Concentration of Measure for the Analysis of Randomized Algorithms

Author: Devdatt P. Dubhashi,Alessandro Panconesi

Publisher: Cambridge University Press

ISBN: 1139480995

Category: Computers

Page: N.A

View: 8179

Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms. It covers the basic toolkit from the Chernoff–Hoeffding bounds to more sophisticated techniques like martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as Chernoff–Hoeffding bounds in dependent settings. The authors emphasise comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.

The Concentration of Measure Phenomenon

Author: Michel Ledoux

Publisher: American Mathematical Soc.

ISBN: 0821837923

Category: Mathematics

Page: 181

View: 3919

It was undoubtedly a necessary task to collect all the results on the concentration of measure during the past years in a monograph. The author did this very successfully and the book is an important contribution to the topic. It will surely influence further research in this area considerably. The book is very well written, and it was a great pleasure for the reviewer to read it. --Mathematical Reviews The observation of the concentration of measure phenomenon is inspired by isoperimetric inequalities. A familiar example is the way the uniform measure on the standard sphere $S^n$ becomes concentrated around the equator as the dimension gets large. This property may be interpreted in terms of functions on the sphere with small oscillations, an idea going back to Levy. The phenomenon also occurs in probability, as a version of the law of large numbers, due to Emile Borel. This book offers the basic techniques and examples of the concentration of measure phenomenon. The concentration of measure phenomenon was put forward in the early seventies by V. Milman in the asymptotic geometry of Banach spaces. It is of powerful interest in applications in various areas, such as geometry, functional analysis and infinite-dimensional integration, discrete mathematics and complexity theory, and probability theory. Particular emphasis is on geometric, functional, and probabilistic tools to reach and describe measure concentration in a number of settings. The book presents concentration functions and inequalities, isoperimetric and functional examples, spectrum and topological applications, product measures, entropic and transportation methods, as well as aspects of M. Talagrand's deep investigation of concentration in product spaces and its application in discrete mathematics and probability theory, supremum of Gaussian and empirical processes, spin glass, random matrices, etc. Prerequisites are a basic background in measure theory, functional analysis, and probability theory.

Probability and Computing

Randomized Algorithms and Probabilistic Analysis

Author: Michael Mitzenmacher,Eli Upfal

Publisher: Cambridge University Press

ISBN: 9780521835404

Category: Computers

Page: 352

View: 831

Randomization and probabilistic techniques play an important role in modern computer science, with applications ranging from combinatorial optimization and machine learning to communication networks and secure protocols.Assuming only an elementary background in discrete mathematics, this textbook is designed to accompany a one- or two-semester course for advanced undergraduates or beginning graduate students in computer science and applied mathematics. It gives an excellent introduction to the probabilistic techniques and paradigms used in the development of probabilistic algorithms and analyses, including random sampling, expectations, Markov's and Chevyshev's inequalities, Chernoff bounds, balls and bins models, the probabilistic method, Markov chains, MCMC, martingales, entropy, and other topics.

Concentration Inequalities

A Nonasymptotic Theory of Independence

Author: Stéphane Boucheron,Gábor Lugosi,Pascal Massart

Publisher: Oxford University Press

ISBN: 0199535256

Category: Mathematics

Page: 481

View: 4622

An accessible account of the rich theory surrounding concentration inequalities in probability theory, with applications from machine learning and statistics to high-dimensional geometry. This book introduces key ideas and presents a detailed summary of the state-of-the-art in the area, making it ideal for independent learning and as a reference.

High-Dimensional Probability

An Introduction with Applications in Data Science

Author: Roman Vershynin

Publisher: Cambridge University Press

ISBN: 1108415199

Category: Business & Economics

Page: 296

View: 4440

An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.

Spectral Algorithms

Author: Ravindran Kannan,Santosh Vempala

Publisher: Now Publishers Inc

ISBN: 1601982747

Category: Computers

Page: 139

View: 739

Spectral methods refer to the use of eigenvalues, eigenvectors, singular values and singular vectors. They are widely used in Engineering, Applied Mathematics and Statistics. More recently, spectral methods have found numerous applications in Computer Science to "discrete" as well "continuous" problems. Spectral Algorithms describes modern applications of spectral methods, and novel algorithms for estimating spectral parameters. The first part of the book presents applications of spectral methods to problems from a variety of topics including combinatorial optimization, learning and clustering. The second part of the book is motivated by efficiency considerations. A feature of many modern applications is the massive amount of input data. While sophisticated algorithms for matrix computations have been developed over a century, a more recent development is algorithms based on "sampling on the y" from massive matrices. Good estimates of singular values and low rank approximations of the whole matrix can be provably derived from a sample. The main emphasis in the second part of the book is to present these sampling methods with rigorous error bounds. It also presents recent extensions of spectral methods from matrices to tensors and their applications to some combinatorial optimization problems.

