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Non Convex Optimization For Machine Learning

Author : Prateek Jain
ISBN : 1680833685
Genre : Machine learning
File Size : 29. 23 MB
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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.

Non Convex Optimization In Machine Learning

Author : Majid Janzamin
ISBN : 133983510X
Genre :
File Size : 33. 96 MB
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In the last decade, machine learning algorithms have been substantially developed and they have gained tremendous empirical success. But, there is limited theoretical understanding about this success. Most real learning problems can be formulated as non-convex optimization problems which are difficult to analyze due to the existence of several local optimal solutions. In this dissertation, we provide simple and efficient algorithms for learning some probabilistic models with provable guarantees on the performance of the algorithm. We particularly focus on analyzing tensor methods which entail non-convex optimization. Furthermore, our main focus is on challenging overcomplete models. Although many existing approaches for learning probabilistic models fail in the challenging overcomplete regime, we provide scalable algorithms for learning such models with low computational and statistical complexity.In probabilistic modeling, the underlying structure which describes the observed variables can be represented by latent variables. In the overcomplete models, these hidden underlying structures are in a higher dimension compared to the dimension of observed variables. A wide range of applications such as speech and image are well-described by overcomplete models. In this dissertation, we propose and analyze overcomplete tensor decomposition methods and exploit them for learning several latent representations and latent variable models in the unsupervised setting. This include models such as multiview mixture model, Gaussian mixtures, Independent Component Analysis, and Sparse Coding (Dictionary Learning). Since latent variables are not observed, we also have the identifiability issue in latent variable modeling and characterizing latent representations. We also propose sufficient conditions for identifiability of overcomplete topic models. In addition to unsupervised setting, we adapt the tensor techniques to supervised setting for learning neural networks and mixtures of generalized linear models.

Algorithms For Nonconvex Optimization Problems In Machine Learning And Statistics

Author : Robert Mohr
ISBN : OCLC:1156871037
Genre : Algorithms
File Size : 20. 69 MB
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First Order And Stochastic Optimization Methods For Machine Learning

Author : Guanghui Lan
ISBN : 9783030395681
Genre : Mathematics
File Size : 90. 42 MB
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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.

Accelerated Optimization For Machine Learning

Author : Zhouchen Lin
ISBN : 9789811529108
Genre : Computers
File Size : 26. 21 MB
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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.

Probabilistic Machine Learning For Civil Engineers

Author : James-A. Goulet
ISBN : 9780262538701
Genre : Computers
File Size : 44. 28 MB
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An introduction to key concepts and techniques in probabilistic machine learning for civil engineering students and professionals; with many step-by-step examples, illustrations, and exercises. This book introduces probabilistic machine learning concepts to civil engineering students and professionals, presenting key approaches and techniques in a way that is accessible to readers without a specialized background in statistics or computer science. It presents different methods clearly and directly, through step-by-step examples, illustrations, and exercises. Having mastered the material, readers will be able to understand the more advanced machine learning literature from which this book draws. The book presents key approaches in the three subfields of probabilistic machine learning: supervised learning, unsupervised learning, and reinforcement learning. It first covers the background knowledge required to understand machine learning, including linear algebra and probability theory. It goes on to present Bayesian estimation, which is behind the formulation of both supervised and unsupervised learning methods, and Markov chain Monte Carlo methods, which enable Bayesian estimation in certain complex cases. The book then covers approaches associated with supervised learning, including regression methods and classification methods, and notions associated with unsupervised learning, including clustering, dimensionality reduction, Bayesian networks, state-space models, and model calibration. Finally, the book introduces fundamental concepts of rational decisions in uncertain contexts and rational decision-making in uncertain and sequential contexts. Building on this, the book describes the basics of reinforcement learning, whereby a virtual agent learns how to make optimal decisions through trial and error while interacting with its environment.

Neural Information Processing

Author : Tingwen Huang
ISBN : 9783642344756
Genre : Computers
File Size : 61. 32 MB
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The five volume set LNCS 7663, LNCS 7664, LNCS 7665, LNCS 7666 and LNCS 7667 constitutes the proceedings of the 19th International Conference on Neural Information Processing, ICONIP 2012, held in Doha, Qatar, in November 2012. The 423 regular session papers presented were carefully reviewed and selected from numerous submissions. These papers cover all major topics of theoretical research, empirical study and applications of neural information processing research. The 5 volumes represent 5 topical sections containing articles on theoretical analysis, neural modeling, algorithms, applications, as well as simulation and synthesis.

