Machine learning refers to the process of enabling computer systems to learn with data using statistical techniques without being explicitly programmed. It is the process of active engagement with algorithms in order to enable them to learn from and make predictions on data. Machine learning is closely associated with computational statistics, mathematical optimization, and data learning. It is associated with predictive analysis, which allows producing reliable and fast results by learning from historical trends. Supervised learning: The computer is presented with some example inputs, based on which the desired outputs are to be formed.

This paper considers stochastic optimization problems for a large class of objective functions, including convex and continuous submodular. Stochastic proximal gradient methods have been widely used to solve such problems; however, their applicability remains limited when the problem dimension is large and the projection onto a convex set is costly. Instead, stochastic conditional gradient methods are proposed as an alternative solution relying on (i) Approximating gradients via a simple averaging technique requiring a single stochastic gradient evaluation per iteration; (ii) Solving a linear program to compute the descent/ascent direction. The averaging technique reduces the noise of gradient approximations as time progresses, and replacing projection step in proximal methods by a linear program lowers the computational complexity of each iteration. We show that under convexity and smoothness assumptions, our proposed method converges to the optimal objective function value at a sublinear rate of $O(1/t^{1/3})$. Further, for a monotone and continuous DR-submodular function and subject to a general convex body constraint, we prove that our proposed method achieves a $((1-1/e)OPT-\eps)$ guarantee with $O(1/\eps^3)$ stochastic gradient computations. This guarantee matches the known hardness results and closes the gap between deterministic and stochastic continuous submodular maximization. Additionally, we obtain $((1/e)OPT -\eps)$ guarantee after using $O(1/\eps^3)$ stochastic gradients for the case that the objective function is continuous DR-submodular but non-monotone and the constraint set is down-closed. By using stochastic continuous optimization as an interface, we provide the first $(1-1/e)$ tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a matroid constraint and $(1/e)$ approximation guarantee for the non-monotone case.

Here we describe well-known chaotic sequences, including new generalizations, with application to random number generation, highly non-linear auto-regressive models for times series, simulation, random permutations, and the use of big numbers (libraries available in programming languages to work with numbers with hundreds of decimals) as standard computer precision almost always produces completely erroneous results after a few iterations -- a fact rarely if ever mentioned in the scientific literature, but illustrated here, together with a solution. It is possible that all scientists who published on chaotic processes, used faulty numbers because of this issue. This article is accessible to non-experts, even though we solve a special stochastic equation for the first time, providing an unexpected exact solution, for a new chaotic process that generalizes the logistic map. We also describe a general framework for continuous random number generators, and investigate the interesting auto-correlation structure associated with some of these sequences. References are provided, as well as fast source code to process big numbers accurately, and even an elegant mathematical proof in the last section.

Multiple classifier systems focus on the combination of classifiers to obtain better performance than a single robust one. These systems unfold three major phases: pool generation, selection and integration. One of the most promising MCS approaches is Dynamic Selection (DS), which relies on finding the most competent classifier or ensemble of classifiers to predict each test sample. The majority of the DS techniques are based on the K-Nearest Neighbors (K-NN) definition, and the quality of the neighborhood has a huge impact on the performance of DS methods. In this paper, we perform an analysis comparing the classification results of DS techniques and the K-NN classifier under different conditions. Experiments are performed on 18 state-of-the-art DS techniques over 30 classification datasets and results show that DS methods present a significant boost in classification accuracy even though they use the same neighborhood as the K-NN. The reasons behind the outperformance of DS techniques over the K-NN classifier reside in the fact that DS techniques can deal with samples with a high degree of instance hardness (samples that are located close to the decision border) as opposed to the K-NN. In this paper, not only we explain why DS techniques achieve higher classification performance than the K-NN but also when DS should be used.

You may even call it post-doctorship, as the level is beyond the traditional PhD degree. It is not a degree, not competing with university programs, but instead, akin to a fellowship or apprenticeship to learn doing state-of-the-art applied research, discover ground-breaking results or applications, and translate your discoveries into seminal material suitable for a broad audience. It is intended for professionals with substantial experience, perhaps to people who already have a PhD in a different field. It is mentored by well connected, world-class recognized scientists (not necessarily affiliated with a university) with broad domain of expertise in many environments. The focus is on real-world problems and applications to help you get a high-level position in the industry or as an independent researcher.

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It is well known that contemporary computers don't like to randomly access data in an unpredictible manner in memory. However, not all forms of random accesses are equally harmful. Suppose that the array is large. Take an array made of 100 million elements. It far exceeds the CPU cache on the machines I own.

This chapter is mainly on the dot product (vector and/or matrix multiplication). We will also see some of its properties. Then, we will see how to synthesize a system of linear equations using matrix notation. This is a major process for the following chapters.

You will find here nine interesting topics that you won't learn in college classes. Most have interesting applications in business and elsewhere. They are not especially difficult, and I explain them in simple English. Yet they are not part of the traditional statistical curriculum, and even many experienced data scientists with a PhD degree have not heard about some of these concepts. This is a well known model, used as a base stochastic process to model the logarithm of stock prices, yet it has interesting properties (depending on dimension) that few people know about.

Across a variety of scientific disciplines, sparse inverse covariance estimation is a popular tool for capturing the underlying dependency relationships in multivariate data. Unfortunately, most estimators are not scalable enough to handle the sizes of modern high-dimensional data sets (often on the order of terabytes), and assume Gaussian samples. To address these deficiencies, we introduce HP-CONCORD, a highly scalable optimization method for estimating a sparse inverse covariance matrix based on a regularized pseudolikelihood framework, without assuming Gaussianity. Our parallel proximal gradient method uses a novel communication-avoiding linear algebra algorithm and runs across a multi-node cluster with up to 1k nodes (24k cores), achieving parallel scalability on problems with up to ~819 billion parameters (1.28 million dimensions); even on a single node, HP-CONCORD demonstrates scalability, outperforming a state-of-the-art method. We also use HP-CONCORD to estimate the underlying dependency structure of the brain from fMRI data, and use the result to identify functional regions automatically. The results show good agreement with a clustering from the neuroscience literature.