We provide a general theory of the expectation-maximization (EM) algorithm for inferring high dimensional latent variable models. In particular, we make two contributions: (i) For parameter estimation, we propose a novel high dimensional EM algorithm which naturally incorporates sparsity structure into parameter estimation. With an appropriate initialization, this algorithm converges at a geometric rate and attains an estimator with the (near-)optimal statistical rate of convergence. (ii) Based on the obtained estimator, we propose new inferential procedures for testing hypotheses and constructing confidence intervals for low dimensional components of high dimensional parameters. For a broad family of statistical models, our framework establishes the first computationally feasible approach for optimal estimation and asymptotic inference in high dimensions. Our theory is supported by thorough numerical results.

This paper establishes minimax rates for online regression with arbitrary classes of functions and general losses. We show that below a certain threshold for the complexity of the function class, the minimax rates depend on both the curvature of the loss function and the sequential complexities of the class. Above this threshold, the curvature of the loss does not affect the rates. Furthermore, for the case of square loss, our results point to the interesting phenomenon: whenever sequential and i.i.d. empirical entropies match, the rates for statistical and online learning are the same. In addition to the study of minimax regret, we derive a generic forecaster that enjoys the established optimal rates. We also provide a recipe for designing online prediction algorithms that can be computationally efficient for certain problems. We illustrate the techniques by deriving existing and new forecasters for the case of finite experts and for online linear regression.

We focus on the distribution regression problem: regressing to a real-valued response from a probability distribution. Although there exist a large number of similarity measures between distributions, very little is known about their generalization performance in specific learning tasks. Learning problems formulated on distributions have an inherent two-stage sampled difficulty: in practice only samples from sampled distributions are observable, and one has to build an estimate on similarities computed between sets of points. To the best of our knowledge, the only existing method with consistency guarantees for distribution regression requires kernel density estimation as an intermediate step (which suffers from slow convergence issues in high dimensions), and the domain of the distributions to be compact Euclidean. In this paper, we provide theoretical guarantees for a remarkably simple algorithmic alternative to solve the distribution regression problem: embed the distributions to a reproducing kernel Hilbert space, and learn a ridge regressor from the embeddings to the outputs. Our main contribution is to prove the consistency of this technique in the two-stage sampled setting under mild conditions (on separable, topological domains endowed with kernels). For a given total number of observations, we derive convergence rates as an explicit function of the problem difficulty. As a special case, we answer a 15-year-old open question: we establish the consistency of the classical set kernel [Haussler, 1999; Gartner et. al, 2002] in regression, and cover more recent kernels on distributions, including those due to [Christmann and Steinwart, 2010].

C. COOPER 21 3 Data representation--the key to conceptualisation: D. B. VIGOR 33 MECHANISED MATHEMATICS 45 4 An approach to analytic integration using ordered algebraic expressions: L. I. HODGSON 47 5 Some theorem-proving strategies based on the resolution principle: J. L DARLINGTON 57 MACHINE LEARNING AND HEURISTIC PROGRAMMING 73 6 Automatic description and recognition of board patterns in Go-Moku: A. M. MURRAY and E. W. Etcomc

The two outstanding figures in the history of computer science are Alan Turing and John von Neumann, and they shared the view that logic was the key to understanding and automating computation. In particular, it was Turing who gave us in the mid-1930s the fundamental analysis, and the logical definition, of the concept of'computability by machine' and who discovered the surprising and beautiful basic fact that there exist universal machines which by suitable programming can be made to t This essay is an expanded and revised version of one entitled The Role of Logic in Computer Science and Artificial Intelligence, which was completed in January 1992 (and was later published in the Proceedings of the Fifth Generation computer Systems 1992 Conference). Since completing that essay I have had the benefit of extremely helpful discussions on many of the details with Professor Donald Michie and Professor I. J. Good, both of whom knew Turing well during the war years at Bletchley Park. Professor J. A. N. Lee, whose knowledge of the literature and archives of the history of computing is encyclopedic, also provided additional information, some of which is still unpublished. Further light has very recently been shed on the von Neumann side of the story by Norman Macrae's excellent biography John von Neumann (Macrae 1992). Accordingly, it seemed appropriate to undertake a more complete and thorough version of the FGCS'92 essay, focussing somewhat more on the interesting historical and biographical issues. I am grateful to Donald Michie and Stephen Muggleton for inviting me to contribute such a'second edition' to the present volume, and I would also like to thank the Institute for New Computer Technology (ICOT) for kind permission to make use of the FGCS'92 essay in this way. 1 LOGIC, COMPUTERS, TURING, AND VON NEUMANN

The aim of the Stat Log project is to compare the performance of statistical, machine learning, and neural network algorithms, on large real world problems. This paper describes the completed work on classification in the StatLog project. Classification is here defined to be the problem, given a set of multivariate data with assigned classes, of estimating the probability from a set of attributes describing a new example sampled from the same source that it has a pre-defined class. We gathered together a representative collection of algorithms from statistics (Naive Bayes, K-nearest Neighbour, Kernel density, Linear discriminant, Quadratic discriminant, Logistic regression, Projection pursuit, Bayesian networks), machine learning (CART, C4.5, NewID, AC2, CAL5, CN2, ITrule -- only propositional symbolic algorithms were considered), and neural networks (Backpropagation, Radial basis functions, Kohonen).

Networks as models of word storage: G. R. Kiss

We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter \theta, i.e. we want to construct a confidence set for theta that contains theta with high probability, and that is as small as possible. The l_2 diameter of a such confidence set should depend on the sparsity S of \theta - the larger S, the wider the confidence set. However, in practice, S is unknown. This paper focuses on constructing a confidence set for \theta which contains \theta with high probability, whose diameter is adaptive to the unknown sparsity S, and which is implementable in practice.

To improve accuracy and speed of regressions and classifications, we present a data-based prediction method, Random Bits Regression (RBR). This method first generates a large number of random binary intermediate/derived features based on the original input matrix, and then performs regularized linear/logistic regression on those intermediate/derived features to predict the outcome. Benchmark analyses on a simulated dataset, UCI machine learning repository datasets and a GWAS dataset showed that RBR outperforms other popular methods in accuracy and robustness. RBR (available on https://sourceforge.net/projects/rbr/) is very fast and requires reasonable memories, therefore, provides a strong, robust and fast predictor in the big data era.

Learning sparse combinations is a frequent theme in machine learning. In this paper, we study its associated optimization problem in the distributed setting where the elements to be combined are not centrally located but spread over a network. We address the key challenges of balancing communication costs and optimization errors. To this end, we propose a distributed Frank-Wolfe (dFW) algorithm. We obtain theoretical guarantees on the optimization error $\epsilon$ and communication cost that do not depend on the total number of combining elements. We further show that the communication cost of dFW is optimal by deriving a lower-bound on the communication cost required to construct an $\epsilon$-approximate solution. We validate our theoretical analysis with empirical studies on synthetic and real-world data, which demonstrate that dFW outperforms both baselines and competing methods. We also study the performance of dFW when the conditions of our analysis are relaxed, and show that dFW is fairly robust.