| convex optimization for machine learning: Non-convex Optimization for Machine Learning Prateek Jain, Purushottam Kar, 2017-12-04 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. |
| convex optimization for machine learning: Algorithms for Convex Optimization Nisheeth K. Vishnoi, 2021-10-07 In the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself. |
| convex optimization for machine learning: Convex Optimization Algorithms Dimitri Bertsekas, 2015-02-01 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 2009Convex 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. |
| convex optimization for machine learning: Convex Optimization Sébastien Bubeck, 2015-11-12 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. |
| convex optimization for machine learning: Convex Optimization Stephen P. Boyd, Lieven Vandenberghe, 2004-03-08 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. |
| convex optimization for machine learning: Online Learning and Online Convex Optimization Shai Shalev-Shwartz, 2012 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. |
| convex optimization for machine learning: Introduction to Online Convex Optimization, second edition Elad Hazan, 2022-09-06 New edition of a graduate-level textbook on that focuses on online convex optimization, a machine learning framework that views optimization as a process. In many practical applications, the environment is so complex that it is not feasible to lay out a comprehensive theoretical model and use classical algorithmic theory and/or mathematical optimization. Introduction to Online Convex Optimization presents a robust machine learning approach that contains elements of mathematical optimization, game theory, and learning theory: an optimization method that learns from experience as more aspects of the problem are observed. This view of optimization as a process has led to some spectacular successes in modeling and systems that have become part of our daily lives. Based on the “Theoretical Machine Learning” course taught by the author at Princeton University, the second edition of this widely used graduate level text features: Thoroughly updated material throughout New chapters on boosting, adaptive regret, and approachability and expanded exposition on optimization Examples of applications, including prediction from expert advice, portfolio selection, matrix completion and recommendation systems, SVM training, offered throughout Exercises that guide students in completing parts of proofs |
| convex optimization for machine learning: Accelerated Optimization for Machine Learning Zhouchen Lin, Huan Li, Cong Fang, 2020-05-29 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. |
| convex optimization for machine learning: Convex Optimization Theory Dimitri Bertsekas, 2009-06-01 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). |
| convex optimization for machine learning: Optimization for Machine Learning Suvrit Sra, Sebastian Nowozin, Stephen J. Wright, 2012 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. |
| convex optimization for machine learning: Lectures on Convex Optimization Yurii Nesterov, 2018-11-19 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. |
| convex optimization for machine learning: First-order and Stochastic Optimization Methods for Machine Learning Guanghui Lan, 2020-05-15 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. |
| convex optimization for machine learning: Selected Applications of Convex Optimization Li Li, 2015-03-26 This book focuses on the applications of convex optimization and highlights several topics, including support vector machines, parameter estimation, norm approximation and regularization, semi-definite programming problems, convex relaxation, and geometric problems. All derivation processes are presented in detail to aid in comprehension. The book offers concrete guidance, helping readers recognize and formulate convex optimization problems they might encounter in practice. |
| convex optimization for machine learning: Convex Analysis and Optimization Dimitri Bertsekas, Angelia Nedic, Asuman Ozdaglar, 2003-03-01 A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization. The book provides a comprehensive development of convexity theory, and its rich applications in optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. It is an excellent supplement to several of our books: Convex Optimization Theory (Athena Scientific, 2009), Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2016), Network Optimization (Athena Scientific, 1998), and Introduction to Linear Optimization (Athena Scientific, 1997). Aside from a thorough account of convex analysis and optimization, the book aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis, including: 1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. 2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. 3) A unification of the major constraint qualifications allowing the use of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions. Among its features the book: a) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar b) Provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality c) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality d) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming e) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted at the publisher's web site http://www.athenasc.com/convexity.html |
| convex optimization for machine learning: Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers Stephen Boyd, Neal Parikh, Eric Chu, 2011 Surveys the theory and history of the alternating direction method of multipliers, and discusses its applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. |
