Spectral clustering and co-clustering are well-known techniques in data analysis, and recent work has extended spectral clustering to square, symmetric tensors and hypermatrices derived from a network. We develop a new tensor spectral co-clustering method that simultaneously clusters the rows, columns, and slices of a nonnegative three-mode tensor and generalizes to tensors with any number of modes. The algorithm is based on a new random walk model which we call the super-spacey random surfer. We show that our method out-performs state-of-the-art co-clustering methods on several synthetic datasets with ground truth clusters and then use the algorithm to analyze several real-world datasets.

Nystr\"{o}m method has been used successfully to improve the computational efficiency of kernel ridge regression (KRR). Recently, theoretical analysis of Nystr\"{o}m KRR, including generalization bound and convergence rate, has been established based on reproducing kernel Hilbert space (RKHS) associated with the symmetric positive semi-definite kernel. However, in real world applications, RKHS is not always optimal and kernel function is not necessary to be symmetric or positive semi-definite. In this paper, we consider the generalized Nystr\"{o}m kernel regression (GNKR) with $\ell_2$ coefficient regularization, where the kernel just requires the continuity and boundedness. Error analysis is provided to characterize its generalization performance and the column norm sampling is introduced to construct the refined hypothesis space. In particular, the fast learning rate with polynomial decay is reached for the GNKR. Experimental analysis demonstrates the satisfactory performance of GNKR with the column norm sampling.

Dropout has been witnessed with great success in training deep neural networks by independently zeroing out the outputs of neurons at random. It has also received a surge of interest for shallow learning, e.g., logistic regression. However, the independent sampling for dropout could be suboptimal for the sake of convergence. In this paper, we propose to use multinomial sampling for dropout, i.e., sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow learning with multinomial dropout and establish the risk bound for stochastic optimization. By minimizing a sampling dependent factor in the risk bound, we obtain a distribution-dependent dropout with sampling probabilities dependent on the second order statistics of the data distribution. To tackle the issue of evolving distribution of neurons in deep learning, we propose an efficient adaptive dropout (named \textbf{evolutional dropout}) that computes the sampling probabilities on-the-fly from a mini-batch of examples. Empirical studies on several benchmark datasets demonstrate that the proposed dropouts achieve not only much faster convergence and but also a smaller testing error than the standard dropout. For example, on the CIFAR-100 data, the evolutional dropout achieves relative improvements over 10\% on the prediction performance and over 50\% on the convergence speed compared to the standard dropout.

We present a general theoretical analysis of structured prediction with a series of new results. We give new data-dependent margin guarantees for structured prediction for a very wide family of loss functions and a general family of hypotheses, with an arbitrary factor graph decomposition. These are the tightest margin bounds known for both standard multi-class and general structured prediction problems. Our guarantees are expressed in terms of a data-dependent complexity measure, \emph{factor graph complexity}, which we show can be estimated from data and bounded in terms of familiar quantities for several commonly used hypothesis sets, and a sparsity measure for features and graphs. Our proof techniques include generalizations of Talagrand's contraction lemma that can be of independent interest. We further extend our theory by leveraging the principle of Voted Risk Minimization (VRM) and show that learning is possible even with complex factor graphs. We present new learning bounds for this advanced setting, which we use to devise two new algorithms, \emph{Voted Conditional Random Field} (VCRF) and \emph{Voted Structured Boosting} (StructBoost). These algorithms can make use of complex features and factor graphs and yet benefit from favorable learning guarantees. We also report the results of experiments with VCRF on several datasets to validate our theory.

