth at initialization.
Download or read book Non convex Optimization for Machine Learning written by Prateek Jain and published by Foundations and Trends in Machine Learning. This book was released on 2017-12-04 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: Non-convex Optimization for Machine Learning takes an in-depth look at the basics of non-convex optimization with applications to machine learning. It introduces the rich literature in this area, as well as equips the reader with the tools and techniques needed to apply and analyze simple but powerful procedures for non-convex problems. Non-convex Optimization for Machine Learning is as self-contained as possible while not losing focus of the main topic of non-convex optimization techniques. The monograph initiates the discussion with entire chapters devoted to presenting a tutorial-like treatment of basic concepts in convex analysis and optimization, as well as their non-convex counterparts. The monograph concludes with a look at four interesting applications in the areas of machine learning and signal processing, and exploring how the non-convex optimization techniques introduced earlier can be used to solve these problems. The monograph also contains, for each of the topics discussed, exercises and figures designed to engage the reader, as well as extensive bibliographic notes pointing towards classical works and recent advances. Non-convex Optimization for Machine Learning can be used for a semester-length course on the basics of non-convex optimization with applications to machine learning. On the other hand, it is also possible to cherry pick individual portions, such the chapter on sparse recovery, or the EM algorithm, for inclusion in a broader course. Several courses such as those in machine learning, optimization, and signal processing may benefit from the inclusion of such topics.
Download or read book Topics on Nonconvex Learning written by Bingyuan Liu and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Many machine learning models need to solve nonconvex and nonsmooth optimization problems. Compared with convex optimization, nonconvex optimization captures the intrinsic structure of the learning problem more accurately. But, there are usually no well-developed algorithms with convergence guarantees for solving nonconvex and nonsmooth optimization problems. This thesis investigates how to design efficient algorithms with convergence guarantees and establish statistical properties for the computed solutions in these nonconvex learning problems. In the first part of this thesis, we study three nonconvex high-dimensional statistical learning problems. In chapter 3, we propose a robust high-dimensional regression estimator with coefficient thresholding. The coefficient thresholding is imposed in the loss function to handle the strong dependence between predictors but leads to a nonconvex loss function. We propose an efficient composite gradient descent algorithm to solve the optimization with convergence guarantee and prove the estimation consistency of our proposed estimator. In chapter 4, we propose a sparse estimation of semiparametric covariate-adjusted graphical models. In chapter 5, we study sparse sufficient dimension reduction estimators. We study the theoretical property of nonconvex penalize estimators for both chapters and propose nonconvex ADMM algorithms to solve them with computational guarantees efficiently. In the second part of this thesis, we study nonconvex neural network models. First, we study the loss landscape of attention mechanisms, which is a widely used module in deep learning. Theoretically and empirically, we show that neural network models with attention mechanisms have lower sample complexity, better generalization, and maintain a good loss landscape structure. Second, we propose a novel neural network layer that improved model robustness against adversarial attacks through neighborhood preservation. We show that despite a highly nonconvex nature, our layer has a lower Lipschitz bound, thus more robust against adversarial attacks.
Download or read book Modern Nonconvex Nondifferentiable Optimization written by Ying Cui and published by Society for Industrial and Applied Mathematics (SIAM). This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This monograph serves present and future needs where nonconvexity and nondifferentiability are inevitably present in the faithful modeling of real-world applications of optimization"--
Download or read book Convex Optimization written by Stephen P. Boyd and published by Cambridge University Press. This book was released on 2004-03-08 with total page 744 pages. Available in PDF, EPUB and Kindle. Book excerpt: Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
Download or read book Nonsmooth Optimization and Related Topics written by F.H. Clarke and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the edited texts of the lect. nres presented at the International School of Mathematics devoted to Nonsmonth Optimization, held from . June 20 to July I, 1988. The site for the meeting was the "Ettore ~Iajorana" Centre for Sci entific Culture in Erice, Sicily. In the tradition of these meetings the main purpose was to give the state-of-the-art of an important and growing field of mathematics, and to stimulate interactions between finite-dimensional and infinite-dimensional op timization. The School was attended by approximately 80 people from 23 countries; in particular it was possible to have some distinguished lecturers from the SO\·iet Union, whose research institutions are here gratt-fnlly acknowledged. Besides the lectures, several seminars were delivered; a special s·~ssion was devoted to numerical computing aspects. The result was a broad exposure. gi ·. ring a deep knowledge of the present research tendencies in the field. We wish to express our appreciation to all the participants. Special mention 5hould be made of the Ettorc ;. . Iajorana Centre in Erice, which helped provide a stimulating and rewarding experience, and of its staff which was fundamental for the success of the meeting. j\, loreover, WP want to extend uur deep appreci
Download or read book Online Learning and Online Convex Optimization written by Shai Shalev-Shwartz and published by Foundations & Trends. This book was released on 2012 with total page 88 pages. Available in PDF, EPUB and Kindle. Book excerpt: Online Learning and Online Convex Optimization is a modern overview of online learning. Its aim is to provide the reader with a sense of some of the interesting ideas and in particular to underscore the centrality of convexity in deriving efficient online learning algorithms.