| convex optimization for machine learning: Convex Analysis and Nonlinear Optimization Jonathan Borwein, Adrian S. Lewis, 2010-05-05 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. |
| convex optimization for machine learning: Learning with Submodular Functions Francis Bach, Now Publishers, 2013 Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the Lovász extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this monograph, we present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, we show how submodular function minimization is equivalent to solving a wide variety of convex optimization problems. This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance. By listing many examples of submodular functions, we review various applications to machine learning, such as clustering, experimental design, sensor placement, graphical model structure learning or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions. |
| convex optimization for machine learning: Optimization in Machine Learning and Applications Anand J. Kulkarni, Suresh Chandra Satapathy, 2019-11-29 This book discusses one of the major applications of artificial intelligence: the use of machine learning to extract useful information from multimodal data. It discusses the optimization methods that help minimize the error in developing patterns and classifications, which further helps improve prediction and decision-making. The book also presents formulations of real-world machine learning problems, and discusses AI solution methodologies as standalone or hybrid approaches. Lastly, it proposes novel metaheuristic methods to solve complex machine learning problems. Featuring valuable insights, the book helps readers explore new avenues leading toward multidisciplinary research discussions. |
| convex optimization for machine learning: Optimization Models Giuseppe C. Calafiore, Laurent El Ghaoui, 2014-10-31 This accessible textbook demonstrates how to recognize, simplify, model and solve optimization problems - and apply these principles to new projects. |
| convex optimization for machine learning: Convex Optimization of Power Systems Joshua Adam Taylor, 2015-02-12 A mathematically rigorous guide to convex optimization for power systems engineering. |
| convex optimization for machine learning: Optimization for Data Analysis Stephen J. Wright, Benjamin Recht, 2022-04-21 A concise text that presents and analyzes the fundamental techniques and methods in optimization that are useful in data science. |
| convex optimization for machine learning: Introduction to Applied Linear Algebra Stephen Boyd, Lieven Vandenberghe, 2018-06-07 A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples. |
| convex optimization for machine learning: Optimization with Sparsity-Inducing Penalties Francis Bach, Rodolphe Jenatton, Julien Mairal, 2011-12-23 Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate nonsmooth norms. Optimization with Sparsity-Inducing Penalties presents optimization tools and techniques dedicated to such sparsity-inducing penalties from a general perspective. It covers proximal methods, block-coordinate descent, reweighted ?2-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provides an extensive set of experiments to compare various algorithms from a computational point of view. The presentation of Optimization with Sparsity-Inducing Penalties is essentially based on existing literature, but the process of constructing a general framework leads naturally to new results, connections and points of view. It is an ideal reference on the topic for anyone working in machine learning and related areas. |
| convex optimization for machine learning: Optimization for Machine Learning Jason Brownlee, 2021-09-22 Optimization happens everywhere. Machine learning is one example of such and gradient descent is probably the most famous algorithm for performing optimization. Optimization means to find the best value of some function or model. That can be the maximum or the minimum according to some metric. Using clear explanations, standard Python libraries, and step-by-step tutorial lessons, you will learn how to find the optimum point to numerical functions confidently using modern optimization algorithms. |
| convex optimization for machine learning: Online Learning and Online Convex Optimization Shai Shalev-Shwartz, 2012 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. |
| convex optimization for machine learning: Introduction to Multi-Armed Bandits Aleksandrs Slivkins, 2019-10-31 Multi-armed bandits is a rich, multi-disciplinary area that has been studied since 1933, with a surge of activity in the past 10-15 years. This is the first book to provide a textbook like treatment of the subject. |
| convex optimization for machine learning: Statistical Inference Via Convex Optimization Anatoli Juditsky, Arkadi Nemirovski, 2020-04-07 This authoritative book draws on the latest research to explore the interplay of high-dimensional statistics with optimization. Through an accessible analysis of fundamental problems of hypothesis testing and signal recovery, Anatoli Juditsky and Arkadi Nemirovski show how convex optimization theory can be used to devise and analyze near-optimal statistical inferences. Statistical Inference via Convex Optimization is an essential resource for optimization specialists who are new to statistics and its applications, and for data scientists who want to improve their optimization methods. Juditsky and Nemirovski provide the first systematic treatment of the statistical techniques that have arisen from advances in the theory of optimization. They focus on four well-known statistical problems—sparse recovery, hypothesis testing, and recovery from indirect observations of both signals and functions of signals—demonstrating how they can be solved more efficiently as convex optimization problems. The emphasis throughout is on achieving the best possible statistical performance. The construction of inference routines and the quantification of their statistical performance are given by efficient computation rather than by analytical derivation typical of more conventional statistical approaches. In addition to being computation-friendly, the methods described in this book enable practitioners to handle numerous situations too difficult for closed analytical form analysis, such as composite hypothesis testing and signal recovery in inverse problems. Statistical Inference via Convex Optimization features exercises with solutions along with extensive appendixes, making it ideal for use as a graduate text. |
