Entropic Graph-based Posterior Regularization: Extended Version

Graph smoothness objectives have achieved great success in semi-supervised learning but have not yet been applied extensively to unsupervised generative models. We define a new class of entropic graph-based posterior regularizers that augment a probabilistic model by encouraging pairs of nearby variables in a regularization graph to have similar posterior distributions. We present a three-way alternating optimization algorithm with closed-form updates for performing inference on this joint model and learning its parameters. This method admits updates linear in the degree of the regularization graph, exhibits monotone convergence, and is easily parallelizable. We are motivated by applications in computational biology in which temporal models such as hidden Markov models are used to learn a human-interpretable representation of genomic data. On a synthetic problem, we show that our method outperforms existing methods for graph-based regularization and a comparable strategy for incorporating long-range interactions using existing methods for approximate inference. Using genome-scale functional genomics data, we integrate genome 3D interaction data into existing models for genome annotation and demonstrate significant improvements in predicting genomic activity. Graph-based methods have recently been successful in solving many types of semi-supervised learning problems (Chapelle et al., 2006; Das & Smith, 2011; Joachims, 1999; Subramanya et al., 2010; ?; Subramanya & Bilmes, 2011; Zhu et al., 2004; Zhu & Ghahramani, 2002). These methods assume that data instances lie in a low-dimensional manifold that may be represented as a graph. They optimize a graph smoothness criterion, which states that data instances nearby in the graph should be more likely to receive the same label. In a semi-supervised learning setting, optimizing this criterion has the effect of spreading labels from labeled to unlabeled instances. Despite the success of graph-based methods for semi-supervised learning, there has not been as much study of the use of graph smoothness objectives in an unsupervised setting. In unsupervised problems, we do not have labels but instead have a generative model that is assumed to explain the observed data given the latent labels. While some types of relationships between instances (for example, the relationship between neighboring words in a sentence or neighboring bases in a genome) can easily be incorporated into the generative model, it is often inappropriate to encode a graph smoothness assumption into the model this way, for two reasons. First, in some cases, it is not clear what probabilistic process generated the labels with respect to the graph. Some objectives and distance measures that are successful for semi-supervised learning do not have probabilistic analogues. Second, large models must obey factorization properties (e.g., a tree or chain as in hidden Markov models) to facilitate the use of efficient dynamic programming algorithms such as belief propagation. Graphs representing similarity between variables do not in general satisfy these structure requirements because they tend to be densely clustered, leading to very high-order factors. In this paper, therefore, we propose a new regularization approach for expressing a graph smoothness objective over a probabilistic model. We employ the posterior regularization (PR) framework of Ganchev et al. (2010), in which a probabilistic model is regularized through a term defined on an auxiliary posterior distribution variable. We define a powerful posterior regularizer which encourages pairs of variables to have similar posterior distributions by adding a penalty based on their Kullback-Leibler (KL) divergence. The pairs of penalized variables are encoded in a regularization graph which may be entirely different from the graphical model on which inference is performed. This regularizer graph need not have low treewidth and admits efficient optimization even when fully connected. We call our strategy of adding KL regularization penalties entropic graph-based posterior regularization (EGPR). We show that inference and learning using this regularizer can be performed efficiently using a three-way alternating optimization algorithm with closed-form updates. This algorithm alternates between (1) smoothing marginal posteriors according to a regularization similarity graph, (2) performing probabilistic inference in a graphical model with the same dependence structure as the unregularized model, and (3) updating model parameters. The updates are linear in the degree 1So that the supplementary material appears in context, this supplement includes the text that appears in the main version in grey. The text that appears only in the extended version is shown in black.