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Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

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A tutorial on spectral clustering

Kernel methods provide a powerful and unified framework for pattern discovery, motivating algorithms that can act on general types of data (e.g. strings, vectors or text) and look for general types of relations (e.g. rankings, classifications, regressions, clusters). The application areas range from neural networks and pattern recognition to machine learning and data mining. This book, developed from lectures and tutorials, fulfils two major roles: firstly it provides practitioners with a large toolkit of algorithms, kernels and solutions ready to use for standard pattern discovery problems in fields such as bioinformatics, text analysis, image analysis. Secondly it provides an easy introduction for students and researchers to the growing field of kernel-based pattern analysis, demonstrating with examples how to handcraft an algorithm or a kernel for a new specific application, and covering all the necessary conceptual and mathematical tools to do so.

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Kernel Methods for Pattern Analysis

In the field of machine learning, semi-supervised learning (SSL) occupies the middle ground, between supervised learning (in which all training examples are labeled) and unsupervised learning (in which no label data are given). Interest in SSL has increased in recent years, particularly because of application domains in which unlabeled data are plentiful, such as images, text, and bioinformatics. This first comprehensive overview of SSL presents state-of-the-art algorithms, a taxonomy of the field, selected applications, benchmark experiments, and perspectives on ongoing and future research. Semi-Supervised Learning first presents the key assumptions and ideas underlying the field: smoothness, cluster or low-density separation, manifold structure, and transduction. The core of the book is the presentation of SSL methods, organized according to algorithmic strategies. After an examination of generative models, the book describes algorithms that implement the low-density separation assumption, graph-based methods, and algorithms that perform two-step learning. The book then discusses SSL applications and offers guidelines for SSL practitioners by analyzing the results of extensive benchmark experiments. Finally, the book looks at interesting directions for SSL research. The book closes with a discussion of the relationship between semi-supervised learning and transduction. Adaptive Computation and Machine Learning series

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Semi-Supervised Learning

This paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m1 × m2 matrix A0 corrupted by noise are observed. We propose a new nuclear-norm penalized estimator of A0 and establish a general sharp oracle inequality for this estimator for arbitrary values of n, m1, m2 under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form, and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting m1m2 ≫ n. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix A0, a nonminimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of A0 with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by A0, and the aim is to find the best trace regression model approximating the data. As a by-product, we show that, under the restricted eigenvalue condition, the usual vector Lasso estimator satisfies a sharp oracle inequality (i.e., an oracle inequality with leading constant 1).

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Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion

This paper deals with the trace regression model where $n$ entries or linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by noise are observed. We propose a new nuclear norm penalized estimator of $A_0$ and establish a general sharp oracle inequality for this estimator for arbitrary values of $n,m_1,m_2$ under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting $m_1m_2\gg n$. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix $A_0$, a non-minimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of $A_0$ with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by $A_0$ and the aim is to find the best trace regression model approximating the data.

Nuclear norm penalization and optimal rates for noisy low rank matrix completion

We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers. Such combinations could be implemented by neural networks or by voting methods of combining the classifiers, such as boosting and bagging. The bounds are in terms of the empirical distribution of the margin of the combined classifier. They are based on the methods of the theory of Gaussian and empirical processes (comparison inequalities, symmetrization method, concentration inequalities) and they improve previous results of Bartlett (1998) on bounding the generalization error of neural networks in terms of $\ell_1$-norms of the weights of neurons and of Schapire, Freund, Bartlett and Lee (1998) on bounding the generalization error of boosting. We also obtain rates of convergence in Levy distance of empirical margin distribution to the true margin distribution uniformly over the classes of classifiers and prove the optimality of these rates.

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Empirical margin distributions and bounding the generalization error of combined classifiers

The purpose of these lecture notes is to provide an introduction to the general theory of empirical risk minimization with an emphasis on excess risk bounds and oracle inequalities in penalized problems. In recent years, there have been new developments in this area motivated by the study of new classes of methods in machine learning such as large margin classification methods (boosting, kernel machines). The main probabilistic tools involved in the analysis of these problems are concentration and deviation inequalities by Talagrand along with other methods of empirical processes theory (symmetrization inequalities, contraction inequality for Rademacher sums, entropy and generic chaining bounds). Sparse recovery based on l_1-type penalization and low rank matrix recovery based on the nuclear norm penalization are other active areas of research, where the main problems can be stated in the framework of penalized empirical risk minimization, and concentration inequalities and empirical processes tools have proved to be very useful.

Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems: École d'Été de Probabilités de Saint-Flour XXXVIII-2008

We suggest a penalty function to be used in various problems of structural risk minimization. This penalty is data dependent and is based on the sup-norm of the so-called Rademacher process indexed by the underlying class of functions (sets). The standard complexity penalties, used in learning problems and based on the VC-dimensions of the classes, are conservative upper bounds (in a probabilistic sense, uniformly over the set of all underlying distributions) for the penalty we suggest. Thus, for a particular distribution of training examples, one can expect better performance of learning algorithms with the data-driven Rademacher penalties. We obtain oracle inequalities for the theoretical risk of estimators, obtained by structural minimization of the empirical risk with Rademacher penalties. The inequalities imply some form of optimality of the empirical risk minimizers. We also suggest an iterative approach to structural risk minimization with Rademacher penalties, in which the hierarchy of classes is not given in advance, but is determined in the data-driven iterative process of risk minimization. We prove probabilistic oracle inequalities for the theoretical risk of the estimators based on this approach as well.

Vladimir Koltchinskii

Papers

Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion

Nuclear norm penalization and optimal rates for noisy low rank matrix completion

Empirical margin distributions and bounding the generalization error of combined classifiers

Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems: École d'Été de Probabilités de Saint-Flour XXXVIII-2008

Rademacher penalties and structural risk minimization