| convex optimization for machine learning: Introductory Lectures on Convex Optimization Y. Nesterov, 2013-12-01 It was in the middle of the 1980s, when the seminal paper by Kar markar opened a new epoch in nonlinear optimization. The importance of this paper, containing a new polynomial-time algorithm for linear op timization problems, was not only in its complexity bound. At that time, the most surprising feature of this algorithm was that the theoretical pre diction of its high efficiency was supported by excellent computational results. This unusual fact dramatically changed the style and direc tions of the research in nonlinear optimization. Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments. In a new rapidly develop ing field, which got the name polynomial-time interior-point methods, such a justification was obligatory. Afteralmost fifteen years of intensive research, the main results of this development started to appear in monographs [12, 14, 16, 17, 18, 19]. Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students. The idea was to create a course which would reflect the new developments in the field. Actually, this was a major challenge. At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students. The general theory of self-concordant functions had appeared in print only once in the form of research monograph [12]. |
| convex optimization for machine learning: Lectures on Modern Convex Optimization Aharon Ben-Tal, Arkadi Nemirovski, 2001-01-01 Here is a book devoted to well-structured and thus efficiently solvable convex optimization problems, with emphasis on conic quadratic and semidefinite programming. The authors present the basic theory underlying these problems as well as their numerous applications in engineering, including synthesis of filters, Lyapunov stability analysis, and structural design. The authors also discuss the complexity issues and provide an overview of the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming. The book's focus on well-structured convex problems in conic form allows for unified theoretical and algorithmical treatment of a wide spectrum of important optimization problems arising in applications. |
| convex optimization for machine learning: A Probabilistic Theory of Pattern Recognition Luc Devroye, Laszlo Györfi, Gabor Lugosi, 2013-11-27 A self-contained and coherent account of probabilistic techniques, covering: distance measures, kernel rules, nearest neighbour rules, Vapnik-Chervonenkis theory, parametric classification, and feature extraction. Each chapter concludes with problems and exercises to further the readers understanding. Both research workers and graduate students will benefit from this wide-ranging and up-to-date account of a fast- moving field. |
| convex optimization for machine learning: Real Time Convex Optimisation for 5G Networks and Beyond Long D. Nguyen, Trung Q. Duong, Hoang D. Tuan, 2022-02-11 This book considers advanced real-time optimisation methods for 5G and beyond networks. The authors discuss the fundamentals, technologies, practical questions and challenges around real-time optimisation of 5G and beyond communications, providing insights into relevant theories, models and techniques. |
| convex optimization for machine learning: Linear Matrix Inequalities in System and Control Theory Stephen Boyd, Laurent El Ghaoui, Eric Feron, Venkataramanan Balakrishnan, 1994-01-01 In this book the authors reduce a wide variety of problems arising in system and control theory to a handful of convex and quasiconvex optimization problems that involve linear matrix inequalities. These optimization problems can be solved using recently developed numerical algorithms that not only are polynomial-time but also work very well in practice; the reduction therefore can be considered a solution to the original problems. This book opens up an important new research area in which convex optimization is combined with system and control theory, resulting in the solution of a large number of previously unsolved problems. |
| convex optimization for machine learning: Mathematics for Machine Learning Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong, 2020-04-23 The fundamental mathematical tools needed to understand machine learning include linear algebra, analytic geometry, matrix decompositions, vector calculus, optimization, probability and statistics. These topics are traditionally taught in disparate courses, making it hard for data science or computer science students, or professionals, to efficiently learn the mathematics. This self-contained textbook bridges the gap between mathematical and machine learning texts, introducing the mathematical concepts with a minimum of prerequisites. It uses these concepts to derive four central machine learning methods: linear regression, principal component analysis, Gaussian mixture models and support vector machines. For students and others with a mathematical background, these derivations provide a starting point to machine learning texts. For those learning the mathematics for the first time, the methods help build intuition and practical experience with applying mathematical concepts. Every chapter includes worked examples and exercises to test understanding. Programming tutorials are offered on the book's web site. |
