We are interested in a general data - based prediction task: g iven a train ing data matrix ( TrX), a training outcome vector ( TrY) and a test data matrix ( TeX), predict test outcome vector (). In the era of big data, two practically conflicting challenges are eminent: (1) the prior knowledge on the subject (a lso known as domain specific knowledge) is largely insufficient; (2) computation and storage cost of big data is unaffordable. To meet these aforementioned challenge s, this paper is devoted to modeling large number of observations without domain specific k nowledge, using regression and classification. The methods widely used for regression and classification can be classified as: linear regression, k nearest neighbor (KNN) [1], support vector machine (SVM) [2], neural network (NN) [3, 4], extreme learning machine (ELM) [5], deep learning (DL) [6], random forest (RF) [7] and boosting (GBM) [8] among others . Each method performs well on some types of datasets but has its own limitations on others [9 - 12] . A method with reasonable performance on boarder, if not universe, datasets is highly desired .

E consider a stationary discrete-time vector process or time series. Such a process could model, e.g., the time evolution of air pollutant concentrations [1], [2] or medical diagnostic data obtained in electrocorticography (ECoG) [3]. One specific way of representing the dependence structure of a vector process is via a graphical model [4], where the nodes of the graph represent the individual scalar process components, and the edges represent statistical relations between the individual process components. More precisely, the (undirected) edges of a conditional independence graph (CIG) associated with a process represent conditional independence statements about the process components [4], [1]. In particular, two nodes in the CIG are connected by an edge if and only if the two corresponding process components are conditionally dependent, given the remaining process components. Note that the so defined CIG for time series extends the basic notion of a CIG for random vectors by considering dependencies between entire time series instead of dependencies between scalar random variables [5], [6]. In this work, we investigate the problem of graphical model selection (GMS), i.e., that of inferring the CIG of a time series, given a finite-length observation. A. Jung is with the Institute of Telecommunications, Vienna University of Technology, 1040-Vienna, Austria email: ajung@nt.tuwien.ac.at.

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In a context that does not allow for indecision, this leads to an approach that is mathematically equivalent to working with coherent conditional probabilities. If we do allow for indecision, this leads to a more general foundation for coherent (imprecise-)probabilistic inference. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finetti's Representation Theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps. We discuss, as particular examples, two important inference principles: representation insensitivitya strengthened version of Walley's representation invarianceand specificity. We show that there is an infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the skeptically cautious inference system, the inference systems corresponding to (a modified version of) Walley and Bernard's Imprecise Dirichlet Multinomial Models (IDMM), the skeptical IDMM inference systems, and the Haldane inference system. We also prove that the latter produces the same posterior inferences as would be obtained using Haldane's improper prior, implying that there is an infinity of proper priors that produce the same coherent posterior inferences as Haldane's improper one. Finally, we impose an additional inference principle that allows us to characterise uniquely the immediate predictions for the IDMM inference systems.

In certain situations that shall be undoubtedly more and more common in the Big Data era, the datasets available are so massive that computing statistics over the full sample is hardly feasible, if not unfeasible. A natural approach in this context consists in using survey schemes and substituting the "full data" statistics with their counterparts based on the resulting random samples, of manageable size. It is the main purpose of this paper to investigate the impact of survey sampling with unequal inclusion probabilities on stochastic gradient descent-based M-estimation methods in large-scale statistical and machine-learning problems. Precisely, we prove that, in presence of some a priori information, one may significantly increase asymptotic accuracy when choosing appropriate first order inclusion probabilities, without affecting complexity. These striking results are described here by limit theorems and are also illustrated by numerical experiments.

In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic. My aim is to describe the most successful methods from this area along with some interesting examples that these techniques can illuminate. I hope that the results in these pages will inspire future work on applications of random matrices as well as refinements of the matrix concentration inequalities discussed herein. I have chosen to present a coherent body of results based on a generalization of the Laplace transform method for establishing scalar concentration inequalities. In the last two years, Lester Mackey and I, together with our coauthors, have developed an alternative approach to matrix concentration using exchangeable pairs and Markov chain couplings. With some regret, I have chosen to omit this theory because the ideas seem less accessible to a broad audience of researchers. The interested reader will find pointers to these articles in the annotated bibliography. The work described in these notes reflects the influence of many researchers.

This article is an overview of the "SP theory of intelligence". The theory aims to simplify and integrate concepts across artificial intelligence, mainstream computing and human perception and cognition, with information compression as a unifying theme. It is conceived as a brain-like system that receives 'New' information and stores some or all of it in compressed form as 'Old' information. It is realised in the form of a computer model -- a first version of the SP machine. The concept of "multiple alignment" is a powerful central idea. Using heuristic techniques, the system builds multiple alignments that are 'good' in terms of information compression. For each multiple alignment, probabilities may be calculated. These provide the basis for calculating the probabilities of inferences. The system learns new structures from partial matches between patterns. Using heuristic techniques, the system searches for sets of structures that are 'good' in terms of information compression. These are normally ones that people judge to be 'natural', in accordance with the 'DONSVIC' principle -- the discovery of natural structures via information compression. The SP theory may be applied in several areas including 'computing', aspects of mathematics and logic, representation of knowledge, natural language processing, pattern recognition, several kinds of reasoning, information storage and retrieval, planning and problem solving, information compression, neuroscience, and human perception and cognition. Examples include the parsing and production of language including discontinuous dependencies in syntax, pattern recognition at multiple levels of abstraction and its integration with part-whole relations, nonmonotonic reasoning and reasoning with default values, reasoning in Bayesian networks including 'explaining away', causal diagnosis, and the solving of a geometric analogy problem.

