Markov Random Fields (MRFs) have proven very powerful both as density estimators and feature extractors for classification. However, their use is often limited by an inability to estimate the partition function $Z$. In this paper, we exploit the gradient descent training procedure of restricted Boltzmann machines (a type of MRF) to {\bf track} the log partition function during learning. Our method relies on two distinct sources of information: (1) estimating the change $\Delta Z$ incurred by each gradient update, (2) estimating the difference in $Z$ over a small set of tempered distributions using bridge sampling. The two sources of information are then combined using an inference procedure similar to Kalman filtering.

The performance of Markov chain Monte Carlo methods is often sensitive to the scaling and correlations between the random variables of interest. An important source of information about the local correlation and scale is given by the Hessian matrix of the target distribution, but this is often either computationally expensive or infeasible. In this paper we propose MCMC samplers that make use of quasi-Newton approximations from the optimization literature, that approximate the Hessian of the target distribution from previous samples and gradients generated by the sampler. A key issue is that MCMC samplers that depend on the history of previous states are in general not valid. We address this problem by using limited memory quasi-Newton methods, which depend only on a fixed window of previous samples.

We describe how the pre-training algorithm for Deep Boltzmann Machines (DBMs) is related to the pre-training algorithm for Deep Belief Networks and we show that under certain conditions, the pre-training procedure improves the variational lower bound of a two-hidden-layer DBM. Based on this analysis, we develop a different method of pre-training DBMs that distributes the modelling work more evenly over the hidden layers. Our results on the MNIST and NORB datasets demonstrate that the new pre-training algorithm allows us to learn better generative models. Papers published at the Neural Information Processing Systems Conference.

We propose a Deep Boltzmann Machine for learning a generative model of multimodal data. We show how to use the model to extract a meaningful representation of multimodal data. We find that the learned representation is useful for classification and information retreival tasks, and hence conforms to some notion of semantic similarity. The model defines a probability density over the space of multimodal inputs. By sampling from the conditional distributions over each data modality, it possible to create the representation even when some data modalities are missing.

This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linear-Gaussian models, hidden Markov models, Gaussian mixture models, and Markov evolutionary trees. The setting is one where we only have samples from certain observed variables in the tree, and our goal is to estimate the tree structure (i.e., the graph of how the underlying hidden variables are connected to each other and to the observed variables). We propose the Spectral Recursive Grouping algorithm, an efficient and simple bottom-up procedure for recovering the tree structure from independent samples of the observed variables. Our finite sample size bounds for exact recovery of the tree structure reveal certain natural dependencies on underlying statistical and structural properties of the underlying joint distribution. Furthermore, our sample complexity guarantees have no explicit dependence on the dimensionality of the observed variables, making the algorithm applicable to many high-dimensional settings.

Factored Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs) form a powerful framework for multiagent planning under uncertainty, but optimal solutions require a rigid history-based policy representation. In this paper we allow inter-agent communication which turns the problem in a centralized Multiagent POMDP (MPOMDP). We map belief distributions over state factors to an agent's local actions by exploiting structure in the joint MPOMDP policy. The key point is that when sparse dependencies between the agents' decisions exist, often the belief over its local state factors is sufficient for an agent to unequivocally identify the optimal action, and communication can be avoided. We formalize these notions by casting the problem into convex optimization form, and present experimental results illustrating the savings in communication that we can obtain.

Markov logic is a widely used tool in statistical relational learning, which uses a weighted first-order logic knowledge base to specify a Markov random field (MRF) or a conditional random field (CRF). In many applications, a Markov logic network (MLN) is trained in one domain, but used in a different one. This paper focuses on dynamic Markov logic networks, where the domain of time points typically varies between training and testing. It has been previously pointed out that the marginal probabilities of truth assignments to ground atoms can change if one extends or reduces the domains of predicates in an MLN. We show that in addition to this problem, the standard way of unrolling a Markov logic theory into a MRF may result in time-inhomogeneity of the underlying Markov chain. Furthermore, even if these representational problems are not significant for a given domain, we show that the more practical problem of generating samples in a sequential conditional random field for the next slice relying on the samples from the previous slice has high computational cost in the general case, due to the need to estimate a normalization factor for each sample.

We consider infinite-horizon stationary $\gamma$-discounted Markov Decision Processes, for which it is known that there exists a stationary optimal policy. Using Value and Policy Iteration with some error $\epsilon$ at each iteration, it is well-known that one can compute stationary policies that are $\frac{2\gamma{(1-\gamma) 2}\epsilon$-optimal. After arguing that this guarantee is tight, we develop variations of Value and Policy Iteration for computing non-stationary policies that can be up to $\frac{2\gamma}{1-\gamma}\epsilon$-optimal, which constitutes a significant improvement in the usual situation when $\gamma$ is close to $1$. Surprisingly, this shows that the problem of computing near-optimal non-stationary policies'' is much simpler than that of computing near-optimal stationary policies''. Papers published at the Neural Information Processing Systems Conference.

Conditional Markov Chains (also known as Linear-Chain Conditional Random Fields in the literature) are a versatile class of discriminative models for the distribution of a sequence of hidden states conditional on a sequence of observable variables. Large-sample properties of Conditional Markov Chains have been first studied by Sinn and Poupart [1]. The paper extends this work in two directions: first, mixing properties of models with unbounded feature functions are being established; second, necessary conditions for model identifiability and the uniqueness of maximum likelihood estimates are being given. Papers published at the Neural Information Processing Systems Conference.

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy.