We reduce a broad class of machine learning problems, usually addressed by EM or sampling, to the problem of finding the $k$ extremal rays spanning the conical hull of a data point set. These $k$ ``anchors'' lead to a global solution and a more interpretable model that can even outperform EM and sampling on generalization error. To find the $k$ anchors, we propose a novel divide-and-conquer learning scheme ``DCA'' that distributes the problem to $\mathcal O(k\log k)$ same-type sub-problems on different low-D random hyperplanes, each can be solved by any solver. For the 2D sub-problem, we present a non-iterative solver that only needs to compute an array of cosine values and its max/min entries. DCA also provides a faster subroutine for other methods to check whether a point is covered in a conical hull, which improves algorithm design in multiple dimensions and brings significant speedup to learning. We apply our method to GMM, HMM, LDA, NMF and subspace clustering, then show its competitive performance and scalability over other methods on rich datasets.

In this paper we consider the problem of learning online what is the information to consider when making sequential decisions. We formalize this as a contextual multi-armed bandit problem where a high dimensional ($D$-dimensional) context vector arrives to a learner which needs to select an action to maximize its expected reward at each time step. Each dimension of the context vector is called a type. We assume that there exists an unknown relation between actions and types, called the relevance relation, such that the reward of an action only depends on the contexts of the relevant types. When the relation is a function, i.e., the reward of an action only depends on the context of a single type, and the expected reward of an action is Lipschitz continuous in the context of its relevant type, we propose an algorithm that achieves $\tilde{O}(T^{\gamma})$ regret with a high probability, where $\gamma=2/(1+\sqrt{2})$. Our algorithm achieves this by learning the unknown relevance relation, whereas prior contextual bandit algorithms that do not exploit the existence of a relevance relation will have $\tilde{O}(T^{(D+1)/(D+2)})$ regret. Our algorithm alternates between exploring and exploiting, it does not require reward observations in exploitations, and it guarantees with a high probability that actions with suboptimality greater than $\epsilon$ are never selected in exploitations. Our proposed method can be applied to a variety of learning applications including medical diagnosis, recommender systems, popularity prediction from social networks, network security etc., where at each instance of time vast amounts of different types of information are available to the decision maker, but the effect of an action depends only on a single type.

The recently established RPCA method provides a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because RPCA ignores clustering structures of the data which are ubiquitous in applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades. We show that the challenges raised by coherent data (i.e., data with high coherence) could be alleviated by Low-Rank Representation (LRR)~\cite{tpami_2013_lrr}, provided that the dictionary in LRR is configured appropriately. More precisely, we mathematically prove that if the dictionary itself is low-rank then LRR is immune to the coherence parameter which increases with the underlying cluster number. This provides an elementary principle for dealing with coherent data and naturally leads to a practical algorithm for obtaining proper dictionaries in unsupervised environments. Experiments on randomly generated matrices and real motion sequences verify our claims. See the full paper at arXiv:1404.4032.

Tensor factorizations have become popular methods for learning from multi-relational data. In this context, the rank of a factorization is an important parameter that determines runtime as well as generalization ability. To determine conditions under which factorization is an efficient approach for learning from relational data, we derive upper and lower bounds on the rank required to recover adjacency tensors. Based on our findings, we propose a novel additive tensor factorization model for learning from latent and observable patterns in multi-relational data and present a scalable algorithm for computing the factorization. Experimentally, we show that the proposed approach does not only improve the predictive performance over pure latent variable methods but that it also reduces the required rank --- and therefore runtime and memory complexity --- significantly.

Performing interventions is a major challenge in economic policy-making. We propose \emph{causal strategic inference} as a framework for conducting interventions and apply it to large, networked microfinance economies. The basic solution platform consists of modeling a microfinance market as a networked economy, learning the parameters of the model from the real-world microfinance data, and designing algorithms for various computational problems in question. We adopt Nash equilibrium as the solution concept for our model. For a special case of our model, we show that an equilibrium point always exists and that the equilibrium interest rates are unique. For the general case, we give a constructive proof of the existence of an equilibrium point. Our empirical study is based on the microfinance data from Bangladesh and Bolivia, which we use to first learn our models. We show that causal strategic inference can assist policy-makers by evaluating the outcomes of various types of interventions, such as removing a loss-making bank from the market, imposing an interest rate cap, and subsidizing banks.