Consider samples from two different data sources $\{\mathbf{x_s^i}\} \sim P_{\rm source}$ and $\{\mathbf{x_t^i}\} \sim P_{\rm target}$. We only observe their transformed versions $h(\mathbf{x_s^i})$ and $g(\mathbf{x_t^i})$, for some known function class $h(\cdot)$ and $g(\cdot)$. Our goal is to perform a statistical test checking if $P_{\rm source}$ = $P_{\rm target}$ while removing the distortions induced by the transformations. This problem is closely related to concepts underlying numerous domain adaptation algorithms, and in our case, is motivated by the need to combine clinical and imaging based biomarkers from multiple sites and/or batches, where this problem is fairly common and an impediment in the conduct of analyses with much larger sample sizes. We develop a framework that addresses this problem using ideas from hypothesis testing on the transformed measurements, where in the distortions need to be estimated {\it in tandem} with the testing. We derive a simple algorithm and study its convergence and consistency properties in detail, and we also provide lower-bound strategies based on recent work in continuous optimization. On a dataset of individuals at risk for neurological disease, our results are competitive with alternative procedures that are twice as expensive and in some cases operationally infeasible to implement.

We develop tools for selective inference in the setting of group sparsity, including the construction of confidence intervals and p-values for testing selected groups of variables. Our main technical result gives the precise distribution of the magnitude of the projection of the data onto a given subspace, and enables us to develop inference procedures for a broad class of group-sparse selection methods, including the group lasso, iterative hard thresholding, and forward stepwise regression. We give numerical results to illustrate these tools on simulated data and on health record data.

Estimators of information theoretic measures such as entropy and mutual information from samples are a basic workhorse for many downstream applications in modern data science. State of the art approaches have been either geometric (nearest neighbor (NN) based) or kernel based (with bandwidth chosen to be data independent and vanishing sub linearly in the sample size). In this paper we combine both these approaches to design new estimators of entropy and mutual information that strongly outperform all state of the art methods. Our estimator uses bandwidth choice of fixed $k$-NN distances; such a choice is both data dependent and linearly vanishing in the sample size and necessitates a bias cancellation term that is universal and independent of the underlying distribution. As a byproduct, we obtain a unified way of obtaining both kernel and NN estimators. The corresponding theoretical contribution relating the geometry of NN distances to asymptotic order statistics is of independent mathematical interest.

Models for collecting and aggregating categorical data on crowdsourcing platforms typically fall into two broad categories: those assuming agents honest and consistent but with heterogeneous error rates, and those assuming agents strategic and seek to maximize their expected reward. The former often leads to tractable aggregation of elicited data, while the latter usually focuses on optimal elicitation and does not consider aggregation. In this paper, we develop a Bayesian model, wherein agents have differing quality of information, but also respond to incentives. Our model generalizes both categories and enables the joint exploration of optimal elicitation and aggregation. This model enables our exploration, both analytically and experimentally, of optimal aggregation of categorical data and optimal multiple-choice interface design.

Predicting the behavior of human participants in strategic settings is an important problem in many domains. Most existing work either assumes that participants are perfectly rational, or attempts to directly model each participant's cognitive processes based on insights from cognitive psychology and experimental economics. In this work, we present an alternative, a deep learning approach that automatically performs cognitive modeling without relying on such expert knowledge. We introduce a novel architecture that allows a single network to generalize across different input and output dimensions by using matrix units rather than scalar units, and show that its performance significantly outperforms that of the previous state of the art, which relies on expert-constructed features.

We study an online decision making problem where on each round a learner chooses a list of items based on some side information, receives a scalar feedback value for each individual item, and a reward that is linearly related to this feedback. These problems, known as contextual semibandits, arise in crowdsourcing, recommendation, and many other domains. This paper reduces contextual semibandits to supervised learning, allowing us to leverage powerful supervised learning methods in this partial-feedback setting. Our first reduction applies when the mapping from feedback to reward is known and leads to a computationally efficient algorithm with near-optimal regret. We show that this algorithm outperforms state-of-the-art approaches on real-world learning-to-rank datasets, demonstrating the advantage of oracle-based algorithms. Our second reduction applies to the previously unstudied setting when the linear mapping from feedback to reward is unknown. Our regret guarantees are superior to prior techniques that ignore the feedback.