We present a new recommendation setting for picking out two items from a given set to be highlighted to a user, based on contextual input. These two items are presented to a user who chooses one of them, possibly stochastically, with a bias that favours the item with the higher value. We propose a second-order algorithm framework that members of it use uses relative upper-confidence bounds to trade off exploration and exploitation, and some explore via sampling. We analyze one algorithm in this framework in an adversarial setting with only mild assumption on the data, and prove a regret bound of $O(Q_T + \sqrt{TQ_T\log T} + \sqrt{T}\log T)$, where $T$ is the number of rounds and $Q_T$ is the cumulative approximation error of item values using a linear model. Experiments with product reviews from 33 domains show the advantage of our methods over algorithms designed for related settings, and that UCB based algorithms are inferior to greed or sampling based algorithms.

Under the framework of spectral clustering, the key of subspace clustering is building a similarity graph which describes the neighborhood relations among data points. Some recent works build the graph using sparse, low-rank, and $\ell_2$-norm-based representation, and have achieved state-of-the-art performance. However, these methods have suffered from the following two limitations. First, the time complexities of these methods are at least proportional to the cube of the data size, which make those methods inefficient for solving large-scale problems. Second, they cannot cope with out-of-sample data that are not used to construct the similarity graph. To cluster each out-of-sample datum, the methods have to recalculate the similarity graph and the cluster membership of the whole data set. In this paper, we propose a unified framework which makes representation-based subspace clustering algorithms feasible to cluster both out-of-sample and large-scale data. Under our framework, the large-scale problem is tackled by converting it as out-of-sample problem in the manner of "sampling, clustering, coding, and classifying". Furthermore, we give an estimation for the error bounds by treating each subspace as a point in a hyperspace. Extensive experimental results on various benchmark data sets show that our methods outperform several recently-proposed scalable methods in clustering large-scale data set.

In this paper we study nonconvex penalization using Bernstein functions whose first-order derivatives are completely monotone. The Bernstein function can induce a class of nonconvex penalty functions for high-dimensional sparse estimation problems. We derive a thresholding function based on the Bernstein penalty and discuss some important mathematical properties in sparsity modeling. We show that a coordinate descent algorithm is especially appropriate for regression problems penalized by the Bernstein function. We also consider the application of the Bernstein penalty in classification problems and devise a proximal alternating linearized minimization method. Based on theory of the Kurdyka-Lojasiewicz inequality, we conduct convergence analysis of these alternating iteration procedures. We particularly exemplify a family of Bernstein nonconvex penalties based on a generalized Gamma measure and conduct empirical analysis for this family.

#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of satisfiability modulo theories (SMT) there is a growing need for model counting solvers, coming from several application domains (quantitative information flow, static analysis of probabilistic programs). In this paper, we show a reduction from an approximate version of #SMT to SMT. We focus on the theories of integer arithmetic and linear real arithmetic. We propose model counting algorithms that provide approximate solutions with formal bounds on the approximation error. They run in polynomial time and make a polynomial number of queries to the SMT solver for the underlying theory, exploiting "for free" the sophisticated heuristics implemented within modern SMT solvers. We have implemented the algorithms and used them to solve the value problem for a model of loop-free probabilistic programs with nondeterminism.

Since Pearl's seminal work on providing a formal language for causality, the subject has garnered a lot of interest among philosophers and researchers in artificial intelligence alike. One of the most debated topics in this context regards the notion of actual causation, which concerns itself with specific - as opposed to general - causal claims. The search for a proper formal definition of actual causation has evolved into a controversial debate, that is pervaded with ambiguities and confusion. The goal of our research is twofold. First, we wish to provide a clear way to compare competing definitions. Second, we also want to improve upon these definitions so they can be applied to a more diverse range of instances, including non-deterministic ones. To achieve these goals we will provide a general, abstract definition of actual causation, formulated in the context of the expressive language of CP-logic (Causal Probabilistic logic). We will then show that three recent definitions by Ned Hall (originally formulated for structural models) and a definition of our own (formulated for CP-logic directly) can be viewed and directly compared as instantiations of this abstract definition, which allows them to deal with a broader range of examples.

Monte Carlo sampling for Bayesian posterior inference is a common approach used in machine learning. The Markov Chain Monte Carlo procedures that are used are often discrete-time analogues of associated stochastic differential equations (SDEs). These SDEs are guaranteed to leave invariant the required posterior distribution. An area of current research addresses the computational benefits of stochastic gradient methods in this setting. Existing techniques rely on estimating the variance or covariance of the subsampling error, and typically assume constant variance. In this article, we propose a covariance-controlled adaptive Langevin thermostat that can effectively dissipate parameter-dependent noise while maintaining a desired target distribution. The proposed method achieves a substantial speedup over popular alternative schemes for large-scale machine learning applications.

The minimum description length (MDL) principle provides a general information-theoretic approach to model selection and other forms of statistical inference [5, 17]. The MDL criterion for model selection is consistent, meaning that it will select the data-generating model from a countable set of competing parametric models with probability approaching 1 as the sample size n goes to infinity [4]. For example, if each of the parametric models is a logistic regression model with predictor variables taken from a fixed set of potential predictors, then the MDL model-selection criterion will choose the correct combination of predictors with probability approaching 1 as n . The MDL model-selection criterion also has a number of strong optimality properties, which greatly extend Shannon's noiseless coding theorem [5, §III.E]. In its simplest form, the MDL principle advocates choosing the model for which the observed data has the shortest message length under a particular prefix code defined by a minimax condition [11, §2.4.3]. Shtarkov [19] showed that this is equivalent to choosing the model with the largest normalized maximum likelihood (NML) for the observed data.

We introduce principal differences analysis (PDA) for analyzing differences between high-dimensional distributions. The method operates by finding the projection that maximizes the Wasserstein divergence between the resulting univariate populations. Relying on the Cramer-Wold device, it requires no assumptions about the form of the underlying distributions, nor the nature of their inter-class differences. A sparse variant of the method is introduced to identify features responsible for the differences. We provide algorithms for both the original minimax formulation as well as its semidefinite relaxation. In addition to deriving some convergence results, we illustrate how the approach may be applied to identify differences between cell populations in the somatosensory cortex and hippocampus as manifested by single cell RNA-seq. Our broader framework extends beyond the specific choice of Wasserstein divergence.

Many new methods for solving Boolean Satisfiability Problem (SAT) were introduced in the past two decades. These methods make it possible to solve combinatorial problems from various areas [2]. One can use different approaches to encode an original problem to SAT [14]. Often each particular problem requires researchers to develop and implement special encoding technique. Recently a number of systems that automate procedures of encoding combinatorial problems to SAT were developed [6,7,12,17,20].

Gaussian Graphical Models (GGMs) are popular tools for studying network structures. However, many modern applications such as gene network discovery and social interactions analysis often involve high-dimensional noisy data with outliers or heavier tails than the Gaussian distribution. In this paper, we propose the Trimmed Graphical Lasso for robust estimation of sparse GGMs. Our method guards against outliers by an implicit trimming mechanism akin to the popular Least Trimmed Squares method used for linear regression. We provide a rigorous statistical analysis of our estimator in the high-dimensional setting. In contrast, existing approaches for robust sparse GGMs estimation lack statistical guarantees. Our theoretical results are complemented by experiments on simulated and real gene expression data which further demonstrate the value of our approach.