We develop convergent minimization algorithms for Bethe variational approximations which explicitly constrain marginal estimates to families of valid distributions. While existing message passing algorithms define fixed point iterations corresponding to stationary points of the Bethe free energy, their greedy dynamics do not distinguish between local minima and maxima, and can fail to converge. For continuous estimation problems, this instability is linked to the creation of invalid marginal estimates, such as Gaussians with negative variance. Conversely, our approach leverages multiplier methods with well-understood convergence properties, and uses bound projection methods to ensure that marginal approximations are valid at all iterations. We derive general algorithms for discrete and Gaussian pairwise Markov random fields, showing improvements over standard loopy belief propagation. We also apply our method to a hybrid model with both discrete and continuous variables, showing improvements over expectation propagation.

By developing data augmentation methods unique to the negative binomial (NB) distribution, we unite seemingly disjoint count and mixture models under the NB process framework. We develop fundamental properties of the models and derive efficient Gibbs sampling inference. We show that the gamma-NB process can be reduced to the hierarchical Dirichlet process with normalization, highlighting its unique theoretical, structural and computational advantages. A variety of NB processes with distinct sharing mechanisms are constructed and applied to topic modeling, with connections to existing algorithms, showing the importance of inferring both the NB dispersion and probability parameters.

Several machine learning methods allow for abstaining from uncertain predictions. While being common for settings like conventional classification, abstention has been studied much less in learning to rank. We address abstention for the label ranking setting, allowing the learner to declare certain pairs of labels as being incomparable and, thus, to predict partial instead of total orders. In our method, such predictions are produced via thresholding the probabilities of pairwise preferences between labels, as induced by a predicted probability distribution on the set of all rankings. We formally analyze this approach for the Mallows and the Plackett-Luce model, showing that it produces proper partial orders as predictions and characterizing the expressiveness of the induced class of partial orders. These theoretical results are complemented by experiments demonstrating the practical usefulness of the approach.

The question of aggregating pairwise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time: be it ranking of online gamers (e.g. MSR’s TrueSkill system) and chess players, aggregating social opinions, or deciding which product to sell based on transactions. In most settings, in addition to obtaining ranking, finding ‘scores’ for each object (e.g. player’s rating) is of interest to understanding the intensity of the preferences. In this paper, we propose a novel iterative rank aggregation algorithm for discovering scores for objects from pairwise comparisons. The algorithm has a natural random walk interpretation over the graph of objects with edges present between two objects if they are compared; the scores turn out to be the stationary probability of this random walk. The algorithm is model independent. To establish the efficacy of our method, however, we consider the popular Bradley-Terry-Luce (BTL) model in which each object has an associated score which determines the probabilistic outcomes of pairwise comparisons between objects. We bound the finite sample error rates between the scores assumed by the BTL model and those estimated by our algorithm. This, in essence, leads to order-optimal dependence on the number of samples required to learn the scores well by our algorithm. Indeed, the experimental evaluation shows that our (model independent) algorithm performs as well as the Maximum Likelihood Estimator of the BTL model and outperforms a recently proposed algorithm by Ammar and Shah [1].

We consider the estimation of an i.i.d.\ vector $\xbf \in \R^n$ from measurements $\ybf \in \R^m$ obtained by a general cascade model consisting of a known linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. We present a method, called adaptive generalized approximate message passing (Adaptive GAMP), that enables joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector $\xbf$. The proposed algorithm is a generalization of a recently-developed method by Vila and Schniter that uses expectation-maximization (EM) iterations where the posteriors in the E-steps are computed via approximate message passing. The techniques can be applied to a large class of learning problems including the learning of sparse priors in compressed sensing or identification of linear-nonlinear cascade models in dynamical systems and neural spiking processes. We prove that for large i.i.d.\ Gaussian transform matrices the asymptotic componentwise behavior of the adaptive GAMP algorithm is predicted by a simple set of scalar state evolution equations. This analysis shows that the adaptive GAMP method can yield asymptotically consistent parameter estimates, which implies that the algorithm achieves a reconstruction quality equivalent to the oracle algorithm that knows the correct parameter values. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of complex linear-nonlinear models with provable guarantees.

