Bayesian optimization is a powerful global optimization technique for expensive black-box functions. One of its shortcomings is that it requires auxiliary optimization of an acquisition function at each iteration. This auxiliary optimization can be costly and very hard to carry out in practice. Moreover, it creates serious theoretical concerns, as most of the convergence results assume that the exact optimum of the acquisition function can be found. In this paper, we introduce a new technique for efficient global optimization that combines Gaussian process confidence bounds and treed simultaneous optimistic optimization to eliminate the need for auxiliary optimization of acquisition functions. The experiments with global optimization benchmarks and a novel application to automatic information extraction demonstrate that the resulting technique is more efficient than the two approaches from which it draws inspiration. Unlike most theoretical analyses of Bayesian optimization with Gaussian processes, our finite-time convergence rate proofs do not require exact optimization of an acquisition function. That is, our approach eliminates the unsatisfactory assumption that a difficult, potentially NP-hard, problem has to be solved in order to obtain vanishing regret rates.

We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response $y$, conditional on the model being selected (``condition on selection" framework). This allows us to construct valid confidence intervals and hypothesis tests for regression coefficients that account for the selection procedure. In contrast to recent work in high-dimensional statistics, our results are exact (non-asymptotic) and require no eigenvalue-like assumptions on the design matrix $X$. Furthermore, the computational cost of marginal regression, constructing confidence intervals and hypothesis testing is negligible compared to the cost of linear regression, thus making our methods particularly suitable for extremely large datasets. Although we focus on marginal screening to illustrate the applicability of the condition on selection framework, this framework is much more broadly applicable. We show how to apply the proposed framework to several other selection procedures including orthogonal matching pursuit, non-negative least squares, and marginal screening+Lasso.

We consider a family of learning strategies for online optimization problems that evolve in continuous time and we show that they lead to no regret. From a more traditional, discrete-time viewpoint, this continuous-time approach allows us to derive the no-regret properties of a large class of discrete-time algorithms including as special cases the exponential weight algorithm, online mirror descent, smooth fictitious play and vanishingly smooth fictitious play. In so doing, we obtain a unified view of many classical regret bounds, and we show that they can be decomposed into a term stemming from continuous-time considerations and a term which measures the disparity between discrete and continuous time. As a result, we obtain a general class of infinite horizon learning strategies that guarantee an $\mathcal{O}(n^{-1/2})$ regret bound without having to resort to a doubling trick.

In this work we compute lower Lipschitz bounds of $\ell_p$ pooling operators for $p=1, 2, \infty$ as well as $\ell_p$ pooling operators preceded by half-rectification layers. These give sufficient conditions for the design of invertible neural network layers. Numerical experiments on MNIST and image patches confirm that pooling layers can be inverted with phase recovery algorithms. Moreover, the regularity of the inverse pooling, controlled by the lower Lipschitz constant, is empirically verified with a nearest neighbor regression.

Numerous algorithms are used for nonnegative matrix factorization under the assumption that the matrix is nearly separable. In this paper, we show how to make these algorithms efficient for data matrices that have many more rows than columns, so-called "tall-and-skinny matrices". One key component to these improved methods is an orthogonal matrix transformation that preserves the separability of the NMF problem. Our final methods need a single pass over the data matrix and are suitable for streaming, multi-core, and MapReduce architectures. We demonstrate the efficacy of these algorithms on terabyte-sized synthetic matrices and real-world matrices from scientific computing and bioinformatics.

HEX-programs extend logic programs under the answer set semantics with external computations through external atoms. As reasoning from ground Horn programs with nonmonotonic external atoms of polynomial complexity is already on the second level of the polynomial hierarchy, minimality checking of answer set candidates needs special attention. To this end, we present an approach based on unfounded sets as a generalization of related techniques for ASP programs. The unfounded set detection is expressed as a propositional SAT problem, for which we provide two different encodings and optimizations to them. We then integrate our approach into a previously developed evaluation framework for HEX-programs, which is enriched by additional learning techniques that aim at avoiding the reconstruction of the same or related unfounded sets. Furthermore, we provide a syntactic criterion that allows one to skip the minimality check in many cases. An experimental evaluation shows that the new approach significantly decreases runtime.

Contaminated mixture models are developed for model-based clustering of data with asymmetric clusters as well as spurious points, outliers, and/or noise. Specifically, we introduce a contaminated mixture of contaminated shifted asymmetric Laplace distributions and a contaminated mixture of contaminated skew-normal distributions. In each case, mixture components have a parameter controlling the proportion of bad points (i.e., spurious points, outliers, and/or noise) and one specifying the degree of contamination. A very important feature of our approaches is that these parameters do not have to be specified a priori. Expectation-conditional maximization algorithms are outlined for parameter estimation and the number of components is selected using the Bayesian information criterion. The performance of our approaches is illustrated on artificial and real data.

Sample size determination for a data set is an important statistical process for analyzing the data to an optimum level of accuracy and using minimum computational work. The applications of this process are credible in every domain which deals with large data sets and high computational work. This study uses Bayesian analysis for determination of minimum sample size of vibration signals to be considered for fault diagnosis of a bearing using pre-defined parameters such as the inverse standard probability and the acceptable margin of error. Thus an analytical formula for sample size determination is introduced. The fault diagnosis of the bearing is done using a machine learning approach using an entropy-based J48 algorithm. The following method will help researchers involved in fault diagnosis to determine minimum sample size of data for analysis for a good statistical stability and precision.

It takes skill to build a meaningful predictive model even with the abundance of implementations of modern machine learning algorithms and readily available computing resources. Building a model becomes challenging if hundreds of terabytes of data need to be processed to produce the training data set. In a digital advertising technology setting, we are faced with the need to build thousands of such models that predict user behavior and power advertising campaigns in a 24/7 chaotic real-time production environment. As data scientists, we also have to convince other internal departments critical to implementation success, our management, and our customers that our machine learning system works. In this paper, we present the details of the design and implementation of an automated, robust machine learning platform that impacts billions of advertising impressions monthly. This platform enables us to continuously optimize thousands of campaigns over hundreds of millions of users, on multiple continents, against varying performance objectives.

We study the robustness properties of $\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\ell_1$ estimator for measurements errors consisting of outliers coupled with noise. We introduce a new estimation technique resulting from a regularization of $\ell_1$ minimization by inf-convolution with the $\ell_2$ norm. Concerning robustness to large outliers, the proposed estimator keeps the breakdown point of the $\ell_1$ estimator, and reduces to least squares when there are not outliers. We present a globally convergent forward-backward algorithm for computing our estimator and some numerical experiments confirming its theoretical properties.