Hierarchical clustering methods offer an intuitive and powerful way to model a wide variety of data sets. However, the assumption of a fixed hierarchy is often overly restrictive when working with data generated over a period of time: We expect both the structure of our hierarchy, and the parameters of the clusters, to evolve with time. In this paper, we present a distribution over collections of time-dependent, infinite-dimensional trees that can be used to model evolving hierarchies, and present an efficient and scalable algorithm for performing approximate inference in such a model. We demonstrate the efficacy of our model and inference algorithm on both synthetic data and real-world document corpora.

Deriving from the gradient vector of a generative model of local features, Fisher vector coding (FVC) has been identified as an effective coding method for image classification. Most, if not all, FVC implementations employ the Gaussian mixture model (GMM) to characterize the generation process of local features. This choice has shown to be sufficient for traditional low dimensional local features, e.g., SIFT; and typically, good performance can be achieved with only a few hundred Gaussian distributions. However, the same number of Gaussians is insufficient to model the feature space spanned by higher dimensional local features, which have become popular recently. In order to improve the modeling capacity for high dimensional features, it turns out to be inefficient and computationally impractical to simply increase the number of Gaussians. In this paper, we propose a model in which each local feature is drawn from a Gaussian distribution whose mean vector is sampled from a subspace. With certain approximation, this model can be converted to a sparse coding procedure and the learning/inference problems can be readily solved by standard sparse coding methods. By calculating the gradient vector of the proposed model, we derive a new fisher vector encoding strategy, termed Sparse Coding based Fisher Vector Coding (SCFVC). Moreover, we adopt the recently developed Deep Convolutional Neural Network (CNN) descriptor as a high dimensional local feature and implement image classification with the proposed SCFVC. Our experimental evaluations demonstrate that our method not only significantly outperforms the traditional GMM based Fisher vector encoding but also achieves the state-of-the-art performance in generic object recognition, indoor scene, and fine-grained image classification problems.

To keep up with the Big Data challenge, parallelized algorithms based on dual decomposition have been proposed to perform inference in Markov random fields. Despite this parallelization, current algorithms struggle when the energy has high order terms and the graph is densely connected. In this paper we propose a partitioning strategy followed by a message passing algorithm which is able to exploit pre-computations. It only updates the high-order factors when passing messages across machines. We demonstrate the effectiveness of our approach on the task of joint layout and semantic segmentation estimation from single images, and show that our approach is orders of magnitude faster than current methods.

Stochastic gradient descent algorithms for training linear and kernel predictors are gaining more and more importance, thanks to their scalability. While various methods have been proposed to speed up their convergence, the model selection phase is often ignored. In fact, in theoretical works most of the time assumptions are made, for example, on the prior knowledge of the norm of the optimal solution, while in the practical world validation methods remain the only viable approach. In this paper, we propose a new kernel-based stochastic gradient descent algorithm that performs model selection while training, with no parameters to tune, nor any form of cross-validation. The algorithm builds on recent advancement in online learning theory for unconstrained settings, to estimate over time the right regularization in a data-dependent way. Optimal rates of convergence are proved under standard smoothness assumptions on the target function as well as preliminary empirical results.

We propose a new provable method for robust PCA, where the task is to recover a low-rank matrix, which is corrupted with sparse perturbations. Our method consists of simple alternating projections onto the set of low rank and sparse matrices with intermediate de-noising steps. We prove correct recovery of the low rank and sparse components under tight recovery conditions, which match those for the state-of-art convex relaxation techniques. Our method is extremely simple to implement and has low computational complexity. For a $m \times n$ input matrix (say m \geq n), our method has O(r^2 mn\log(1/\epsilon)) running time, where $r$ is the rank of the low-rank component and $\epsilon$ is the accuracy. In contrast, the convex relaxation methods have a running time O(mn^2/\epsilon), which is not scalable to large problem instances. Our running time nearly matches that of the usual PCA (i.e. non robust), which is O(rmn\log (1/\epsilon)). Thus, we achieve ``best of both the worlds'', viz low computational complexity and provable recovery for robust PCA. Our analysis represents one of the few instances of global convergence guarantees for non-convex methods.