Foundations Of Machine Learning

Author : Mehryar Mohri
ISBN : 9780262039406
Genre : Computers
File Size : 57. 54 MB
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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.

Machine Learning For Signal Processing

Author : Max A. Little
ISBN : 9780198714934
Genre : Machine learning
File Size : 38. 1 MB
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This book describes in detail the fundamental mathematics and algorithms of machine learning (an example of artificial intelligence) and signal processing, two of the most important and exciting technologies in the modern information economy. Taking a gradual approach, it builds up concepts in a solid, step-by-step fashion so that the ideas and algorithms can be implemented in practical software applications. Digital signal processing (DSP) is one of the 'foundational' engineering topics of the modern world, without which technologies such the mobile phone, television, CD and MP3 players, WiFi and radar, would not be possible. A relative newcomer by comparison, statistical machine learning is the theoretical backbone of exciting technologies such as automatic techniques for car registration plate recognition, speech recognition, stock market prediction, defect detection on assembly lines, robot guidance, and autonomous car navigation. Statistical machine learning exploits the analogy between intelligent information processing in biological brains and sophisticated statistical modelling and inference. DSP and statistical machine learning are of such wide importance to the knowledge economy that both have undergone rapid changes and seen radical improvements in scope and applicability. Both make use of key topics in applied mathematics such as probability and statistics, algebra, calculus, graphs and networks. Intimate formal links between the two subjects exist and because of this many overlaps exist between the two subjects that can be exploited to produce new DSP tools of surprising utility, highly suited to the contemporary world of pervasive digital sensors and high-powered, yet cheap, computing hardware. This book gives a solid mathematical foundation to, and details the key concepts and algorithms in this important topic.

On Optimization And Scalability In Deep Learning

Author : Kenji Kawaguchi (Ph. D.)
ISBN : OCLC:1227519815
Genre :
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Deep neural networks have achieved significant empirical success in many fields, including computer vision, machine learning, and artificial intelligence. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. That is, neural networks with one hidden layer can approximate any continuous function, and deeper neural networks can approximate functions of certain classes with fewer parameters. Expressivity theory states that there exist optimal parameter vectors for neural networks of certain sizes to approximate desired target functions. However, the expressivity theory does not ensure that we can find such an optimal vector efficiently during optimization of a neural network. Optimization is one of the key steps in deep learning because learning from data is achieved through optimization, i.e., the process of optimizing the parameters of a deep neural network to make the network consistent with the data. This process typically requires nonconvex optimization, which is not scalable for high-dimensional problems in general. Indeed, in general, optimization of a neural network is not scalable without additional assumptions on its architecture. This thesis studies the non-convex optimization of various architectures of deep neural networks by focusing on some fundamental bottlenecks in the scalability, such as suboptimal local minima and saddle points. In particular, for deep neural networks, we present various guarantees for the values of local minima and critical points, as well as for points found by gradient descent. We prove that mild over-parameterization of practical degrees can ensure that gradient descent will find a global minimum for non-convex optimization of deep neural networks. Furthermore, even without over-parameterization, we show, both theoretically and empirically, that increasing the number of parameters improves the values of critical points and local minima towards the global minimum value. We also prove theoretical guarantees on the values of local minima for residual neural networks. Moreover, this thesis presents a unified theory to analyze the critical points and local minima of various deep neural networks beyond these specific architectures. These results suggest that, whereas there is the issue of scalability in the theoretical worst-case and worst architectures, we can avoid the issue and scale well for large problems with various useful architectures in practice.

A Geometric Approach To Dynamical System

Author : Ji Xu
ISBN : OCLC:1158586621
Genre :
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The sharp analyses in this paper enable us to compare the performance of our method with other phase recovery schemes. Finally, the convergence analysis of the iterative algorithms are done by a geometric approach to dynamical systems. By analyzing the movements from iteration to iteration, we provide a general tool that can show global convergence for many two dimensional dynamical systems. We hope this can shed light on convergence analysis for general dynamical systems.