| convex optimization for machine learning: Numerical Optimization Jorge Nocedal, Stephen Wright, 2006-12-11 Optimization is an important tool used in decision science and for the analysis of physical systems used in engineering. One can trace its roots to the Calculus of Variations and the work of Euler and Lagrange. This natural and reasonable approach to mathematical programming covers numerical methods for finite-dimensional optimization problems. It begins with very simple ideas progressing through more complicated concepts, concentrating on methods for both unconstrained and constrained optimization. |
| convex optimization for machine learning: Algorithms for Optimization Mykel J. Kochenderfer, Tim A. Wheeler, 2019-03-12 A comprehensive introduction to optimization with a focus on practical algorithms for the design of engineering systems. This book offers a comprehensive introduction to optimization with a focus on practical algorithms. The book approaches optimization from an engineering perspective, where the objective is to design a system that optimizes a set of metrics subject to constraints. Readers will learn about computational approaches for a range of challenges, including searching high-dimensional spaces, handling problems where there are multiple competing objectives, and accommodating uncertainty in the metrics. Figures, examples, and exercises convey the intuition behind the mathematical approaches. The text provides concrete implementations in the Julia programming language. Topics covered include derivatives and their generalization to multiple dimensions; local descent and first- and second-order methods that inform local descent; stochastic methods, which introduce randomness into the optimization process; linear constrained optimization, when both the objective function and the constraints are linear; surrogate models, probabilistic surrogate models, and using probabilistic surrogate models to guide optimization; optimization under uncertainty; uncertainty propagation; expression optimization; and multidisciplinary design optimization. Appendixes offer an introduction to the Julia language, test functions for evaluating algorithm performance, and mathematical concepts used in the derivation and analysis of the optimization methods discussed in the text. The book can be used by advanced undergraduates and graduate students in mathematics, statistics, computer science, any engineering field, (including electrical engineering and aerospace engineering), and operations research, and as a reference for professionals. |
| convex optimization for machine learning: Proximal Algorithms Neal Parikh, Stephen Boyd, 2013-11 Proximal Algorithms discusses proximal operators and proximal algorithms, and illustrates their applicability to standard and distributed convex optimization in general and many applications of recent interest in particular. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Proximal Algorithms discusses different interpretations of proximal operators and algorithms, looks at their connections to many other topics in optimization and applied mathematics, surveys some popular algorithms, and provides a large number of examples of proximal operators that commonly arise in practice. |
| convex optimization for machine learning: Nonsmooth Optimization and Related Topics F.H. Clarke, Vladimir F. Dem'yanov, F. Giannessi, 2013-11-11 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 |
| convex optimization for machine learning: Foundations of Machine Learning, second edition Mehryar Mohri, Afshin Rostamizadeh, Ameet Talwalkar, 2018-12-25 A new edition of a graduate-level machine learning textbook that focuses on the analysis and theory of algorithms. This book is a general introduction to machine learning that can serve as a textbook for graduate students and a reference for researchers. It covers fundamental modern topics in machine learning while providing the theoretical basis and conceptual tools needed for the discussion and justification of algorithms. It also describes several key aspects of the application of these algorithms. The authors aim to present novel theoretical tools and concepts while giving concise proofs even for relatively advanced topics. Foundations of Machine Learning is unique in its focus on the analysis and theory of algorithms. The first four chapters lay the theoretical foundation for what follows; subsequent chapters are mostly self-contained. Topics covered include the Probably Approximately Correct (PAC) learning framework; generalization bounds based on Rademacher complexity and VC-dimension; Support Vector Machines (SVMs); kernel methods; boosting; on-line learning; multi-class classification; ranking; regression; algorithmic stability; dimensionality reduction; learning automata and languages; and reinforcement learning. Each chapter ends with a set of exercises. Appendixes provide additional material including concise probability review. This second edition offers three new chapters, on model selection, maximum entropy models, and conditional entropy models. New material in the appendixes includes a major section on Fenchel duality, expanded coverage of concentration inequalities, and an entirely new entry on information theory. More than half of the exercises are new to this edition. |