Download or read book Convex Optimization Algorithms written by Dimitri Bertsekas and published by Athena Scientific. This book was released on 2015-02-01 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive and accessible presentation of algorithms for solving convex optimization problems. It relies on rigorous mathematical analysis, but also aims at an intuitive exposition that makes use of visualization where possible. This is facilitated by the extensive use of analytical and algorithmic concepts of duality, which by nature lend themselves to geometrical interpretation. The book places particular emphasis on modern developments, and their widespread applications in fields such as large-scale resource allocation problems, signal processing, and machine learning. The book is aimed at students, researchers, and practitioners, roughly at the first year graduate level. It is similar in style to the author's 2009"Convex Optimization Theory" book, but can be read independently. The latter book focuses on convexity theory and optimization duality, while the present book focuses on algorithmic issues. The two books share notation, and together cover the entire finite-dimensional convex optimization methodology. To facilitate readability, the statements of definitions and results of the "theory book" are reproduced without proofs in Appendix B.
Download or read book Convex Optimization written by Sébastien Bubeck and published by Foundations and Trends (R) in Machine Learning. This book was released on 2015-11-12 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. It begins with the fundamental theory of black-box optimization and proceeds to guide the reader through recent advances in structural optimization and stochastic optimization. The presentation of black-box optimization, strongly influenced by the seminal book by Nesterov, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. Special attention is also given to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging), and discussing their relevance in machine learning. The text provides a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization it discusses stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. It also briefly touches upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.
Download or read book Optimization for Learning and Control written by Anders Hansson and published by John Wiley & Sons. This book was released on 2023-06-20 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization for Learning and Control Comprehensive resource providing a masters’ level introduction to optimization theory and algorithms for learning and control Optimization for Learning and Control describes how optimization is used in these domains, giving a thorough introduction to both unsupervised learning, supervised learning, and reinforcement learning, with an emphasis on optimization methods for large-scale learning and control problems. Several applications areas are also discussed, including signal processing, system identification, optimal control, and machine learning. Today, most of the material on the optimization aspects of deep learning that is accessible for students at a Masters’ level is focused on surface-level computer programming; deeper knowledge about the optimization methods and the trade-offs that are behind these methods is not provided. The objective of this book is to make this scattered knowledge, currently mainly available in publications in academic journals, accessible for Masters’ students in a coherent way. The focus is on basic algorithmic principles and trade-offs. Optimization for Learning and Control covers sample topics such as: Optimization theory and optimization methods, covering classes of optimization problems like least squares problems, quadratic problems, conic optimization problems and rank optimization. First-order methods, second-order methods, variable metric methods, and methods for nonlinear least squares problems. Stochastic optimization methods, augmented Lagrangian methods, interior-point methods, and conic optimization methods. Dynamic programming for solving optimal control problems and its generalization to reinforcement learning. How optimization theory is used to develop theory and tools of statistics and learning, e.g., the maximum likelihood method, expectation maximization, k-means clustering, and support vector machines. How calculus of variations is used in optimal control and for deriving the family of exponential distributions. Optimization for Learning and Control is an ideal resource on the subject for scientists and engineers learning about which optimization methods are useful for learning and control problems; the text will also appeal to industry professionals using machine learning for different practical applications.
Download or read book Convex Analysis and Nonlinear Optimization written by Jonathan Borwein and published by Springer Science & Business Media. This book was released on 2010-05-05 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization is a rich and thriving mathematical discipline, and the underlying theory of current computational optimization techniques grows ever more sophisticated. This book aims to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. Each section concludes with an often extensive set of optional exercises. This new edition adds material on semismooth optimization, as well as several new proofs.
Download or read book Introduction to Global Optimization written by R. Horst and published by Springer Science & Business Media. This book was released on 2000-12-31 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: A textbook for an undergraduate course in mathematical programming for students with a knowledge of elementary real analysis, linear algebra, and classical linear programming (simple techniques). Focuses on the computation and characterization of global optima of nonlinear functions, rather than the locally optimal solutions addressed by most books on optimization. Incorporates the theoretical, algorithmic, and computational advances of the past three decades that help solve globally multi-extreme problems in the mathematical modeling of real world systems. Annotation copyright by Book News, Inc., Portland, OR
Download or read book Accelerated Optimization for Machine Learning written by Zhouchen Lin and published by Springer Nature. This book was released on 2020-05-29 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book on optimization includes forewords by Michael I. Jordan, Zongben Xu and Zhi-Quan Luo. Machine learning relies heavily on optimization to solve problems with its learning models, and first-order optimization algorithms are the mainstream approaches. The acceleration of first-order optimization algorithms is crucial for the efficiency of machine learning. Written by leading experts in the field, this book provides a comprehensive introduction to, and state-of-the-art review of accelerated first-order optimization algorithms for machine learning. It discusses a variety of methods, including deterministic and stochastic algorithms, where the algorithms can be synchronous or asynchronous, for unconstrained and constrained problems, which can be convex or non-convex. Offering a rich blend of ideas, theories and proofs, the book is up-to-date and self-contained. It is an excellent reference resource for users who are seeking faster optimization algorithms, as well as for graduate students and researchers wanting to grasp the frontiers of optimization in machine learning in a short time.
