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Book Non convex Optimization for Machine Learning

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.

Book Convex 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.

Book First order and Stochastic Optimization Methods for Machine Learning

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.

Book Convex Optimization

    Book Details:
  • Author : Sébastien Bubeck
  • Publisher : Foundations and Trends (R) in Machine Learning
  • Release : 2015-11-12
  • ISBN : 9781601988607
  • Pages : 142 pages

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.

Book Accelerated Optimization for Machine Learning

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.

Book Optimization for Machine Learning

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.

Book Modern Nonconvex Nondifferentiable Optimization

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"--

Book Convex Optimization for Machine Learning

Download or read book Convex Optimization for Machine Learning written by Changho Suh and published by . This book was released on 2022-09-27 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers an introduction to convex optimization, one of the powerful and tractable optimization problems that can be efficiently solved on a computer. The goal of the book is to help develop a sense of what convex optimization is, and how it can be used in a widening array of practical contexts with a particular emphasis on machine learning. The first part of the book covers core concepts of convex sets, convex functions, and related basic definitions that serve understanding convex optimization and its corresponding models. The second part deals with one very useful theory, called duality, which enables us to: (1) gain algorithmic insights; and (2) obtain an approximate solution to non-convex optimization problems which are often difficult to solve. The last part focuses on modern applications in machine learning and deep learning. A defining feature of this book is that it succinctly relates the "story" of how convex optimization plays a role, via historical examples and trending machine learning applications. Another key feature is that it includes programming implementation of a variety of machine learning algorithms inspired by optimization fundamentals, together with a brief tutorial of the used programming tools. The implementation is based on Python, CVXPY, and TensorFlow. This book does not follow a traditional textbook-style organization, but is streamlined via a series of lecture notes that are intimately related, centered around coherent themes and concepts. It serves as a textbook mainly for a senior-level undergraduate course, yet is also suitable for a first-year graduate course. Readers benefit from having a good background in linear algebra, some exposure to probability, and basic familiarity with Python.

Book Non convex Optimization in Machine Learning

Download or read book Non convex Optimization in Machine Learning written by Majid Janzamin and published by . This book was released on 2016 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Book Topics in Non convex Optimization and Learning

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.

Book Algorithms for Convex Optimization

Download or read book Algorithms for Convex Optimization written by Nisheeth K. Vishnoi and published by Cambridge University Press. This book was released on 2021-10-07 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Book Sample Efficient Nonconvex Optimization Algorithms in Machine Learning and Reinforcement Learning

Download or read book Sample Efficient Nonconvex Optimization Algorithms in Machine Learning and Reinforcement Learning written by Pan Xu and published by . This book was released on 2021 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: Machine learning and reinforcement learning have achieved tremendous success in solving problems in various real-world applications. Many modern learning problems boil down to a nonconvex optimization problem, where the objective function is the average or the expectation of some loss function over a finite or infinite dataset. Solving such nonconvex optimization problems, in general, can be NP-hard. Thus one often tackles such a problem through incremental steps based on the nature and the goal of the problem: finding a first-order stationary point, finding a second-order stationary point (or a local optimum), and finding a global optimum. With the size and complexity of the machine learning datasets rapidly increasing, it has become a fundamental challenge to design efficient and scalable machine learning algorithms that can improve the performance in terms of accuracy and save computational cost in terms of sample efficiency at the same time. Though many algorithms based on stochastic gradient descent have been developed and widely studied theoretically and empirically for nonconvex optimization, it has remained an open problem whether we can achieve the optimal sample complexity for finding a first-order stationary point and for finding local optima in nonconvex optimization. In this thesis, we start with the stochastic nested variance reduced gradient (SNVRG) algorithm, which is developed based on stochastic gradient descent methods and variance reduction techniques. We prove that SNVRG achieves the near-optimal convergence rate among its type for finding a first-order stationary point of a nonconvex function. We further build algorithms to efficiently find the local optimum of a nonconvex objective function by examining the curvature information at the stationary point found by SNVRG. With the ultimate goal of finding the global optimum in nonconvex optimization, we then provide a unified framework to analyze the global convergence of stochastic gradient Langevin dynamics-based algorithms for a nonconvex objective function. In the second part of this thesis, we generalize the aforementioned sample-efficient stochastic nonconvex optimization methods to reinforcement learning problems, including policy gradient, actor-critic, and Q-learning. For these problems, we propose novel algorithms and prove that they enjoy state-of-the-art theoretical guarantees on the sample complexity. The works presented in this thesis form an incomplete collection of the recent advances and developments of sample-efficient nonconvex optimization algorithms for both machine learning and reinforcement learning.

Book Nonsmooth Optimization and Related Topics

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

Book Convex Analysis and Nonlinear Optimization

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.

