In many modern applications from, for example, bioinformatics and computer vision, samples have multiple feature representations coming from different data sources. Multiview learning algorithms try to exploit all these available information to obtain a better learner in such scenarios. In this paper, we propose a novel multiple kernel learning algorithm that extends kernel k-means clustering to the multiview setting, which combines kernels calculated on the views in a localized way to better capture sample-specific characteristics of the data. We demonstrate the better performance of our localized data fusion approach on a human colon and rectal cancer data set by clustering patients. Our method finds more relevant prognostic patient groups than global data fusion methods when we evaluate the results with respect to three commonly used clinical biomarkers.

The problem of f-divergence estimation is important in the fields of machine learning, information theory, and statistics. While several divergence estimators exist, relatively few have known convergence properties. In particular, even for those estimators whose MSE convergence rates are known, the asymptotic distributions are unknown. We establish the asymptotic normality of a recently proposed ensemble estimator of f-divergence between two distributions from a finite number of samples. This estimator has MSE convergence rate of O(1/T), is simple to implement, and performs well in high dimensions. This theory enables us to perform divergence-based inference tasks such as testing equality of pairs of distributions based on empirical samples. We experimentally validate our theoretical results and, as an illustration, use them to empirically bound the best achievable classification error.

We introduce a natural extension of the notion of swap regret, conditional swap regret, that allows for action modifications conditioned on the player’s action history. We prove a series of new results for conditional swap regret minimization. We present algorithms for minimizing conditional swap regret with bounded conditioning history. We further extend these results to the case where conditional swaps are considered only for a subset of actions. We also define a new notion of equilibrium, conditional correlated equilibrium, that is tightly connected to the notion of conditional swap regret: when all players follow conditional swap regret minimization strategies, then the empirical distribution approaches this equilibrium. Finally, we extend our results to the multi-armed bandit scenario.

We address the problem of image collection summarization by learning mixtures of submodular functions. We argue that submodularity is very natural to this problem, and we show that a number of previously used scoring functions are submodular — a property not explicitly mentioned in these publications. We provide classes of submodular functions capturing the necessary properties of summaries, namely coverage, likelihood, and diversity. To learn mixtures of these submodular functions as scoring functions, we formulate summarization as a supervised learning problem using large-margin structured prediction. Furthermore, we introduce a novel evaluation metric, which we call V-ROUGE, for automatic summary scoring. While a similar metric called ROUGE has been successfully applied to document summarization [14], no such metric was known for quantifying the quality of image collection summaries. We provide a new dataset consisting of 14 real-world image collections along with many human-generated ground truth summaries collected using mechanical turk. We also extensively compare our method with previously explored methods for this problem and show that our learning approach outperforms all competitors on this new dataset. This paper provides, to our knowledge, the first systematic approach for quantifying the problem of image collection summarization, along with a new dataset of image collections and human summaries.

In this paper, we consider a multi-step version of the stochastic ADMM method with efficient guarantees for high-dimensional problems. We first analyze the simple setting, where the optimization problem consists of a loss function and a single regularizer (e.g. sparse optimization), and then extend to the multi-block setting with multiple regularizers and multiple variables (e.g. matrix decomposition into sparse and low rank components). For the sparse optimization problem, our method achieves the minimax rate of $O(s\log d/T)$ for $s$-sparse problems in $d$ dimensions in $T$ steps, and is thus, unimprovable by any method up to constant factors. For the matrix decomposition problem with a general loss function, we analyze the multi-step ADMM with multiple blocks. We establish $O(1/T)$ rate and efficient scaling as the size of matrix grows. For natural noise models (e.g. independent noise), our convergence rate is minimax-optimal. Thus, we establish tight convergence guarantees for multi-block ADMM in high dimensions. Experiments show that for both sparse optimization and matrix decomposition problems, our algorithm outperforms the state-of-the-art methods.

We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses the sets to estimate the subspaces. As the geometric structure of the clusters (linear subspaces) forbids proper performance of general distance based approaches such as K-means, many model-specific methods have been proposed. In this paper, we provide new simple and efficient algorithms for this problem. Our statistical analysis shows that the algorithms are guaranteed exact (perfect) clustering performance under certain conditions on the number of points and the affinity be- tween subspaces. These conditions are weaker than those considered in the standard statistical literature. Experimental results on synthetic data generated from the standard unions of subspaces model demonstrate our theory. We also show that our algorithm performs competitively against state-of-the-art algorithms on real-world applications such as motion segmentation and face clustering, with much simpler implementation and lower computational cost.

We introduce a new multi-task framework, in which $K$ online learners are sharing a single annotator with limited bandwidth. On each round, each of the $K$ learners receives an input, and makes a prediction about the label of that input. Then, a shared (stochastic) mechanism decides which of the $K$ inputs will be annotated. The learner that receives the feedback (label) may update its prediction rule, and we proceed to the next round. We develop an online algorithm for multi-task binary classification that learns in this setting, and bound its performance in the worst-case setting. Additionally, we show that our algorithm can be used to solve two bandits problems: contextual bandits, and dueling bandits with context, both allowed to decouple exploration and exploitation. Empirical study with OCR data, vowel prediction (VJ project) and document classification, shows that our algorithm outperforms other algorithms, one of which uses uniform allocation, and essentially makes more (accuracy) for the same labour of the annotator.

We consider the challenging practical problem of optimizing the power production of a complex of hydroelectric power plants, which involves control over three continuous action variables, uncertainty in the amount of water inflows and a variety of constraints that need to be satisfied. We propose a policy-search-based approach coupled with predictive modelling to address this problem. This approach has some key advantages compared to other alternatives, such as dynamic programming: the policy representation and search algorithm can conveniently incorporate domain knowledge; the resulting policies are easy to interpret, and the algorithm is naturally parallelizable. Our algorithm obtains a policy which outperforms the solution found by dynamic programming both quantitatively and qualitatively.

Many machine learning problems can be reduced to the maximization of submodular functions. Although well understood in the serial setting, the parallel maximization of submodular functions remains an open area of research with recent results only addressing monotone functions. The optimal algorithm for maximizing the more general class of non-monotone submodular functions was introduced by Buchbinder et al. and follows a strongly serial double-greedy logic and program analysis. In this work, we propose two methods to parallelize the double-greedy algorithm. The first, coordination-free approach emphasizes speed at the cost of a weaker approximation guarantee. The second, concurrency control approach guarantees a tight 1/2-approximation, at the quantifiable cost of additional coordination and reduced parallelism. As a consequence we explore the trade off space between guaranteed performance and objective optimality. We implement and evaluate both algorithms on multi-core hardware and billion edge graphs, demonstrating both the scalability and tradeoffs of each approach.

This paper studies the following problem: given samples from a high dimensional discrete distribution, we want to estimate the leading $(\delta,\rho)$-modes of the underlying distributions. A point is defined to be a $(\delta,\rho)$-mode if it is a local optimum of the density within a $\delta$-neighborhood under metric $\rho$. As we increase the ``scale'' parameter $\delta$, the neighborhood size increases and the total number of modes monotonically decreases. The sequence of the $(\delta,\rho)$-modes reveal intrinsic topographical information of the underlying distributions. Though the mode finding problem is generally intractable in high dimensions, this paper unveils that, if the distribution can be approximated well by a tree graphical model, mode characterization is significantly easier. An efficient algorithm with provable theoretical guarantees is proposed and is applied to applications like data analysis and multiple predictions.