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Author: Zhouchen Lin Publisher: Springer Nature ISBN: 9811529108 Category : Computers Languages : en Pages : 286
Book Description
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.
Author: Zhouchen Lin Publisher: Springer Nature ISBN: 9811529108 Category : Computers Languages : en Pages : 286
Book Description
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.
Author: Guanghui Lan Publisher: Springer Nature ISBN: 3030395685 Category : Mathematics Languages : en Pages : 591
Book Description
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.
Author: Anand J. Kulkarni Publisher: Springer Nature ISBN: 9811509948 Category : Technology & Engineering Languages : en Pages : 202
Book Description
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.
Author: Alireza Fallah Publisher: ISBN: Category : Languages : en Pages : 99
Book Description
In this thesis, we study the problem of minimizing a smooth and strongly convex function, which arises in different areas, including regularized regression problems in machine learning. To solve this optimization problem, we consider using first order methods which are popular due to their scalability with large data sets, and we study the case that the exact gradient information is not available. In this setting, a naive implementation of classical first order algorithms need not converge and even accumulate noise. This motivates consideration of robustness of algorithms to noise as another metric in designing fast algorithms. To address this problem, we first propose a definition for the robustness of an algorithm in terms of the asymptotic expected suboptimality of its iterate sequence to input noise power. We focus on Gradient Descent and Accelerated Gradient methods and develop a framework based on a dynamical system representation of these algorithms to characterize their convergence rate and robustness to noise using tools from control theory and optimization. We provide explicit expressions for the convergence rate and robustness of both algorithms for the quadratic case, and also derive tractable and tight upper bounds for general smooth and strongly convex functions. We also develop a computational framework for choosing parameters of these algorithms to achieve a particular trade-off between robustness and rate. As a second contribution, we consider algorithms that can reach optimality (obtaining perfect robustness). The past literature provided lower bounds on the rate of decay of suboptimality in term of initial distance to optimality (in the deterministic case) and error due to gradient noise (in the stochastic case). We design a novel multistage and accelerated universally optimal algorithm that can achieve both of these lower bounds simultaneously without knowledge of initial optimality gap or noise characterization. We finally illustrate the behavior of our algorithm through numerical experiments.
Book Description
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.
Author: Yurii Nesterov Publisher: Springer ISBN: 3319915789 Category : Mathematics Languages : en Pages : 603
Book Description
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.
Author: Suvrit Sra Publisher: MIT Press ISBN: 026201646X Category : Computers Languages : en Pages : 509
Book Description
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.
Author: Anant Raj Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
With the advent of massive datasets and increasingly complex tasks, modern machine learning systems pose several new challenges in terms of scalability to high dimensional data as well as to large datasets. In this thesis, we consider to study scalable descent methods such as coordinate descent and stochastic coordinate descent which are based on the stochastic approximation of full gradient. In the first part of the thesis, we propose faster and scalable coordinate based opti- mization which scales to high dimensional problems. As a first step to achieve scalable coordinate based descent approaches, we propose a new framework to derive screening rules for convex optimization problems based on duality gap which covers a large class of constrained and penalized optimization formulations. In later stages, we develop new approximately greedy coordinate selection strategy in coordinate descent for large-scale optimization. This novel coordinate selection strategy provavbly works better than uni- formly random selection, and can reach the efficiency of steepest coordinate descent (SCD) in the best case. In best case scenario, this may enable an acceleration of a factor of up to n, the number of coordinates. Having similar objective in mind, we further propose an adaptive sampling strategy for sampling in stochastic gradient based optimization. The proposed safe sampling scheme provably achieves faster convergence than any fixed deterministic sampling schemes for coordinate descent and stochastic gradient descent methods. Exploiting the connection between matching pursuit where a more generalized notion of directions is considered and greedy coordinate descent where all the moving directions are orthogonal, we also propose a unified analysis for both the approaches and extend it to get the accelerated rate. In the second part of this thesis, we focus on providing provably faster and scalable mini batch stochastic gradient descent (SGD) algorithms. Variance reduced SGD methods converge significantly faster than the vanilla SGD counterpart. We propose a variance reduce algorithm k-SVRG that addresses issues of SVRG [98] and SAGA[54] by making best use of the available memory and minimizes the stalling phases without progress. In later part of the work, we provide a simple framework which utilizes the idea of optimistic update to obtain accelerated stochastic algorithms. We obtain accelerated variance reduced algorithm as well as accelerated universal algorithm as a direct consequence of this simple framework. Going further, we also employ the idea of local sensitivity based importance sampling in an iterative optimization method and analyze its convergence while optimizing over the selected subset. In the final part of the thesis, we connect the dots between coordinate descent method and stochastic gradient descent method in the interpolation regime. We show that better stochastic gradient based dual algorithms with fast rate of convergence can be obtained to optimize the convex objective in the interpolation regime.
Author: Zhouchen Lin
Author: Zhouchen Lin
Author: Guanghui Lan
Author: Anand J. Kulkarni
Author: Alireza Fallah
Author: Sébastien Bubeck
Author: Chenchao Shou
Author: Yurii Nesterov
Author: Suvrit Sra
Author: Anant Raj
Author: Stephen J. Wright