Probability and Computing

Randomization and Probabilistic Techniques in Algorithms and Data Analysis

Author: Michael Mitzenmacher,Eli Upfal

Publisher: Cambridge University Press

ISBN: 110715488X

Category: Computers

Page: 488

View: 1010

Greatly expanded, this new edition requires only an elementary background in discrete mathematics and offers a comprehensive introduction to the role of randomization and probabilistic techniques in modern computer science. Newly added chapters and sections cover topics including normal distributions, sample complexity, VC dimension, Rademacher complexity, power laws and related distributions, cuckoo hashing, and the Lovasz Local Lemma. Material relevant to machine learning and big data analysis enables students to learn modern techniques and applications. Among the many new exercises and examples are programming-related exercises that provide students with excellent training in solving relevant problems. This book provides an indispensable teaching tool to accompany a one- or two-semester course for advanced undergraduate students in computer science and applied mathematics.

Probabilistic Methods for Algorithmic Discrete Mathematics

Author: Michel Habib,Colin McDiarmid,Jorge Ramirez-Alfonsin,Bruce Reed

Publisher: Springer Science & Business Media

ISBN: 3662127881

Category: Mathematics

Page: 325

View: 9827

Leave nothing to chance. This cliche embodies the common belief that ran domness has no place in carefully planned methodologies, every step should be spelled out, each i dotted and each t crossed. In discrete mathematics at least, nothing could be further from the truth. Introducing random choices into algorithms can improve their performance. The application of proba bilistic tools has led to the resolution of combinatorial problems which had resisted attack for decades. The chapters in this volume explore and celebrate this fact. Our intention was to bring together, for the first time, accessible discus sions of the disparate ways in which probabilistic ideas are enriching discrete mathematics. These discussions are aimed at mathematicians with a good combinatorial background but require only a passing acquaintance with the basic definitions in probability (e.g. expected value, conditional probability). A reader who already has a firm grasp on the area will be interested in the original research, novel syntheses, and discussions of ongoing developments scattered throughout the book. Some of the most convincing demonstrations of the power of these tech niques are randomized algorithms for estimating quantities which are hard to compute exactly. One example is the randomized algorithm of Dyer, Frieze and Kannan for estimating the volume of a polyhedron. To illustrate these techniques, we consider a simple related problem. Suppose S is some region of the unit square defined by a system of polynomial inequalities: Pi (x. y) ~ o.

Understanding Machine Learning

From Theory to Algorithms

Author: Shai Shalev-Shwartz,Shai Ben-David

Publisher: Cambridge University Press

ISBN: 1107057132

Category: Computers

Page: 409

View: 5693

Introduces machine learning and its algorithmic paradigms, explaining the principles behind automated learning approaches and the considerations underlying their usage.

An Introduction to Data Structures and Algorithms

Author: J.A. Storer

Publisher: Springer Science & Business Media

ISBN: 146120075X

Category: Computers

Page: 599

View: 6687

Data structures and algorithms are presented at the college level in a highly accessible format that presents material with one-page displays in a way that will appeal to both teachers and students. The thirteen chapters cover: Models of Computation, Lists, Induction and Recursion, Trees, Algorithm Design, Hashing, Heaps, Balanced Trees, Sets Over a Small Universe, Graphs, Strings, Discrete Fourier Transform, Parallel Computation. Key features: Complicated concepts are expressed clearly in a single page with minimal notation and without the "clutter" of the syntax of a particular programming language; algorithms are presented with self-explanatory "pseudo-code." * Chapters 1-4 focus on elementary concepts, the exposition unfolding at a slower pace. Sample exercises with solutions are provided. Sections that may be skipped for an introductory course are starred. Requires only some basic mathematics background and some computer programming experience. * Chapters 5-13 progress at a faster pace. The material is suitable for undergraduates or first-year graduates who need only review Chapters 1 -4. * This book may be used for a one-semester introductory course (based on Chapters 1-4 and portions of the chapters on algorithm design, hashing, and graph algorithms) and for a one-semester advanced course that starts at Chapter 5. A year-long course may be based on the entire book. * Sorting, often perceived as rather technical, is not treated as a separate chapter, but is used in many examples (including bubble sort, merge sort, tree sort, heap sort, quick sort, and several parallel algorithms). Also, lower bounds on sorting by comparisons are included with the presentation of heaps in the context of lower bounds for comparison-based structures. * Chapter 13 on parallel models of computation is something of a mini-book itself, and a good way to end a course. Although it is not clear what parallel

Data Streams

Algorithms and Applications

Author: S. Muthukrishnan

Publisher: Now Publishers Inc

ISBN: 9781933019147

Category: Computers

Page: 126

View: 6204

In the data stream scenario, input arrives very rapidly and there is limited memory to store the input. Algorithms have to work with one or few passes over the data, space less than linear in the input size or time significantly less than the input size. In the past few years, a new theory has emerged for reasoning about algorithms that work within these constraints on space, time, and number of passes. Some of the methods rely on metric embeddings, pseudo-random computations, sparse approximation theory and communication complexity. The applications for this scenario include IP network traffic analysis, mining text message streams and processing massive data sets in general. Researchers in Theoretical Computer Science, Databases, IP Networking and Computer Systems are working on the data stream challenges

Mining of Massive Datasets

Author: Jure Leskovec,Anand Rajaraman,Jeffrey David Ullman

Publisher: Cambridge University Press

ISBN: 1107077230

Category: Computers

Page: 476

View: 4126

Now in its second edition, this book focuses on practical algorithms for mining data from even the largest datasets.

