Download or read book Gradient based Optimization and Implicit Regularization Over Non convex Landscapes written by Xiaoxia (Shirley) Wu and published by . This book was released on 2020 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: Large-scale machine learning problems can be reduced to non-convex optimization problems if state-of-the-art models such as deep neural networks are applied. One of the most widely-used algorithms is the first-order iterative gradient-based algorithm, i.e., (stochastic) gradient descent method. Two main challenges arise from understanding the gradient-based algorithm over the non-convex landscapes: the convergence complexity and the algorithm's solutions. This thesis aims to tackle the two challenges by providing a theoretical framework and empirical investigation on three popular gradient-based algorithms, namely, adaptive gradient methods [39], weight normalization [138] and curriculum learning [18]. For convergence, the stepsize or learning rate plays a pivotal role in the iteration complexity. However, it depends crucially on the (generally unknown) Lipschitz smoothness constant and noise level on the stochastic gradient. A popular stepsize auto-tuning method is the adaptive gradient methods such as AdaGrad that update the learning rate on the fly according to the gradients received along the way; Yet, the theoretical guarantees to date for AdaGrad are for online and convex optimization. We bridge this gap by providing theoretical guarantees for the convergence of AdaGrad for smooth, non-convex functions; we show that it converges to a stationary point at the (log(N)/ √N) rate in the stochastic setting and at the optimal (1/N) rate in the batch (non-stochastic) setting. Extensive numerical experiments are provided to corroborate our theory. For the gradient-based algorithm solution, we study weight normalization (WN) methods in the setting of an over-parameterized linear regression problem where WN decouples the weight vector with a scale and a unit vector. We show that this reparametrization has beneficial regularization effects compared to gradient descent on the original objective. WN adaptively regularizes the weights and converges close to the minimum l2 norm solution, even for initializations far from zero. To further understand the stochastic gradient-based algorithm, we study the continuation method -- curriculum learning (CL) -- inspired by cognitive science that humans learn from simple to complex order. CL has proposed ordering examples during training based on their difficulty, while anti-CL proposed the opposite ordering. Both CL and anti-CL have been suggested as improvements to the standard i.i.d. training. We set out to investigate the relative benefits of ordered learning in three settings: standard-time, short-time, and noisy label training. We find that both orders have only marginal benefits for standard benchmark datasets. However, with limited training time budget or noisy data, curriculum, but not anti-curriculum ordering, can improve the performance

Download or read book The Complexity of Optimization Beyond Convexity written by Yair Menachem Carmon and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Gradient descent variants are the workhorse of modern machine learning and large-scale optimization more broadly, where objective functions are often non-convex. Could there be better general-purpose optimization methods than gradient descent, or is it in some sense unimprovable? This thesis addresses this question from the perspective of the worst-case oracle complexity of finding near-stationary points (i.e., points with small gradient norm) of smooth and possibly non-convex functions. On the negative side, we prove a lower bound showing that gradient descent is unimprovable for a natural class of problems. We further prove the worst-case optimality of stochastic gradient descent, recursive variance reduction, cubic regularization of Newton's method and high-order tensor methods, in each case under the set of assumptions for which the method was designed. To prove our lower bounds we extend theory of information-based oracle complexity to the realm of non-convex optimization. On the positive side, we use classical techniques from optimization (namely Nesterov momentum and Krylov subspace methods) to accelerate gradient descent in a large subclass of non-convex problems with higher-order smoothness. Furthermore, we show how recently proposed variance reduction techniques can further improve stochastic gradient descent when stochastic Hessian-vector products available.

Download or read book Beyond the Worst Case Analysis of Algorithms written by Tim Roughgarden and published by Cambridge University Press. This book was released on 2021-01-14 with total page 705 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces exciting new methods for assessing algorithms for problems ranging from clustering to linear programming to neural networks.