Download or read book Optimization for Machine Learning written by Suvrit Sra and published by MIT Press. This book was released on 2012 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: An up-to-date account of the interplay between optimization and machine learning, accessible to students and researchers in both communities. The interplay between optimization and machine learning is one of the most important developments in modern computational science. Optimization formulations and methods are proving to be vital in designing algorithms to extract essential knowledge from huge volumes of data. Machine learning, however, is not simply a consumer of optimization technology but a rapidly evolving field that is itself generating new optimization ideas. This book captures the state of the art of the interaction between optimization and machine learning in a way that is accessible to researchers in both fields. Optimization approaches have enjoyed prominence in machine learning because of their wide applicability and attractive theoretical properties. The increasing complexity, size, and variety of today's machine learning models call for the reassessment of existing assumptions. This book starts the process of reassessment. It describes the resurgence in novel contexts of established frameworks such as first-order methods, stochastic approximations, convex relaxations, interior-point methods, and proximal methods. It also devotes attention to newer themes such as regularized optimization, robust optimization, gradient and subgradient methods, splitting techniques, and second-order methods. Many of these techniques draw inspiration from other fields, including operations research, theoretical computer science, and subfields of optimization. The book will enrich the ongoing cross-fertilization between the machine learning community and these other fields, and within the broader optimization community.
Download or read book First order and Stochastic Optimization Methods for Machine Learning written by Guanghui Lan and published by Springer Nature. This book was released on 2020-05-15 with total page 591 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers not only foundational materials but also the most recent progresses made during the past few years on the area of machine learning algorithms. In spite of the intensive research and development in this area, there does not exist a systematic treatment to introduce the fundamental concepts and recent progresses on machine learning algorithms, especially on those based on stochastic optimization methods, randomized algorithms, nonconvex optimization, distributed and online learning, and projection free methods. This book will benefit the broad audience in the area of machine learning, artificial intelligence and mathematical programming community by presenting these recent developments in a tutorial style, starting from the basic building blocks to the most carefully designed and complicated algorithms for machine learning.
Download or read book Lectures on Convex Optimization written by Yurii Nesterov and published by Springer. This book was released on 2018-11-19 with total page 589 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive, modern introduction to convex optimization, a field that is becoming increasingly important in applied mathematics, economics and finance, engineering, and computer science, notably in data science and machine learning. Written by a leading expert in the field, this book includes recent advances in the algorithmic theory of convex optimization, naturally complementing the existing literature. It contains a unified and rigorous presentation of the acceleration techniques for minimization schemes of first- and second-order. It provides readers with a full treatment of the smoothing technique, which has tremendously extended the abilities of gradient-type methods. Several powerful approaches in structural optimization, including optimization in relative scale and polynomial-time interior-point methods, are also discussed in detail. Researchers in theoretical optimization as well as professionals working on optimization problems will find this book very useful. It presents many successful examples of how to develop very fast specialized minimization algorithms. Based on the author’s lectures, it can naturally serve as the basis for introductory and advanced courses in convex optimization for students in engineering, economics, computer science and mathematics.
Download or read book Convex Optimization Theory written by Dimitri Bertsekas and published by Athena Scientific. This book was released on 2009-06-01 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: An insightful, concise, and rigorous treatment of the basic theory of convex sets and functions in finite dimensions, and the analytical/geometrical foundations of convex optimization and duality theory. Convexity theory is first developed in a simple accessible manner, using easily visualized proofs. Then the focus shifts to a transparent geometrical line of analysis to develop the fundamental duality between descriptions of convex functions in terms of points, and in terms of hyperplanes. Finally, convexity theory and abstract duality are applied to problems of constrained optimization, Fenchel and conic duality, and game theory to develop the sharpest possible duality results within a highly visual geometric framework. This on-line version of the book, includes an extensive set of theoretical problems with detailed high-quality solutions, which significantly extend the range and value of the book. The book may be used as a text for a theoretical convex optimization course; the author has taught several variants of such a course at MIT and elsewhere over the last ten years. It may also be used as a supplementary source for nonlinear programming classes, and as a theoretical foundation for classes focused on convex optimization models (rather than theory). It is an excellent supplement to several of our books: Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2017), Network Optimization(Athena Scientific, 1998), Introduction to Linear Optimization (Athena Scientific, 1997), and Network Flows and Monotropic Optimization (Athena Scientific, 1998).