Book Convex and Non convex Optimization Methods for Machine Learning

Download or read book Convex and Non convex Optimization Methods for Machine Learning written by Fariba Zohrizadeh and published by . This book was released on 2019 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation is concerned with modeling fundamental and challenging machine learning tasks as convex/non-convex optimization problems and designing a mechanism that could solve them in a cost and time-effective manner. Extensive theoretical and practical studies are carried out to give deeper insights into the robustness and effectiveness of the formulated problems. In what follows, we investigate some well-known tasks that frequently arise in machine learning applications. Image Segmentation: Image segmentation is a fundamental and challenging task in computer vision with diverse applications in various areas. One of the major challenges in image segmentation is to determine the optimal number of coherent regions. This dissertation develops a novel and highly parallelizable convex model which takes into account the spatial relationship between the image features and simultaneously determines the number of clusters and their associated members. To solve the model, a computationally efficient algorithm is presented based on the alternating direction method of multiplier. Extensive experiments on benchmark image segmentation datasets demonstrate that the proposed method can provide high quality and competitive results compared to the existing state-of-the-art methods. Convex Relaxation for Solving Optimization Problems with Orthogonality Constraints: A class of optimization problems with orthogonality constraints has been used to model various applications in machine learning such as discriminative dimensionality reduction, graph matching, dictionary learning, etc. Such optimization problems include nonconvex nonlinear equations, that substantively increase the computational complexity of the problems. In this dissertation, we develop a sequential approach based on parabolic relaxation which finds an orthogonal matrix that minimizes a non-convex and non-smooth objective function subject to additional quadratic constraints. We prove that under very mild assumptions, the proposed approach is guaranteed to provide a feasible solution for the original non-convex problem. The effectiveness of the proposed scheme is corroborated for the problem of discriminative dimensionality reduction and graph clustering. Convex Relaxation for Training Neural Networks: Training of a neural network is formulated as a complex optimization problem which is non-convex and inherently hard to solve. In this dissertation, we propose a novel convexification approach that reduces the training problem into solving a sequence of polynomial-time solvable convex surrogates. The proposed approach, called convexified neural network (Convex-NN), jointly estimates the network parameters of all layers and can admit a wide range of additional convex constraints. We theoretically prove that Convex-NN is guaranteed to converge under mild conditions and perform empirical experiments to corroborate the effectiveness of the method. Class Subset Selection for Partial Domain Adaptation: Deep neural networks have demonstrated superior performance in a variety of machine learning problems. These impressive achievements often rely on the availability of large amounts of labeled training data. However, in many applications, the acquisition of sufficient labeled data is difficult and time-consuming. This provides a strong motivation to reduce the labeling cost and effort by learning effective predictive models using richly-annotated datasets and transferring the knowledge to unlabeled datasets from different but related domains. This dissertation proposes an adversarial-based method for the problem of partial domain adaptation (PDA) in which the source label space is a subset of the target label space. Empirical results demonstrate the high potential of the proposed approach in addressing different partial domain adaptation tasks.

Book Optimization in Machine Learning and Applications

Download or read book Optimization in Machine Learning and Applications written by Anand J. Kulkarni and published by Springer Nature. This book was released on 2019-11-29 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Book Nonconvex Optimization and Model Representation with Applications in Control Theory and Machine Learning

Download or read book Nonconvex Optimization and Model Representation with Applications in Control Theory and Machine Learning written by Yue Sun and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In control and machine learning, the primary goal is to learn the models that make predictions or decisions and act in the world. This thesis covers two important aspects for control theory and machine learning: the model structure that allows low training and generalization error with few samples (i.e., low sample complexity), and convergence guarantees for first-order optimization algorithms for nonconvex optimization. If the model and the training algorithm apply the knowledge of the structure of data (such as sparsity, low-rankness, etc.), the model can be learned with low sample complexity. We present two results, the Hankel nuclear norm regularization method for learning a low order system, and the overparameterized representation for linear meta-learning. We study dynamical system identification in the first result. We assume the true system order is low. A low system order means that the state can be represented by a low dimensional vector, and the system corresponds to a low rank Hankel matrix. The low-rankness is known to be encouraged by nuclear norm regularized estimator in matrix completion theory. We apply a nuclear norm regularized estimator for Hankel matrix, and show that it requires fewer samples than the ordinary least squares estimator. We study linear meta-learning in the second part. The meta-learning algorithm contains two steps: learning a large model in representation learning stage, and fine tuning the model in few-shot learning stage. The few-shot dataset contains few samples, and to avoid overfitting, we need a fine-tuning algorithm that uses the information from representation learning. We generalize the subspace-based model in prior arts to Gaussian model, and describe the overparameterized meta-learning procedure. We show that the feature-task alignment reduces the sample complexity in representation learning, and the optimal task representation is overparameterized. First order optimization methods such as gradient based method, is widely used in machine learning thanks to its simplicity for implementation and fast convergence. However, the objective function in machine learning can be nonconvex, and the first order method has only the theoretical guarantee that it converges to a stationary point, rather than a local/global minimum. We dive into more refined analysis of the convergence guarantee, and present two results, the convergence of perturbed gradient descent approach to a local minimum on Riemannian manifold, and a unified global convergence result of policy gradient descent for linear system control problems. We study how Riemannian gradient converges to an approximate local minimum in the first part. While it is well-known that the perturbed gradient descent escapes saddle points in Euclidean space, less is known about the concrete convergence rate when we apply Riemannian gradient descent on the manifold. In the first result, we show that the perturbed Riemannian gradient descent converges to an approximate local minimum and reveal the relation between convergence rate and the manifold curvature. We study the policy gradient descent applied in control in the second part. Many control problems are revisited under the context of the recent boom in reinforcement learning (RL), however, there is a gap between the RL and control methodology: The policy gradient in RL applies first-order method on nonconvex landscape, and it is hard to show they converge to global minimum, while control theory invents reparameterization that makes the problem convex and they are proven to find the globally optimal controller in polynomial time. Targeting on interpreting the success of the nonconvex method, in the second result, we connect the nonconvex policy gradient descent applied for a collection of control problems with their convex parameterization, and propose a unified proof for the global convergence of policy gradient descent.