Formalisms for specifying statistical models, such as probabilistic-programming languages, typically consist of two components: a specification of a stochastic process (the prior), and a specification of observations that restrict the probability space to a conditional subspace (the posterior). Use cases of such formalisms include the development of algorithms in machine learning and artificial intelligence. We propose and investigate a declarative framework for specifying statistical models on top of a database, through an appropriate extension of Datalog. By virtue of extending Datalog, our framework offers a natural integration with the database, and has a robust declarative semantics. Our Datalog extension provides convenient mechanisms to include numerical probability functions; in particular, conclusions of rules may contain values drawn from such functions. The semantics of a program is a probability distribution over the possible outcomes of the input database with respect to the program; these outcomes are minimal solutions with respect to a related program with existentially quantified variables in conclusions. Observations are naturally incorporated by means of integrity constraints over the extensional and intensional relations. We focus on programs that use discrete numerical distributions, but even then the space of possible outcomes may be uncountable (as a solution can be infinite). We define a probability measure over possible outcomes by applying the known concept of cylinder sets to a probabilistic chase procedure. We show that the resulting semantics is robust under different chases. We also identify conditions guaranteeing that all possible outcomes are finite (and then the probability space is discrete). We argue that the framework we propose retains the purely declarative nature of Datalog, and allows for natural specifications of statistical models.

We develop a penalized likelihood estimation framework to estimate the structure of Gaussian Bayesian networks from observational data. In contrast to recent methods which accelerate the learning problem by restricting the search space, our main contribution is a fast algorithm for score-based structure learning which does not restrict the search space in any way and works on high-dimensional datasets with thousands of variables. Our use of concave regularization, as opposed to the more popular $\ell_0$ (e.g. BIC) penalty, is new. Moreover, we provide theoretical guarantees which generalize existing asymptotic results when the underlying distribution is Gaussian. Most notably, our framework does not require the existence of a so-called faithful DAG representation, and as a result the theory must handle the inherent nonidentifiability of the estimation problem in a novel way. Finally, as a matter of independent interest, we provide a comprehensive comparison of our approach to several standard structure learning methods using open-source packages developed for the R language. Based on these experiments, we show that our algorithm is significantly faster than other competing methods while obtaining higher sensitivity with comparable false discovery rates for high-dimensional data. In particular, the total runtime for our method to generate a solution path of 20 estimates for DAGs with 8000 nodes is around one hour.

Quantum machine learning has received significant attention in recent years, and promising progress has been made in the development of quantum algorithms to speed up traditional machine learning tasks. In this work, however, we focus on investigating the information-theoretic upper bounds of sample complexity - how many training samples are sufficient to predict the future behaviour of an unknown target function. This kind of problem is, arguably, one of the most fundamental problems in statistical learning theory and the bounds for practical settings can be completely characterised by a simple measure of complexity. Our main result in the paper is that, for learning an unknown quantum measurement, the upper bound, given by the fat-shattering dimension, is linearly proportional to the dimension of the underlying Hilbert space. Learning an unknown quantum state becomes a dual problem to ours, and as a byproduct, we can recover Aaronson's famous result [Proc. R. Soc. A 463:3089-3144 (2007)] solely using a classical machine learning technique. In addition, other famous complexity measures like covering numbers and Rademacher complexities are derived explicitly. We are able to connect measures of sample complexity with various areas in quantum information science, e.g. quantum state/measurement tomography, quantum state discrimination and quantum random access codes, which may be of independent interest. Lastly, with the assistance of general Bloch-sphere representation, we show that learning quantum measurements/states can be mathematically formulated as a neural network. Consequently, classical ML algorithms can be applied to efficiently accomplish the two quantum learning tasks.

Intelligent systems that learn interactively from their end-users are quickly becoming widespread. Until recently, this progress has been fueled mostly by advances in machine learning; however, more and more researchers are realizing the importance of studying users of these systems. In this article we promote this approach and demonstrate how it can result in better user experiences and more effective learning systems. We present a number of case studies that characterize the impact of interactivity, demonstrate ways in which some existing systems fail to account for the user, and explore new ways for learning systems to interact with their users. We argue that the design process for interactive machine learning systems should involve users at all stages: explorations that reveal human interaction patterns and inspire novel interaction methods, as well as refinement stages to tune details of the interface and choose among alternatives. After giving a glimpse of the progress that has been made so far, we discuss the challenges that we face in moving the field forward.