We propose a streaming algorithm for the binary classification of data based on crowdsourcing. The algorithm learns the competence of each labeller by comparing her labels to those of other labellers on the same tasks and uses this information to minimize the prediction error rate on each task. We provide performance guarantees of our algorithm for a fixed population of independent labellers. In particular, we show that our algorithm is optimal in the sense that the cumulative regret compared to the optimal decision with known labeller error probabilities is finite, independently of the number of tasks to label. The complexity of the algorithm is linear in the number of labellers and the number of tasks, up to some logarithmic factors. Numerical experiments illustrate the performance of our algorithm compared to existing algorithms, including simple majority voting and expectation-maximization algorithms, on both synthetic and real datasets.

In unsupervised ensemble learning, one obtains predictions from multiple sources or classifiers, yet without knowing the reliability and expertise of each source, and with no labeled data to assess it. The task is to combine these possibly conflicting predictions into an accurate meta-learner. Most works to date assumed perfect diversity between the different sources, a property known as conditional independence. In realistic scenarios, however, this assumption is often violated, and ensemble learners based on it can be severely sub-optimal. The key challenges we address in this paper are:\ (i) how to detect, in an unsupervised manner, strong violations of conditional independence; and (ii) construct a suitable meta-learner. To this end we introduce a statistical model that allows for dependencies between classifiers. Our main contributions are the development of novel unsupervised methods to detect strongly dependent classifiers, better estimate their accuracies, and construct an improved meta-learner. Using both artificial and real datasets, we showcase the importance of taking classifier dependencies into account and the competitive performance of our approach.

Transductive learning considers situations when a learner observes $m$ labelled training points and $u$ unlabelled test points with the final goal of giving correct answers for the test points. This paper introduces a new complexity measure for transductive learning called Permutational Rademacher Complexity (PRC) and studies its properties. A novel symmetrization inequality is proved, which shows that PRC provides a tighter control over expected suprema of empirical processes compared to what happens in the standard i.i.d. setting. A number of comparison results are also provided, which show the relation between PRC and other popular complexity measures used in statistical learning theory, including Rademacher complexity and Transductive Rademacher Complexity (TRC). We argue that PRC is a more suitable complexity measure for transductive learning. Finally, these results are combined with a standard concentration argument to provide novel data-dependent risk bounds for transductive learning.

We consider a voting setting where candidates have preferences about the outcome of the election and are free to join or leave the election. The corresponding candidacy game, where candidates choose strategically to participate or not, has been studied by Dutta et al. [6], who showed that no non-dictatorial voting procedure satisfying unanimity is candidacy-strategyproof, that is, is such that the joint action where all candidates enter the election is always a pure strategy Nash equilibrium. In [7] Dutta et al. also showed that for some voting tree procedures, there are candidacy games with no pure Nash equilibria, and that for the rule that outputs the sophisticated winner of voting by successive elimination, all games have a pure Nash equilibrium. No results were known about other voting rules. Here we prove several such results. For four candidates, the message is, roughly, that most scoring rules (with the exception of Borda) do not guarantee the existence of a pure Nash equilibrium but that Condorcet-consistent rules, for an odd number of voters, do. For five candidates, most rules we study no longer have this guarantee. Finally, we identify one prominent rule that guarantees the existence of a pure Nash equilibrium for any number of candidates (and for an odd number of voters): the Copeland rule. We also show that under mild assumptions on the voting rule, the existence of strong equilibria cannot be guaranteed.

We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvement over the naive method of combining PCA with regression. To achieve our results, we first observe that ridge regression can be used to obtain a "smooth projection" onto the top principal components. We then sharpen this approximation to true projection using a low-degree polynomial approximation to the matrix step function. Step function approximation is a topic of long-term interest in scientific computing. We extend prior theory by constructing polynomials with simple iterative structure and rigorously analyzing their behavior under limited precision.

Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, the problem is resistant to standard optimization techniques. We construct MIPs with the same solutions whose objective functions are convex. Specifically, we characterize the tightest convex extension of the objective function, given by the Legendre-Fenchel biconjugate. Computing values of this tightest convex extension is NP-hard. However, by applying our characterization to every function in an additive decomposition of the objective function, we obtain a class of looser convex extensions that can be computed efficiently. For some decompositions, common loss and regularization functions, we derive a closed form.

Clustering is an essential data mining tool that aims to discover inherent cluster structure in data. As such, the study of clusterability, which evaluates whether data possesses such structure, is an integral part of cluster analysis. Yet, despite their central role in the theory and application of clustering, current notions of clusterability fall short in two crucial aspects that render them impractical; most are computationally infeasible and others fail to classify the structure of real datasets. In this paper, we propose a novel approach to clusterability evaluation that is both computationally efficient and successfully captures the structure of real data. Our method applies multimodality tests to the (one-dimensional) set of pairwise distances based on the original, potentially high-dimensional data. We present extensive analyses of our approach for both the Dip and Silverman multimodality tests on real data as well as 17,000 simulations, demonstrating the success of our approach as the first practical notion of clusterability.

Despite the importance of sparsity signal models and the increasing prevalence of high-dimensional streaming data, there are relatively few algorithms for dynamic filtering of time-varying sparse signals. Of the existing algorithms, fewer still provide strong performance guarantees. This paper examines two algorithms for dynamic filtering of sparse signals that are based on efficient l1 optimization methods. We first present an analysis for one simple algorithm (BPDN-DF) that works well when the system dynamics are known exactly. We then introduce a novel second algorithm (RWL1-DF) that is more computationally complex than BPDN-DF but performs better in practice, especially in the case where the system dynamics model is inaccurate. Robustness to model inaccuracy is achieved by using a hierarchical probabilistic data model and propagating higher-order statistics from the previous estimate (akin to Kalman filtering) in the sparse inference process. We demonstrate the properties of these algorithms on both simulated data as well as natural video sequences. Taken together, the algorithms presented in this paper represent the first strong performance analysis of dynamic filtering algorithms for time-varying sparse signals as well as state-of-the-art performance in this emerging application.

Large margin classifiers, such as the support vector machine (SVM) [1] and the maximum entropy discrimination (MED) classifier [2], have enjoyed great popularity in the signal processing and machine learning communities due to their broad applicability, robust performance, and the availability of fast software implementations. When the training data is representative of the test data, the performance of MED/SVM has theoretical guarantees that have been validated in practice [1], [3], [4]. Moreover, since the decision boundary of the MED/SVM is solely defined by a few support vectors, the algorithm can tolerate random feature distortions and perturbations. However, in many real applications, anomalous measurements are inherent to the data set due to strong environmental noise or possible sensor failures. Such anomalies arise in industrial process monitoring, video surveillance, tactical multi-modal sensing, and, more generally, any application that involves unattended sensors in difficult environments (Figure 1).

Storytelling algorithms aim to 'connect the dots' between disparate documents by linking starting and ending documents through a series of intermediate documents. Existing storytelling algorithms are based on notions of coherence and connectivity, and thus the primary way by which users can steer the story construction is via design of suitable similarity functions. We present an alternative approach to storytelling wherein the user can interactively and iteratively provide 'must use' constraints to preferentially support the construction of some stories over others. The three innovations in our approach are distance measures based on (inferred) topic distributions, the use of constraints to define sets of linear inequalities over paths, and the introduction of slack and surplus variables to condition the topic distribution to preferentially emphasize desired terms over others. We describe experimental results to illustrate the effectiveness of our interactive storytelling approach over multiple text datasets.