Probabilistic and Randomized Methods for Design under Uncertainty

Author: Giuseppe Calafiore,Fabrizio Dabbene

Publisher: Springer Science & Business Media

ISBN: 1846280958

Category: Technology & Engineering

Page: 458

View: 8483

Probabilistic and Randomized Methods for Design under Uncertainty is a collection of contributions from the world’s leading experts in a fast-emerging branch of control engineering and operations research. The book will be bought by university researchers and lecturers along with graduate students in control engineering and operational research.

Sublinear Algorithms for Big Data Applications

Author: Dan Wang,Zhu Han

Publisher: Springer

ISBN: 3319204483

Category: Computers

Page: 85

View: 3692

The brief focuses on applying sublinear algorithms to manage critical big data challenges. The text offers an essential introduction to sublinear algorithms, explaining why they are vital to large scale data systems. It also demonstrates how to apply sublinear algorithms to three familiar big data applications: wireless sensor networks, big data processing in Map Reduce and smart grids. These applications present common experiences, bridging the theoretical advances of sublinear algorithms and the application domain. Sublinear Algorithms for Big Data Applications is suitable for researchers, engineers and graduate students in the computer science, communications and signal processing communities.

The Algorithmic Foundations of Differential Privacy

Author: Cynthia Dwork,Aaron Roth

Publisher: Now Publishers

ISBN: 9781601988188

Category: Computers

Page: 286

View: 5401

The problem of privacy-preserving data analysis has a long history spanning multiple disciplines. As electronic data about individuals becomes increasingly detailed, and as technology enables ever more powerful collection and curation of these data, the need increases for a robust, meaningful, and mathematically rigorous definition of privacy, together with a computationally rich class of algorithms that satisfy this definition. Differential Privacy is such a definition. The Algorithmic Foundations of Differential Privacy starts out by motivating and discussing the meaning of differential privacy, and proceeds to explore the fundamental techniques for achieving differential privacy, and the application of these techniques in creative combinations, using the query-release problem as an ongoing example. A key point is that, by rethinking the computational goal, one can often obtain far better results than would be achieved by methodically replacing each step of a non-private computation with a differentially private implementation. Despite some powerful computational results, there are still fundamental limitations. Virtually all the algorithms discussed herein maintain differential privacy against adversaries of arbitrary computational power -- certain algorithms are computationally intensive, others are efficient. Computational complexity for the adversary and the algorithm are both discussed. The monograph then turns from fundamentals to applications other than query-release, discussing differentially private methods for mechanism design and machine learning. The vast majority of the literature on differentially private algorithms considers a single, static, database that is subject to many analyses. Differential privacy in other models, including distributed databases and computations on data streams, is discussed. The Algorithmic Foundations of Differential Privacy is meant as a thorough introduction to the problems and techniques of differential privacy, and is an invaluable reference for anyone with an interest in the topic.

Large Networks and Graph Limits

Author: László Lovász

Publisher: American Mathematical Soc.

ISBN: 0821890859

Category: Mathematics

Page: 475

View: 2046

Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs which has emerged over the last decade.

Cognitive Networked Sensing and Big Data

Author: Robert Qiu,Michael Wicks

Publisher: Springer Science & Business Media

ISBN: 1461445442

Category: Technology & Engineering

Page: 614

View: 4741

Wireless Distributed Computing and Cognitive Sensing defines high-dimensional data processing in the context of wireless distributed computing and cognitive sensing. This book presents the challenges that are unique to this area such as synchronization caused by the high mobility of the nodes. The author will discuss the integration of software defined radio implementation and testbed development. The book will also bridge new research results and contextual reviews. Also the author provides an examination of large cognitive radio network; hardware testbed; distributed sensing; and distributed computing.

Tools and Algorithms for the Construction and Analysis of Systems

22nd International Conference, TACAS 2016, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016, Eindhoven, The Netherlands, April 2-8, 2016, Proceedings

Author: Marsha Chechik,Jean-François Raskin

Publisher: Springer

ISBN: 3662496747

Category: Computers

Page: 961

View: 4506

This book constitutes the proceedings of the 22nd International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2016, which took place in Eindhoven, The Netherlands, in April 2016, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2016. The 44 full papers presented in this volume were carefully reviewed and selected from 175 submissions. They were organized in topical sections named: abstraction and verification; probabilistic and stochastic systems; synthesis; tool papers; concurrency; tool demos; languages and automata; security; optimization; and competition on software verification – SV-COMP.