Kernel-Based Just-In-Time Learning for Passing Expectation Propagation Messages

We propose an efficient nonparametric strategy for learning a message operator in expectation propagation (EP), which takes as input the set of incoming messages to a factor node, and produces an outgoing message as output. This learned operator replaces the multivariate integral required in classical EP, which may not have an analytic expression. We use kernel-based regression, which is trained on a set of probability distributions representing the incoming messages, and the associated outgoing messages. The kernel approach has two main advantages: first, it is fast, as it is implemented using a novel two-layer random feature representation of the input message distributions; second, it has principled uncertainty estimates, and can be cheaply updated online, meaning it can request and incorporate new training data when it encounters inputs on which it is uncertain. In experiments, our approach is able to solve learning problems where a single message operator is required for multiple, substantially different data sets (logistic regression for a variety of classification problems), where it is essential to accurately assess uncertainty and to efficiently and robustly update the message operator.Comment: accepted to UAI 2015. Correct typos. Add more content to the appendix. Main results unchange

Generalization of graph network inferences in higher-order probabilistic graphical models

Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we construct iterative message-passing algorithms using Graph Neural Networks defined on factor graphs to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method gains advantage over Belief Propagation.Comment: 9 pages, 2 figure

A Factor Graph Approach to Automated Design of Bayesian Signal Processing Algorithms

The benefits of automating design cycles for Bayesian inference-based algorithms are becoming increasingly recognized by the machine learning community. As a result, interest in probabilistic programming frameworks has much increased over the past few years. This paper explores a specific probabilistic programming paradigm, namely message passing in Forney-style factor graphs (FFGs), in the context of automated design of efficient Bayesian signal processing algorithms. To this end, we developed "ForneyLab" (https://github.com/biaslab/ForneyLab.jl) as a Julia toolbox for message passing-based inference in FFGs. We show by example how ForneyLab enables automatic derivation of Bayesian signal processing algorithms, including algorithms for parameter estimation and model comparison. Crucially, due to the modular makeup of the FFG framework, both the model specification and inference methods are readily extensible in ForneyLab. In order to test this framework, we compared variational message passing as implemented by ForneyLab with automatic differentiation variational inference (ADVI) and Monte Carlo methods as implemented by state-of-the-art tools "Edward" and "Stan". In terms of performance, extensibility and stability issues, ForneyLab appears to enjoy an edge relative to its competitors for automated inference in state-space models.Comment: Accepted for publication in the International Journal of Approximate Reasonin

Distributed Bayesian Learning with Stochastic Natural-gradient Expectation Propagation and the Posterior Server

This paper makes two contributions to Bayesian machine learning algorithms. Firstly, we propose stochastic natural gradient expectation propagation (SNEP), a novel alternative to expectation propagation (EP), a popular variational inference algorithm. SNEP is a black box variational algorithm, in that it does not require any simplifying assumptions on the distribution of interest, beyond the existence of some Monte Carlo sampler for estimating the moments of the EP tilted distributions. Further, as opposed to EP which has no guarantee of convergence, SNEP can be shown to be convergent, even when using Monte Carlo moment estimates. Secondly, we propose a novel architecture for distributed Bayesian learning which we call the posterior server. The posterior server allows scalable and robust Bayesian learning in cases where a data set is stored in a distributed manner across a cluster, with each compute node containing a disjoint subset of data. An independent Monte Carlo sampler is run on each compute node, with direct access only to the local data subset, but which targets an approximation to the global posterior distribution given all data across the whole cluster. This is achieved by using a distributed asynchronous implementation of SNEP to pass messages across the cluster. We demonstrate SNEP and the posterior server on distributed Bayesian learning of logistic regression and neural networks. Keywords: Distributed Learning, Large Scale Learning, Deep Learning, Bayesian Learn- ing, Variational Inference, Expectation Propagation, Stochastic Approximation, Natural Gradient, Markov chain Monte Carlo, Parameter Server, Posterior Server.Comment: 37 pages, 7 figure

Kernel-based distribution features for statistical tests and Bayesian inference

The kernel mean embedding is known to provide a data representation which preserves full information of the data distribution. While typically computationally costly, its nonparametric nature has an advantage of requiring no explicit model specification of the data. At the other extreme are approaches which summarize data distributions into a finite-dimensional vector of hand-picked summary statistics. This explicit finite-dimensional representation offers a computationally cheaper alternative. Clearly, there is a trade-off between cost and sufficiency of the representation, and it is of interest to have a computationally efficient technique which can produce a data-driven representation, thus combining the advantages from both extremes. The main focus of this thesis is on the development of linear-time mean-embedding-based methods to automatically extract informative features of data distributions, for statistical tests and Bayesian inference. In the first part on statistical tests, several new linear-time techniques are developed. These include a new kernel-based distance measure for distributions, a new linear-time nonparametric dependence measure, and a linear-time discrepancy measure between a probabilistic model and a sample, based on a Stein operator. These new measures give rise to linear-time and consistent tests of homogeneity, independence, and goodness of fit, respectively. The key idea behind these new tests is to explicitly learn distribution-characterizing feature vectors, by maximizing a proxy for the probability of correctly rejecting the null hypothesis. We theoretically show that these new tests are consistent for any finite number of features. In the second part, we explore the use of random Fourier features to construct approximate kernel mean embeddings, for representing messages in expectation propagation (EP) algorithm. The goal is to learn a message operator which predicts EP outgoing messages from incoming messages. We derive a novel two-layer random feature representation of the input messages, allowing online learning of the operator during EP inference

Decision trees and forests: a probabilistic perspective

Decision trees and ensembles of decision trees are very popular in machine learning and often achieve state-of-the-art performance on black-box prediction tasks. However, popular variants such as C4.5, CART, boosted trees and random forests lack a probabilistic interpretation since they usually just specify an algorithm for training a model. We take a probabilistic approach where we cast the decision tree structures and the parameters associated with the nodes of a decision tree as a probabilistic model; given labeled examples, we can train the probabilistic model using a variety of approaches (Bayesian learning, maximum likelihood, etc). The probabilistic approach allows us to encode prior assumptions about tree structures and share statistical strength between node parameters; furthermore, it offers a principled mechanism to obtain probabilistic predictions which is crucial for applications where uncertainty quantification is important. Existing work on Bayesian decision trees relies on Markov chain Monte Carlo which can be computationally slow and suffer from poor mixing. We propose a novel sequential Monte Carlo algorithm that computes a particle approximation to the posterior over trees in a top-down fashion. We also propose a novel sampler for Bayesian additive regression trees by combining the above top-down particle filtering algorithm with the Particle Gibbs (Andrieu et al., 2010) framework. Finally, we propose Mondrian forests (MFs), a computationally efficient hybrid solution that is competitive with non-probabilistic counterparts in terms of speed and accuracy, but additionally produces well-calibrated uncertainty estimates. MFs use the Mondrian process (Roy and Teh, 2009) as the randomization mechanism and hierarchically smooth the node parameters within each tree (using a hierarchical probabilistic model and approximate Bayesian updates), but combine the trees in a non-Bayesian fashion. MFs can be grown in an incremental/online fashion and remarkably, the distribution of online MFs is the same as that of batch MFs