Download or read book A Flexible Framework for Cubic Regularization Algorithms for Non convex Optimization in Function Space written by Anton Schiela and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Provable Non convex Optimization for Learning Parametric Models written by Kai Zhong (Ph. D.) and published by . This book was released on 2018 with total page 866 pages. Available in PDF, EPUB and Kindle. Book excerpt: Non-convex optimization plays an important role in recent advances of machine learning. A large number of machine learning tasks are performed by solving a non-convex optimization problem, which is generally NP-hard. Heuristics, such as stochastic gradient descent, are employed to solve non-convex problems and work decently well in practice despite the lack of general theoretical guarantees. In this thesis, we study a series of non-convex optimization strategies and prove that they lead to the global optimal solution for several machine learning problems, including mixed linear regression, one-hidden-layer (convolutional) neural networks, non-linear inductive matrix completion, and low-rank matrix sensing. At a high level, we show that the non-convex objectives formulated in the above problems have a large basin of attraction around the global optima when the data has benign statistical properties. Therefore, local search heuristics, such as gradient descent or alternating minimization, are guaranteed to converge to the global optima if initialized properly. Furthermore, we show that spectral methods can efficiently initialize the parameters such that they fall into the basin of attraction. Experiments on synthetic datasets and real applications are carried out to justify our theoretical analyses and illustrate the superiority of our proposed methods.

Download or read book A Feedback based Regularized Primal dual Gradient Method for Time varying Nonconvex Optimization written by Yujie Tang and published by . This book was released on 2019 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Mathematical Aspects of Deep Learning written by Philipp Grohs and published by Cambridge University Press. This book was released on 2022-12-22 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years the development of new classification and regression algorithms based on deep learning has led to a revolution in the fields of artificial intelligence, machine learning, and data analysis. The development of a theoretical foundation to guarantee the success of these algorithms constitutes one of the most active and exciting research topics in applied mathematics. This book presents the current mathematical understanding of deep learning methods from the point of view of the leading experts in the field. It serves both as a starting point for researchers and graduate students in computer science, mathematics, and statistics trying to get into the field and as an invaluable reference for future research.

Download or read book Introduction to Deep Learning A Beginner s Edition written by Harshitha Raghavan Devarajan and published by INENCE PUBLICATIONS PVT LTD. This book was released on 2024-08-10 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Introduction to Deep Learning: A Beginner’s Edition" is a comprehensive guide designed specifically for newcomers to the field of deep learning. This book provides an accessible introduction to the fundamental concepts, making it an ideal starting point for those who are curious about artificial intelligence and its rapidly expanding applications. The book begins with a clear explanation of what deep learning is and how it differs from traditional machine learning, covering the basics of neural networks and how they are used to recognize patterns and make decisions. One of the key strengths of this book is its practical, hands-on approach. Readers are guided through the process of building, training, and deploying neural networks using popular frameworks like TensorFlow and PyTorch. The step-by-step instructions, along with code snippets, allow even those with little to no programming experience to engage actively with the material. Visual aids, such as diagrams and flowcharts, are used throughout the book to simplify complex topics, making it easier for readers to grasp the inner workings of neural networks. The book also explores real-world applications of deep learning, highlighting its impact across various industries, including healthcare, autonomous vehicles, and natural language processing. By providing context and practical examples, the book demonstrates how deep learning is being used to solve complex problems and transform industries. In addition to the core content, the book includes a glossary of key terms, quizzes, and exercises to reinforce learning. "Introduction to Deep Learning: A Beginner’s Edition" is more than just a textbook; it is a complete learning experience designed to equip beginners with the knowledge and skills needed to embark on a successful journey into the world of deep learning.

Download or read book Patterns Predictions and Actions Foundations of Machine Learning written by Moritz Hardt and published by Princeton University Press. This book was released on 2022-08-23 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: An authoritative, up-to-date graduate textbook on machine learning that highlights its historical context and societal impacts Patterns, Predictions, and Actions introduces graduate students to the essentials of machine learning while offering invaluable perspective on its history and social implications. Beginning with the foundations of decision making, Moritz Hardt and Benjamin Recht explain how representation, optimization, and generalization are the constituents of supervised learning. They go on to provide self-contained discussions of causality, the practice of causal inference, sequential decision making, and reinforcement learning, equipping readers with the concepts and tools they need to assess the consequences that may arise from acting on statistical decisions. Provides a modern introduction to machine learning, showing how data patterns support predictions and consequential actions Pays special attention to societal impacts and fairness in decision making Traces the development of machine learning from its origins to today Features a novel chapter on machine learning benchmarks and datasets Invites readers from all backgrounds, requiring some experience with probability, calculus, and linear algebra An essential textbook for students and a guide for researchers

