

Machine learning has witnessed a resurgence of interest over the last few years, which is a consequence of the rapid development of the information industry. Data is no longer a scarce resource—it is abundant. Methods for "intelligent'' data analysis to extract relevant information are needed. The goal of this book is to give a selfcontained overview of machine learning, particularly of kernel classifiers—both from an algorithmic and a theoretical perspective. Although there exist many excellent textbooks on learning algorithms (see Duda and Hart (1973), Bishop (1995), Vapnik (1995), Mitchell (1997) and Cristianini and ShaweTaylor (2000)) and on learning theory (see Vapnik (1982), Kearns and Vazirani (1994), Wolpert (1995), Vidyasagar (1997) and Anthony and Bartlett (1999)), there is no single book which presents both aspects together in reasonable depth. Instead, these monographs often cover much larger areas of function classes, e.g., neural networks, decision trees or rule sets, or learning tasks (for example regression estimation or unsupervised learning). My motivation in writing this book is to summarize the enormous amount of work that has been done in the specific field of kernel classification over the last years. It is my aim to show how all the work is related to each other. To some extent, I also try to demystify some of the recent developments, particularly in learning theory, and to make them accessible to a larger audience. In the course of reading it will become apparent that many already known results are proven again, and in detail, instead of simply referring to them. The motivation for doing this is to have all these different results together in one place—in particular to see their similarities and (conceptual) differences. The book is structured into a general introduction (Chapter 1) and two parts, which can be read independently. The material is emphasized through many examples and remarks. The book finishes with a comprehensive appendix containing mathematical background and proofs of the main theorems. It is my hope that the level of detail chosen makes this book a useful reference for many researchers working in this field. Since the book uses a very rigorous notation systems, it is perhaps advisable to have a quick look at the background material and list of symbols on page 331. The first part of the book is devoted to the study of algorithms for learning kernel classifiers. This part starts with a chapter introducing the basic concepts of learning from a machine learning point of view. The chapter will elucidate the basic concepts involved in learning kernel classifiers—in particular the kernel technique. It introduces the support vector machine learning algorithm as one of the most prominent examples of a learning algorithm for kernel classifiers. The second chapter presents the Bayesian view of learning. In particular, it covers Gaussian processes, the relevance vector machine algorithm and the classical Fisher discriminant. The first part is complemented by Appendix D, which gives all the pseudo code for the presented algorithms. In order to enhance the understandability of the algorithms presented, all algorithms are implemented in R—a statistical language similar to SPLUS. The source code is publicly available at http://www.kernelmachines.org/. At this web site the interested reader will also find additional software packages and many related publications. The second part of the book is devoted to the theoretical study of learning algorithms, with a focus on kernel classifiers. This part can be read rather independently of the first part, although I refer back to specific algorithms at some stages. The first chapter of this part introduces many seemingly different models of learning. It was my objective to give easytofollow "proving arguments'' for their main results, sometimes presented in a "vanilla'' version. In order to unburden the main body, all technical details are relegated to Appendix B and C. The classical PAC and VC frameworks are introduced as the most prominent examples of mathematical models for the learning task. It turns out that, despite their unquestionable generality, they only justify training error minimization and thus do not fully use the training sample to get better estimates for the generalization error. The following section introduces a very general framework for learning—the luckiness framework. This chapter concludes with a PACstyle analysis for the particular class of realvalued (linear) functions, which qualitatively justifies the support vector machine learning algorithm. Whereas the first chapter was concerned with bounds which hold uniformly for all classifiers, the methods presented in the second chapter provide bounds for specific learning algorithms. I start with the PACBayesian framework for learning, which studies the generalization error of Bayesian learning algorithms. Subsequently, I demonstrate that for all learning algorithms that can be expressed as compression schemes, we can upper bound the generalization error by the fraction of training examples used—a quantity