Fuzzy Machine Learning

Author : Arindam Chaudhuri
ISBN : 3110603586
Genre :
File Size : 90. 63 MB
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When Do Gradient Methods Work Well In Non Convex Learning Problems?

Author : Yu Bai (Statistician)
ISBN : OCLC:1183028164
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Classical learning theory advocates the use of convex losses in machine learning due to its guaranteed computational efficiency, yet recent success in deep learning suggests that highly non-convex losses can often be efficiently minimized by simple gradient-based algorithms, a phenomenon not well explained by existing theory. Towards closing this gap, this thesis proposes and instantiates landscape analyses, a theoretical framework for studying the optimization of non-convex learning problems via analyzing optimization-aware geometric properties of the empirical risk. In the first part of the thesis, we establish a generic connection between the optimization landscape of smooth non-convex empirical risks, namely, that their gradients and Hessians concentrate uniformly around those of the population risk with statistical rate. Consequently, benign landscape properties on the population risk can be carried onto the empirical risk as well when the sample size is sufficiently large. We apply this principle to showing that gradient descent succeeds in solving several non-convex empirical risk minimization problems, including non-convex binary classification (and its sparse version), robust linear regression, and Gaussian mixture estimation. In the second part, we adopt landscape analysis to solving the orthogonal dictionary learning problem via minimizing a non-convex non-smooth L1 risk on the sphere, which was shown to work well empirically but not well understood in theory. Standard uniform convergence analysis fails on this problem as the subgradients are set-valued and non-Lipschitz. We develop tools, building on Hausdorff distance between sets and a novel sign-aware metric customized for the dictionary learning problem, to handle the above issues and establish the uniform convergence of subgradients. We show that the landscape of the non-smooth empirical risk is benign and subgradient descent succeeds in recovering the dictionary with required sample complexity lower than existing approaches based on non-convex optimization.

Distributed Optimization And Statistical Learning Via The Alternating Direction Method Of Multipliers

Author : Stephen Boyd
ISBN : 9781601984609
Genre : Computers
File Size : 77. 94 MB
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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.

Algorithms For Smooth Nonconvex Optimization With Worst Case Guarantees

Author : Michael O'Neill
ISBN : OCLC:1194892189
Genre :
File Size : 21. 18 MB
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The nature of global convergence guarantees for nonconvex optimization algorithms has changed significantly in recent years. New results characterize the maximum computational cost required for algorithms to satisfy approximate optimality conditions, instead of focusing on the limiting behavior of the iterates. In many contexts, such as those arising from machine learning, convergence to approximate second order points is desired. Algorithms designed for these problems must avoid saddle points efficiently to achieve optimal worst-case guarantees. In this dissertation, we develop and analyze a number of nonconvex optimization algorithms. First, we focus on accelerated gradient algorithms and provide results related to the avoidance of "strict saddle points''. In addition, the rate of divergence these accelerated gradient algorithms exhibit when in a neighborhood of strict saddle points is proven. Subsequently, we propose three new algorithms for smooth, nonconvex optimization with worst-case complexity guarantees. The first algorithm is developed for unconstrained optimization and is based on the classical Newton Conjugate Gradient method. This approach is then extended to bound constrained optimization by modifying the primal-log barrier method. Finally, we present a method for a special class of ``strict saddle functions'' which does not require knowledge of the parameters defining the optimization landscape. These algorithms converge to approximate second-order points in the best known computational complexity for their respective problem classes.

Mathematical Theories Of Machine Learning Theory And Applications

Author : Bin Shi
ISBN : 9783030170769
Genre : Technology & Engineering
File Size : 26. 89 MB
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This book studies mathematical theories of machine learning. The first part of the book explores the optimality and adaptivity of choosing step sizes of gradient descent for escaping strict saddle points in non-convex optimization problems. In the second part, the authors propose algorithms to find local minima in nonconvex optimization and to obtain global minima in some degree from the Newton Second Law without friction. In the third part, the authors study the problem of subspace clustering with noisy and missing data, which is a problem well-motivated by practical applications data subject to stochastic Gaussian noise and/or incomplete data with uniformly missing entries. In the last part, the authors introduce an novel VAR model with Elastic-Net regularization and its equivalent Bayesian model allowing for both a stable sparsity and a group selection.