| convex optimization for machine learning: Introduction To Algorithms Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, Clifford Stein, 2001 An extensively revised edition of a mathematically rigorous yet accessible introduction to algorithms. |
| convex optimization for machine learning: First-Order Methods in Optimization Amir Beck, 2017-10-02 The primary goal of this book is to provide a self-contained, comprehensive study of the main ?rst-order methods that are frequently used in solving large-scale problems. First-order methods exploit information on values and gradients/subgradients (but not Hessians) of the functions composing the model under consideration. With the increase in the number of applications that can be modeled as large or even huge-scale optimization problems, there has been a revived interest in using simple methods that require low iteration cost as well as low memory storage. The author has gathered, reorganized, and synthesized (in a unified manner) many results that are currently scattered throughout the literature, many of which cannot be typically found in optimization books. First-Order Methods in Optimization offers comprehensive study of first-order methods with the theoretical foundations; provides plentiful examples and illustrations; emphasizes rates of convergence and complexity analysis of the main first-order methods used to solve large-scale problems; and covers both variables and functional decomposition methods. |
Convex Optimization for Neural Networks - Stanford University
convex optimization and convex regularization methods are well understood and widely used in machine learning and statistics. EE364b, Stanford University 19
Theory of Convex Optimization for Machine Learning …
we proceed to give a few important examples of convex optimization problems in machine learning. 1.1 Some convex optimization problems for machine learning Many fundamental …
Convex Optimization and Machine Learning - University of …
We encounter a lot of constraint minimization problems in Machine Learning. Why We Want Convex Problems? where f0; fi are convex functions, hj are linear functions. The feasible set of …
Convex Optimization: Modeling and Algorithms - seas.ucla.edu
Convex optimization problem minimize f0(x) subject to fi(x) ≤ 0, i= 1,...,m • objective and constraint functions are convex: for 0 ≤ θ≤ 1 fi(θx+(1−θ)y) ≤ θfi(x)+(1−θ)fi(y) • can be solved globally, with …
CONVEX OPTIMIZATION FOR MACHINE LEARNING - OAPEN
I. Convex optimization basics (14 sections and 4 problem sets): A brief history of convex optimization; basic concepts on convex sets and convex functions, and the definition of …
ml_convex_optimization
Definition: let X be a convex set. A function f : X R is said to be convex if for all x, y X and [0, 1] , )f (y). With a strict inequality, f is said to be strictly convex. f is said to be concave when f is …
Machine learning lecture slides - Department of Computer …
How much of yixi to add to w(t−1) is scaled by how far σ(yixT iw(t−1)) currently is from 1. Theorem: Assume f is twice-differentiable and convex, and λmax(∇2f(w)) ≤ β for all w ∈ Rd (“f is β …
Machine Learning 8: Convex Optimization for Machine Learning
Convex functions and optimization Proposition Let f be convex. Then x is local minimum of f i 0 2@f(x), and in that case, x is a global minimum of f; if Xis closed and if f is di erentiable on X, …
Optimization for Machine Learning
A stochastic gradient method with an exponential convergence rate for strongly-convex optimization with finite training sets. In Advances in Neural Information Processing Systems …
Learning Convex Optimization Models - Stanford University
We describe three general classes of convex optimization models, maximum a posteriori (MAP) models, utility maximization models, and agent models, and present a numerical experiment …
Convexity and Optimization - Carnegie Mellon University
Convexity and Optimization Statistical Machine Learning, Spring 2015 Ryan Tibshirani (with Larry Wasserman) 1 An entirely too brief motivation 1.1 Why optimization? Optimization problems are …
Convex Optimization for Statistics and Machine Learning
Optimization underlies almost everything we do in Statistics and Machine Learning. In many settings, you learn how to: Examples of this? Examples of the contrary? Motivation: why do we …
Introduction to the Language of Convex Optimization - EECS …
how convex optimization unifies machine learning concepts related to support vector machines.
1 Convex Optimization - CIS 5200 Machine Learning
Recall the definitions of convex function and sets. Definition 1(Convex function). A function F : Rd →R is convex if for all w,w′∈Rd and α∈[0,1], F αw+ (1 −α)w′ ≤αF(w) + (1 −α)F(w′). Definition …
CPSC 440: Advanced Machine Learning - Convex Optimization
where f is a convex function and C is a convex set. Key property: all local optima are global optima. convex comb. Rd, non-negative orthant, hyper-planes, half-spaces, and norm-balls. …
Optimization for Machine Learning
Optimization for Machine Learning SUVRIT SRA Massachusetts Institute of Technology 16th February, 2021 Lecture 1: Overview; Convex sets & functions
Convex Optimization Overview - Stanford University
Convex Optimization Overview Zico Kolter (updated by Honglak Lee) October 17, 2008 1 Introduction Many situations arise in machine learning where we would like to optimize the …
Convex Optimization for Statistics and Machine Learning
Convex Optimization for Statistics and Machine Learning Part II: First-Order Methods Ryan Tibshirani Depts. of Statistics & Machine Learning http://www.stat.cmu.edu/~ryantibs/talks/ …
Convex Optimization in Machine Learning and Computational …
How do we recognize those convex problems that can be efficiently solved? How to use the software that can solve them?
Introduction to Convex Optimization for Machine Learning
Introduction to Convex Optimization for Machine Learning. What is Optimization (and why do we care?) What is Optimization? Example: Stock market. “Minimize variance of return subject to …
Convex Optimization for Neural Networks - Stanford University
convex optimization and convex regularization methods are well understood and widely used in machine learning and statistics. EE364b, Stanford University 19
Theory of Convex Optimization for Machine Learning …
we proceed to give a few important examples of convex optimization problems in machine learning. 1.1 Some convex optimization problems for machine learning Many fundamental …
Convex Optimization and Machine Learning - University of …
We encounter a lot of constraint minimization problems in Machine Learning. Why We Want Convex Problems? where f0; fi are convex functions, hj are linear functions. The feasible set of …
Convex Optimization: Modeling and Algorithms - seas.ucla.edu
Convex optimization problem minimize f0(x) subject to fi(x) ≤ 0, i= 1,...,m • objective and constraint functions are convex: for 0 ≤ θ≤ 1 fi(θx+(1−θ)y) ≤ θfi(x)+(1−θ)fi(y) • can be solved globally, with …
CONVEX OPTIMIZATION FOR MACHINE LEARNING - OAPEN
I. Convex optimization basics (14 sections and 4 problem sets): A brief history of convex optimization; basic concepts on convex sets and convex functions, and the definition of …
ml_convex_optimization
Definition: let X be a convex set. A function f : X R is said to be convex if for all x, y X and [0, 1] , )f (y). With a strict inequality, f is said to be strictly convex. f is said to be concave when f is …
Machine learning lecture slides - Department of Computer …
How much of yixi to add to w(t−1) is scaled by how far σ(yixT iw(t−1)) currently is from 1. Theorem: Assume f is twice-differentiable and convex, and λmax(∇2f(w)) ≤ β for all w ∈ Rd (“f is β …
Machine Learning 8: Convex Optimization for Machine …
Convex functions and optimization Proposition Let f be convex. Then x is local minimum of f i 0 2@f(x), and in that case, x is a global minimum of f; if Xis closed and if f is di erentiable on X, …
Optimization for Machine Learning
A stochastic gradient method with an exponential convergence rate for strongly-convex optimization with finite training sets. In Advances in Neural Information Processing Systems …
Learning Convex Optimization Models - Stanford University
We describe three general classes of convex optimization models, maximum a posteriori (MAP) models, utility maximization models, and agent models, and present a numerical experiment …
Convexity and Optimization - Carnegie Mellon University
Convexity and Optimization Statistical Machine Learning, Spring 2015 Ryan Tibshirani (with Larry Wasserman) 1 An entirely too brief motivation 1.1 Why optimization? Optimization problems are …
Convex Optimization for Statistics and Machine Learning
Optimization underlies almost everything we do in Statistics and Machine Learning. In many settings, you learn how to: Examples of this? Examples of the contrary? Motivation: why do we …
Introduction to the Language of Convex Optimization - EECS …
how convex optimization unifies machine learning concepts related to support vector machines.
1 Convex Optimization - CIS 5200 Machine Learning
Recall the definitions of convex function and sets. Definition 1(Convex function). A function F : Rd →R is convex if for all w,w′∈Rd and α∈[0,1], F αw+ (1 −α)w′ ≤αF(w) + (1 −α)F(w′). Definition …
CPSC 440: Advanced Machine Learning - Convex …
where f is a convex function and C is a convex set. Key property: all local optima are global optima. convex comb. Rd, non-negative orthant, hyper-planes, half-spaces, and norm-balls. …
Optimization for Machine Learning
Optimization for Machine Learning SUVRIT SRA Massachusetts Institute of Technology 16th February, 2021 Lecture 1: Overview; Convex sets & functions
Convex Optimization Overview - Stanford University
Convex Optimization Overview Zico Kolter (updated by Honglak Lee) October 17, 2008 1 Introduction Many situations arise in machine learning where we would like to optimize the …
Convex Optimization for Statistics and Machine Learning
Convex Optimization for Statistics and Machine Learning Part II: First-Order Methods Ryan Tibshirani Depts. of Statistics & Machine Learning http://www.stat.cmu.edu/~ryantibs/talks/ …
Convex Optimization in Machine Learning and …
How do we recognize those convex problems that can be efficiently solved? How to use the software that can solve them?