which can be viewed as a compression coefficient. The last section of this chapter contains a very recent development known as algorithmic stability bounds. These results apply to all algorithms for which an additional training example has only limited influence. As with every book, this monograph has (almost surely) typing errors as well as other mistakes. Therefore, whenever you find a mistake in this book, I would be very grateful to receive an email at herbrich@kernelmachines.org. The list of errata will be publicly available at http://www.kernelmachines.org. This book is the result of two years' work of a computer scientist with a strong interest in mathematics who stumbled onto the secrets of statistics rather innocently. Being originally fascinated by the the field of artificial intelligence, I started programming different learning algorithms, finally ending up with a giant learning system that was completely unable to generalize. At this stage my interest in learning theory was born—highly motivated by the seminal book by Vapnik (1995). In recent times, my focus has shifted toward theoretical aspects. Taking that into account, this book might at some stages look mathematically overloaded (from a practitioner's point of view) or too focused on algorithmical aspects (from a theoretician's point of view). As it presents a snapshot of the stateoftheart, the book may be difficult to access for people from a completely different field. As complementary texts, I highly recommend the books by Cristianini and ShaweTaylor (2000) and Vapnik 1995). This book is partly based on my doctoral thesis Herbrich (2000), which I wrote at the Technical University of Berlin. I would like to thank the whole statistics group at the Technical University of Berlin with whom I had the pleasure of carrying out research in an excellent environment. In particular, the discussions with Peter BollmannSdorra, Matthias Burger, Jörg Betzin and Jürgen Schweiger were very inspiring. I am particularly grateful to my supervisor, Professor Ulrich Kockelkorn, whose help was invaluable. Discussions with him were always very delightful, and I would like to thank him particularly for the inspiring environment he provided. I am also indebted to my second supervisor, Professor John ShaweTaylor, who made my short visit at the Royal Holloway College a total success. His support went far beyond the short period at the college, and during the many discussions we had, I easily understood most of the recent developments in learning theory. His "anytime availability'' was of uncountable value while writing this book. Thank you very much! Furthermore, I had the opportunity to visit the Department of Engineering at the Australian National University in Canberra. I would like to thank Bob Williamson for this opportunity, for his great hospitality and for the many fruitful discussions. This book would not be as it is without the many suggestions he had. Finally, I would like to thank Chris Bishop for giving all the support I needed to complete the book during my first few months at Microsoft Research Cambridge. During the last three years I have had the good fortune to receive help from many people all over the world. Their views and comments on my work were very influential in leading to the current publication. Some of the many people I am particularly indebted to are David McAllester, Peter Bartlett, Jonathan Baxter, Shai BenDavid, Colin Campbell, Nello Cristianini, Denver Dash, Thomas Hofmann, Neil Lawrence, Jens Matthias, Manfred Opper, Patrick Perez, Gunnar Rätsch, Craig Saunders, Bernhard Schölkopf, Matthias Seeger, Alex Smola, Peter Sollich, Mike Tipping, Jaco Vermaak, Jason Weston and Hugo Zaragoza. In the course of writing the book I highly appreciated the help of many people who proofread previous manuscripts. David McAllester, Jörg Betzin, Peter BollmannSdorra, Matthias Burger, Thore Graepel, Ulrich Kockelkorn, John Krumm, Gary Lee, Craig Saunders, Bernhard Schölkopf, Jürgen Schweiger, John ShaweTaylor, Jason Weston, Bob Williamson and Hugo Zaragoza gave helpful comments on the book and found many errors. I am greatly indebted to Simon Hill, whose help in proofreading the final manuscript was invaluable. Thanks to all of you for your enormous help! Special thanks goes to one person—Thore Graepel. We became very good friends far beyond the level of scientific cooperation. I will never forget the many enlightening discussions we had in several pubs in Berlin and the few excellent conference and research trips we made together, in particular our trip to Australia. Our collaboration and friendship was—and still is—of uncountable value for me. Finally, I would like to thank my wife, Jeannette, and my parents for their patience and moral support during the whole time. I could not have done this work without my wife's enduring love and support. I am very grateful for her patience and reassurance at all times. Finally, I would like to thank Mel Goldsipe, Bob Prior, Katherine Innis and Sharon Deacon Warne at The MIT Press for their continuing support and help during the completion of the book. 