Regularization Optimization Kernels And Support Vector Machines

Author : Johan A.K. Suykens
ISBN : 9781482241396
Genre : Computers
File Size : 90. 52 MB
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Regularization, Optimization, Kernels, and Support Vector Machines offers a snapshot of the current state of the art of large-scale machine learning, providing a single multidisciplinary source for the latest research and advances in regularization, sparsity, compressed sensing, convex and large-scale optimization, kernel methods, and support vector machines. Consisting of 21 chapters authored by leading researchers in machine learning, this comprehensive reference: Covers the relationship between support vector machines (SVMs) and the Lasso Discusses multi-layer SVMs Explores nonparametric feature selection, basis pursuit methods, and robust compressive sensing Describes graph-based regularization methods for single- and multi-task learning Considers regularized methods for dictionary learning and portfolio selection Addresses non-negative matrix factorization Examines low-rank matrix and tensor-based models Presents advanced kernel methods for batch and online machine learning, system identification, domain adaptation, and image processing Tackles large-scale algorithms including conditional gradient methods, (non-convex) proximal techniques, and stochastic gradient descent Regularization, Optimization, Kernels, and Support Vector Machines is ideal for researchers in machine learning, pattern recognition, data mining, signal processing, statistical learning, and related areas.

Convex Optimization

Author : Sébastien Bubeck
ISBN : 1601988605
Genre : Convex domains
File Size : 44. 82 MB
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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.

Optimization For Probabilistic Machine Learning

Author : Ghazal Fazelnia
ISBN : OCLC:1128160871
Genre :
File Size : 44. 32 MB
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Later in this dissertation, I will present my works on designing probabilistic models in combination with deep learning methods for representing sequential data. Sequential datasets comprise a significant portion of resources in the area of machine learning research. Designing models to capture dependencies in sequential datasets are of great interest and have a wide variety of applications in engineering, medicine and statistics. Recent advances in deep learning research has shown exceptional promises in this area. However, they lack interpretability in their general form. To remedy this, I will present my work on mixing probabilistic models with neural network models that results in better performance and expressiveness of the results.

Linear Algebra And Optimization For Machine Learning

Author : Charu C. Aggarwal
ISBN : 9783030403447
Genre : Computers
File Size : 76. 92 MB
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This textbook introduces linear algebra and optimization in the context of machine learning. Examples and exercises are provided throughout this text book together with access to a solution’s manual. This textbook targets graduate level students and professors in computer science, mathematics and data science. Advanced undergraduate students can also use this textbook. The chapters for this textbook are organized as follows: 1. Linear algebra and its applications: The chapters focus on the basics of linear algebra together with their common applications to singular value decomposition, matrix factorization, similarity matrices (kernel methods), and graph analysis. Numerous machine learning applications have been used as examples, such as spectral clustering, kernel-based classification, and outlier detection. The tight integration of linear algebra methods with examples from machine learning differentiates this book from generic volumes on linear algebra. The focus is clearly on the most relevant aspects of linear algebra for machine learning and to teach readers how to apply these concepts. 2. Optimization and its applications: Much of machine learning is posed as an optimization problem in which we try to maximize the accuracy of regression and classification models. The “parent problem” of optimization-centric machine learning is least-squares regression. Interestingly, this problem arises in both linear algebra and optimization, and is one of the key connecting problems of the two fields. Least-squares regression is also the starting point for support vector machines, logistic regression, and recommender systems. Furthermore, the methods for dimensionality reduction and matrix factorization also require the development of optimization methods. A general view of optimization in computational graphs is discussed together with its applications to back propagation in neural networks. A frequent challenge faced by beginners in machine learning is the extensive background required in linear algebra and optimization. One problem is that the existing linear algebra and optimization courses are not specific to machine learning; therefore, one would typically have to complete more course material than is necessary to pick up machine learning. Furthermore, certain types of ideas and tricks from optimization and linear algebra recur more frequently in machine learning than other application-centric settings. Therefore, there is significant value in developing a view of linear algebra and optimization that is better suited to the specific perspective of machine learning.

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