Utilizing the structure of a probabilistic model can significantly increase its learning speed. Motivated by several recent applications, in particular bigram models in language processing, we consider learning low-rank conditional probability matrices under expected KL-risk. This choice makes smoothing, that is the careful handling of low-probability elements, paramount. We derive an iterative algorithm that extends classical non-negative matrix factorization to naturally incorporate additive smoothing and prove that it converges to the stationary points of a penalized empirical risk. We then derive sample-complexity bounds for the global minimzer of the penalized risk and show that it is within a small factor of the optimal sample complexity.

That is, for the negative log-likelihood loss function, we show that the minimization of PAC-Bayesian generalization risk bounds maximizes the Bayesian marginal likelihood. This provides an alternative explanation to the Bayesian Occam's razor criteria, under the assumption that the data is generated by an i.i.d.

We study causal subset selection with Directed Information as the measure of prediction causality. Two typical tasks, causal sensor placement and covariate selection, are correspondingly formulated into cardinality constrained directed information maximizations. To attack the NP-hard problems, we show that the first problem is submodular while not necessarily monotonic. And the second one is "nearly" submodular. To substantiate the idea of approximate submodularity, we introduce a novel quantity, namely submodularity index (SmI), for general set functions. Moreover, we show that based on SmI, greedy algorithm has performance guarantee for the maximization of possibly non-monotonic and non-submodular functions, justifying its usage for a much broader class of problems. We evaluate the theoretical results with several case studies, and also illustrate the application of the subset selection to causal structure learning.

We introduce a new dynamical system for sequentially observed multivariate count data. This model is based on the gamma-Poisson construction--a natural choice for count data--and relies on a novel Bayesian nonparametric prior that ties and shrinks the model parameters, thus avoiding overfitting. We present an efficient MCMC inference algorithm that advances recent work on augmentation schemes for inference in negative binomial models. Finally, we demonstrate the model's inductive bias using a variety of real-world data sets, showing that it exhibits superior predictive performance over other models and infers highly interpretable latent structure.

The softmax representation of probabilities for categorical variables plays a prominent role in modern machine learning with numerous applications in areas such as large scale classification, neural language modeling and recommendation systems. However, softmax estimation is very expensive for large scale inference because of the high cost associated with computing the normalizing constant. Here, we introduce an efficient approximation to softmax probabilities which takes the form of a rigorous lower bound on the exact probability. This bound is expressed as a product over pairwise probabilities and it leads to scalable estimation based on stochastic optimization. It allows us to perform doubly stochastic estimation by subsampling both training instances and class labels. We show that the new bound has interesting theoretical properties and we demonstrate its use in classification problems.

Neuroscience experiments often require training animals to perform tasks designed to elicit various sensory, cognitive, and motor behaviors. Training typically involves a series of gradual adjustments of stimulus conditions and rewards in order to bring about learning. However, training protocols are usually hand-designed, relying on a combination of intuition, guesswork, and trial-and-error, and often require weeks or months to achieve a desired level of task performance. Here we combine ideas from reinforcement learning and adaptive optimal experimental design to formulate methods for adaptive optimal training of animal behavior. Our work addresses two intriguing problems at once: first, it seeks to infer the learning rules underlying an animal's behavioral changes during training; second, it seeks to exploit these rules to select stimuli that will maximize the rate of learning toward a desired objective.

Given a task of predicting Y from X, a loss function L, and a set of probability distributions Γ on (X, Y), what is the optimal decision rule minimizing the worstcase expected loss over Γ? In this paper, we address this question by introducing a generalization of the maximum entropy principle. Applying this principle to sets of distributions with marginal on X constrained to be the empirical marginal, we provide a minimax interpretation of the maximum likelihood problem over generalized linear models as well as some popular regularization schemes. For quadratic and logarithmic loss functions we revisit well-known linear and logistic regression models. Moreover, for the 0-1 loss we derive a classifier which we call the minimax SVM. The minimax SVM minimizes the worst-case expected 0-1 loss over the proposed Γ by solving a tractable optimization problem. We perform several numerical experiments to show the power of the minimax SVM in outperforming the SVM.

Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find stationary points of the likelihood objective, and these points may be far from any maximizer. This article addresses this disconnect between the statistical principles behind EM and its algorithmic properties. Specifically, it provides a global analysis of EM for specific models in which the observations comprise an i.i.d.

We propose two stochastic gradient MCMC methods for sampling from Bayesian posterior distributions defined on Riemann manifolds with a known geodesic flow, e.g.

Neural circuits contain heterogeneous groups of neurons that differ in type, location, connectivity, and basic response properties. However, traditional methods for dimensionality reduction and clustering are ill-suited to recovering the structure underlying the organization of neural circuits. In particular, they do not take advantage of the rich temporal dependencies in multi-neuron recordings and fail to account for the noise in neural spike trains. Here we describe new tools for inferring latent structure from simultaneously recorded spike train data using a hierarchical extension of a multi-neuron point process model commonly known as the generalized linear model (GLM). Our approach combines the GLM with flexible graph-theoretic priors governing the relationship between latent features and neural connectivity patterns. Fully Bayesian inference via Pólya-gamma augmentation of the resulting model allows us to classify neurons and infer latent dimensions of circuit organization from correlated spike trains. We demonstrate the effectiveness of our method with applications to synthetic data and multi-neuron recordings in primate retina, revealing latent patterns of neural types and locations from spike trains alone.