Download or read book Optimization for Machine Learning written by Suvrit Sra and published by MIT Press. This book was released on 2012 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: An up-to-date account of the interplay between optimization and machine learning, accessible to students and researchers in both communities. The interplay between optimization and machine learning is one of the most important developments in modern computational science. Optimization formulations and methods are proving to be vital in designing algorithms to extract essential knowledge from huge volumes of data. Machine learning, however, is not simply a consumer of optimization technology but a rapidly evolving field that is itself generating new optimization ideas. This book captures the state of the art of the interaction between optimization and machine learning in a way that is accessible to researchers in both fields. Optimization approaches have enjoyed prominence in machine learning because of their wide applicability and attractive theoretical properties. The increasing complexity, size, and variety of today's machine learning models call for the reassessment of existing assumptions. This book starts the process of reassessment. It describes the resurgence in novel contexts of established frameworks such as first-order methods, stochastic approximations, convex relaxations, interior-point methods, and proximal methods. It also devotes attention to newer themes such as regularized optimization, robust optimization, gradient and subgradient methods, splitting techniques, and second-order methods. Many of these techniques draw inspiration from other fields, including operations research, theoretical computer science, and subfields of optimization. The book will enrich the ongoing cross-fertilization between the machine learning community and these other fields, and within the broader optimization community.

Download or read book Deep Learning written by Ian Goodfellow and published by MIT Press. This book was released on 2016-11-10 with total page 801 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to a broad range of topics in deep learning, covering mathematical and conceptual background, deep learning techniques used in industry, and research perspectives. “Written by three experts in the field, Deep Learning is the only comprehensive book on the subject.” —Elon Musk, cochair of OpenAI; cofounder and CEO of Tesla and SpaceX Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.

Download or read book Sparse Modeling for Image and Vision Processing written by Julien Mairal and published by Now Publishers. This book was released on 2014-12-19 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sparse Modeling for Image and Vision Processing offers a self-contained view of sparse modeling for visual recognition and image processing. More specifically, it focuses on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.

Download or read book Recent Developments in Electronics and Communication Systems written by KVS Ramachandra Murthy and published by IOS Press. This book was released on 2023-01-31 with total page 746 pages. Available in PDF, EPUB and Kindle. Book excerpt: Often, no single field or expert has all the information necessary to solve complex problems, and this is no less true in the fields of electronics and communications systems. Transdisciplinary engineering solutions can address issues arising when a solution is not evident during the initial development stages in the multidisciplinary area. This book presents the proceedings of RDECS-2022, the 1st international conference on Recent Developments in Electronics and Communication Systems, held on 22 and 23 July 2022 at Aditya Engineering College, Surampalem, India. The primary goal of RDECS-2022 was to challenge existing ideas and encourage interaction between academia and industry to promote the sort of collaborative activities involving scientists, engineers, professionals, researchers, and students that play a major role in almost all fields of scientific growth. The conference also aimed to provide an arena for showcasing advancements and research endeavors being undertaken in all parts of the world. A large number of technical papers with rich content, describing ground-breaking research from participants from various institutes, were submitted for presentation at the conference. This book presents 108 of these papers, which cover a wide range of topics ranging from cloud computing to disease forecasting and from weather reporting to the detection of fake news. Offering a fascinating overview of recent research and developments in electronics and communications systems, the book will be of interest to all those working in the field.

Download or read book Introductory Lectures on Convex Optimization written by Y. Nesterov and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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].

Download or read book Optimization Algorithms on Matrix Manifolds written by P.-A. Absil and published by Princeton University Press. This book was released on 2009-04-11 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

Download or read book Numerical Algorithms written by Justin Solomon and published by CRC Press. This book was released on 2015-06-24 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics presents a new approach to numerical analysis for modern computer scientists. Using examples from a broad base of computational tasks, including data processing, computational photography, and animation, the textbook introduces numerical modeling and algorithmic desig

Download or read book Automated Machine Learning written by Frank Hutter and published by Springer. This book was released on 2019-05-17 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access book presents the first comprehensive overview of general methods in Automated Machine Learning (AutoML), collects descriptions of existing systems based on these methods, and discusses the first series of international challenges of AutoML systems. The recent success of commercial ML applications and the rapid growth of the field has created a high demand for off-the-shelf ML methods that can be used easily and without expert knowledge. However, many of the recent machine learning successes crucially rely on human experts, who manually select appropriate ML architectures (deep learning architectures or more traditional ML workflows) and their hyperparameters. To overcome this problem, the field of AutoML targets a progressive automation of machine learning, based on principles from optimization and machine learning itself. This book serves as a point of entry into this quickly-developing field for researchers and advanced students alike, as well as providing a reference for practitioners aiming to use AutoML in their work.