A challenging problem in hierarchical classification is to leverage the hierarchical relations among classes for improving classification performance. An even greater challenge is to do so in a manner that is computationally feasible for the large scale problems usually encountered in practice. This paper proposes a set of Bayesian methods to model hierarchical dependencies among class labels using multivari- ate logistic regression. Specifically, the parent-child relationships are modeled by placing a hierarchical prior over the children nodes centered around the parame- ters of their parents; thereby encouraging classes nearby in the hierarchy to share similar model parameters. We present new, efficient variational algorithms for tractable posterior inference in these models, and provide a parallel implementa- tion that can comfortably handle large-scale problems with hundreds of thousands of dimensions and tens of thousands of classes. We run a comparative evaluation on multiple large-scale benchmark datasets that highlights the scalability of our approach, and shows a significant performance advantage over the other state-of- the-art hierarchical methods.

Statistical decision theory axiomatically assumes that the relative desirability of different options that humans perceive is well described by assigning them option-specific scalar utility functions. However, this assumption is refuted by observed human behavior, including studies wherein preferences have been shown to change systematically simply through variation in the set of choice options presented. In this paper, we show that interpreting desirability as a relative comparison between available options at any particular decision instance results in a rational theory of value-inference that explains heretofore intractable violations of rational choice behavior in human subjects. Complementarily, we also characterize the conditions under which a rational agent selecting optimal options indicated by dynamic value inference in our framework will behave identically to one whose preferences are encoded using a static ordinal utility function.

Modelling natural images with sparse coding (SC) has faced two main challenges: flexibly representing varying pixel intensities and realistically representing lowlevel imagecomponents. This paper proposes a novel multiple-cause generative model of low-level image statistics that generalizes the standard SC model in two crucial points: (1) it uses a spike-and-slab prior distribution for a more realistic representation of component absence/intensity, and (2) the model uses the highly nonlinear combination rule of maximal causes analysis (MCA) instead of a linear combination.The major challenge is parameter optimization because a model with either (1) or (2) results in strongly multimodal posteriors. We show for the first time that a model combining both improvements can be trained efficiently while retaining the rich structure of the posteriors. We design an exact piecewise Gibbssampling method and combine this with a variational method based on preselection of latent dimensions. This combined training scheme tackles both analytical and computational intractability and enables application of the model to a large number of observed and hidden dimensions. Applying the model to image patches we study the optimal encoding of images by simple cells in V1 and compare the model's predictions with in vivo neural recordings.

In the conventional approaches for supervised parametric learning, relations between data and target variables are provided through training sets consisting of pairs of corresponded data and target variables. In this work, we describe a new learning scheme for parametric learning, in which the target variables $\y$ can be modeled with a prior model $p(\y)$ and the relations between data and target variables are estimated through $p(\y)$ and a set of uncorresponded data $\x$ in training. We term this method as learning with target priors (LTP). Specifically, LTP learning seeks parameter $\t$ that maximizes the log likelihood of $f_\t(\x)$ on a uncorresponded training set with regards to $p(\y)$. Compared to the conventional (semi)supervised learning approach, LTP can make efficient use of prior knowledge of the target variables in the form of probabilistic distributions, and thus removes/reduces the reliance on training data in learning. Compared to the Bayesian approach, the learned parametric regressor in LTP can be more efficiently implemented and deployed in tasks where running efficiency is critical, such as on-line BCI signal decoding. We demonstrate the effectiveness of the proposed approach on parametric regression tasks for BCI signal decoding and pose estimation from video.

Hashing is a common method to reduce large, potentially infinite feature vectors to a fixed-size table. In reinforcement learning, hashing is often used in conjunction withtile coding to represent states in continuous spaces. Hashing is also a promising approach to value function approximation in large discrete domains such as Go and Hearts, where feature vectors can be constructed by exhaustively combining a set of atomic features. Unfortunately, the typical use of hashing in value function approximation results in biased value estimates due to the possibility ofcollisions. Recent work in data stream summaries has led to the development of the tug-of-war sketch, an unbiased estimator for approximating inner products. Our work investigates the application of this new data structure to linear value function approximation. Although in the reinforcement learning setting the use of the tug-of-war sketch leads to biased value estimates, we show that this bias can be orders of magnitude less than that of standard hashing. We provide empirical results on two RL benchmark domains and fifty-five Atari 2600 games to highlight the superior learning performance obtained when using tug-of-war hashing.