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

A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

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Gaussian Processes for Machine Learning

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Preface * Introduction * The Bayes Error * Inequalities and alternatedistance measures * Linear discrimination * Nearest neighbor rules *Consistency * Slow rates of convergence Error estimation * The regularhistogram rule * Kernel rules Consistency of the k-nearest neighborrule * Vapnik-Chervonenkis theory * Combinatorial aspects of Vapnik-Chervonenkis theory * Lower bounds for empirical classifier selection* The maximum likelihood principle * Parametric classification *Generalized linear discrimination * Complexity regularization *Condensed and edited nearest neighbor rules * Tree classifiers * Data-dependent partitioning * Splitting the data * The resubstitutionestimate * Deleted estimates of the error probability * Automatickernel rules * Automatic nearest neighbor rules * Hypercubes anddiscrete spaces * Epsilon entropy and totally bounded sets * Uniformlaws of large numbers * Neural networks * Other error estimates *Feature extraction * Appendix * Notation * References * Index

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A Probabilistic Theory of Pattern Recognition

We describe the maximum-likelihood parameter estimation problem and how the ExpectationMaximization (EM) algorithm can be used for its solution. We first describe the abstract form of the EM algorithm as it is often given in the literature. We then develop the EM parameter estimation procedure for two applications: 1) finding the parameters of a mixture of Gaussian densities, and 2) finding the parameters of a hidden Markov model (HMM) (i.e., the Baum-Welch algorithm) for both discrete and Gaussian mixture observation models. We derive the update equations in fairly explicit detail but we do not prove any convergence properties. We try to emphasize intuition rather than mathematical rigor.

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A gentle tutorial of the em algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models

In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwer's fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorithms have since been proposed for finding Nash equilibria, but none known to run in polynomial time. In 1991 the complexity class PPAD (polynomial parity arguments on directed graphs), for which Brouwer's problem is complete, was introduced [C. Papadimitriou, J. Comput. System Sci., 48 (1994), pp. 489-532], motivated largely by the classification problem for Nash equilibria; but whether the Nash problem is complete for this class remained open. In this paper we resolve these questions: We show that finding a Nash equilibrium in three-player games is indeed PPAD-complete; and we do so by a reduction from Brouwer's problem, thus establishing that the two problems are computationally equivalent. Our reduction simulates a (stylized) Brouwer function by a graphical game [M. Kearns, M. Littman, and S. Singh, Graphical model for game theory, in 17th Conference in Uncertainty in Artificial Intelligence (UAI), 2001], relying on “gadgets,” graphical games performing various arithmetic and logical operations. We then show how to simulate this graphical game by a three-player game, where each of the three players is essentially a color class in a coloring of the underlying graph. Subsequent work [X. Chen and X. Deng, Setting the complexity of 2-player Nash-equilibrium, in 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006] established, by improving our construction, that even two-player games are PPAD-complete; here we show that this result follows easily from our proof.

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The Complexity of Computing a Nash Equilibrium

Genetic and anatomical evidence suggests that Homo sapiens arose in Africa between 200 and 100 thousand years (kyr) ago, and recent evidence indicates symbolic behaviour may have appeared approximately 135-75 kyr ago. From 195-130 kyr ago, the world was in a fluctuating but predominantly glacial stage (marine isotope stage MIS6); much of Africa was cooler and drier, and dated archaeological sites are rare. Here we show that by approximately 164 kyr ago (+/-12 kyr) at Pinnacle Point (on the south coast of South Africa) humans expanded their diet to include marine resources, perhaps as a response to these harsh environmental conditions. The earliest previous evidence for human use of marine resources and coastal habitats was dated to approximately 125 kyr ago. Coincident with this diet and habitat expansion is an early use and modification of pigment, probably for symbolic behaviour, as well as the production of bladelet stone tool technology, previously dated to post-70 kyr ago. Shellfish may have been crucial to the survival of these early humans as they expanded their home ranges to include coastlines and followed the shifting position of the coast when sea level fluctuated over the length of MIS6.

Early human use of marine resources and pigment in South Africa during the Middle Pleistocene

Basic concepts in soil science and current methods field strategies the microscopic approach thin section description sediments anthropogenic features post-depositional processes and relevant features. Case studies: the cave of Vaufrey, Perigard, France Kebara cave, Israel the cave of the pigeons at Taforalt, Eastern Morocco the Acheulean site of Berekhat Ram, Israel the neolithic site of Netiv Hagdud and the Salibya depression, Lower Jordan Valley, Israel the bronze age/neolithic site of Fort Harrouard, France dark earth, England human settlement patterns and holocene environments in the Shaggar Plain, Northwest India rural settlement at Castellaro di Uscio, Liguria, Italy chalcolithic land-use in Italian Appennines and Alps.

Soils and Micromorphology in Archaeology

The complexity of computing a Nash equilibrium

We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recently-established equivalence between polynomial time solvability of normal form games and graphical games, establishing that these kinds of games can simulate a PPAD-complete class of Brouwer functions.

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Paul Goldberg

Papers

The Complexity of Computing a Nash Equilibrium

Early human use of marine resources and pigment in South Africa during the Middle Pleistocene

Soils and Micromorphology in Archaeology

The complexity of computing a Nash equilibrium

The complexity of computing a Nash equilibrium