arXiv subjects

Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 433 records · Page 24Linked to original sources

arXiv2012

Graph partition strategies for generalized mean field inference

An autonomous variational inference algorithm for arbitrary graphical models requires the ability to optimize variational approximations over the space of model parameters as well as over the choice of tractable families used for the variational approximation. In this paper, we present a novel combination of graph partitioning algorithms with a generalized mean field (GMF) inference algorithm. This combination optimizes over disjoint clustering of variables and performs inference using those clusters. We provide a formal analysis of the relationship between the graph cut and the GMF approximation, and explore several graph partition strategies empirically. Our empirical results provide rather clear support for a weighted version of MinCut as a useful clustering algorithm for GMF inference, which is consistent with the implications from the formal analysis.

cs.LG↗

arXiv2012

The DLR Hierarchy of Approximate Inference

We propose a hierarchy for approximate inference based on the Dobrushin, Lanford, Ruelle (DLR) equations. This hierarchy includes existing algorithms, such as belief propagation, and also motivates novel algorithms such as factorized neighbors (FN) algorithms and variants of mean field (MF) algorithms. In particular, we show that extrema of the Bethe free energy correspond to approximate solutions of the DLR equations. In addition, we demonstrate a close connection between these approximate algorithms and Gibbs sampling. Finally, we compare and contrast various of the algorithms in the DLR hierarchy on spin-glass problems. The experiments show that algorithms higher up in the hierarchy give more accurate results when they converge but tend to be less stable.

cs.LG↗

arXiv2012

A Scalable Bootstrap for Massive Data

The bootstrap provides a simple and powerful means of assessing the quality of estimators. However, in settings involving large datasets---which are increasingly prevalent---the computation of bootstrap-based quantities can be prohibitively demanding computationally. While variants such as subsampling and the $m$ out of $n$ bootstrap can be used in principle to reduce the cost of bootstrap computations, we find that these methods are generally not robust to specification of hyperparameters (such as the number of subsampled data points), and they often require use of more prior information (such as rates of convergence of estimators) than the bootstrap. As an alternative, we introduce the Bag of Little Bootstraps (BLB), a new procedure which incorporates features of both the bootstrap and subsampling to yield a robust, computationally efficient means of assessing the quality of estimators. BLB is well suited to modern parallel and distributed computing architectures and furthermore retains the generic applicability and statistical efficiency of the bootstrap. We demonstrate BLB's favorable statistical performance via a theoretical analysis elucidating the procedure's properties, as well as a simulation study comparing BLB to the bootstrap, the $m$ out of $n$ bootstrap, and subsampling. In addition, we present results from a large-scale distributed implementation of BLB demonstrating its computational superiority on massive data, a method for adaptively selecting BLB's hyperparameters, an empirical study applying BLB to several real datasets, and an extension of BLB to time series data.

stat.ME↗

arXiv2012

Bayesian Multicategory Support Vector Machines

We show that the multi-class support vector machine (MSVM) proposed by Lee et. al. (2004), can be viewed as a MAP estimation procedure under an appropriate probabilistic interpretation of the classifier. We also show that this interpretation can be extended to a hierarchical Bayesian architecture and to a fully-Bayesian inference procedure for multi-class classification based on data augmentation. We present empirical results that show that the advantages of the Bayesian formalism are obtained without a loss in classification accuracy.

cs.LG↗

arXiv2012

Revisiting k-means: New Algorithms via Bayesian Nonparametrics

Bayesian models offer great flexibility for clustering applications---Bayesian nonparametrics can be used for modeling infinite mixtures, and hierarchical Bayesian models can be utilized for sharing clusters across multiple data sets. For the most part, such flexibility is lacking in classical clustering methods such as k-means. In this paper, we revisit the k-means clustering algorithm from a Bayesian nonparametric viewpoint. Inspired by the asymptotic connection between k-means and mixtures of Gaussians, we show that a Gibbs sampling algorithm for the Dirichlet process mixture approaches a hard clustering algorithm in the limit, and further that the resulting algorithm monotonically minimizes an elegant underlying k-means-like clustering objective that includes a penalty for the number of clusters. We generalize this analysis to the case of clustering multiple data sets through a similar asymptotic argument with the hierarchical Dirichlet process. We also discuss further extensions that highlight the benefits of our analysis: i) a spectral relaxation involving thresholded eigenvectors, and ii) a normalized cut graph clustering algorithm that does not fix the number of clusters in the graph.

cs.LG↗

arXiv2012

The Phylogenetic Indian Buffet Process: A Non-Exchangeable Nonparametric Prior for Latent Features

Nonparametric Bayesian models are often based on the assumption that the objects being modeled are exchangeable. While appropriate in some applications (e.g., bag-of-words models for documents), exchangeability is sometimes assumed simply for computational reasons; non-exchangeable models might be a better choice for applications based on subject matter. Drawing on ideas from graphical models and phylogenetics, we describe a non-exchangeable prior for a class of nonparametric latent feature models that is nearly as efficient computationally as its exchangeable counterpart. Our model is applicable to the general setting in which the dependencies between objects can be expressed using a tree, where edge lengths indicate the strength of relationships. We demonstrate an application to modeling probabilistic choice.

cs.LG↗

arXiv2012

Optimization of Structured Mean Field Objectives

In intractable, undirected graphical models, an intuitive way of creating structured mean field approximations is to select an acyclic tractable subgraph. We show that the hardness of computing the objective function and gradient of the mean field objective qualitatively depends on a simple graph property. If the tractable subgraph has this property- we call such subgraphs v-acyclic-a very fast block coordinate ascent algorithm is possible. If not, optimization is harder, but we show a new algorithm based on the construction of an auxiliary exponential family that can be used to make inference possible in this case as well. We discuss the advantages and disadvantages of each regime and compare the algorithms empirically.

stat.ML↗

arXiv2012

The Stick-Breaking Construction of the Beta Process as a Poisson Process

We show that the stick-breaking construction of the beta process due to Paisley, et al. (2010) can be obtained from the characterization of the beta process as a Poisson process. Specifically, we show that the mean measure of the underlying Poisson process is equal to that of the beta process. We use this underlying representation to derive error bounds on truncated beta processes that are tighter than those in the literature. We also develop a new MCMC inference algorithm for beta processes, based in part on our new Poisson process construction.

math.ST↗

arXiv2012

EP-GIG Priors and Applications in Bayesian Sparse Learning

In this paper we propose a novel framework for the construction of sparsity-inducing priors. In particular, we define such priors as a mixture of exponential power distributions with a generalized inverse Gaussian density (EP-GIG). EP-GIG is a variant of generalized hyperbolic distributions, and the special cases include Gaussian scale mixtures and Laplace scale mixtures. Furthermore, Laplace scale mixtures can subserve a Bayesian framework for sparse learning with nonconvex penalization. The densities of EP-GIG can be explicitly expressed. Moreover, the corresponding posterior distribution also follows a generalized inverse Gaussian distribution. These properties lead us to EM algorithms for Bayesian sparse learning. We show that these algorithms bear an interesting resemblance to iteratively re-weighted $\ell_2$ or $\ell_1$ methods. In addition, we present two extensions for grouped variable selection and logistic regression.

stat.ML↗

arXiv2012

Coherence Functions with Applications in Large-Margin Classification Methods

Support vector machines (SVMs) naturally embody sparseness due to their use of hinge loss functions. However, SVMs can not directly estimate conditional class probabilities. In this paper we propose and study a family of coherence functions, which are convex and differentiable, as surrogates of the hinge function. The coherence function is derived by using the maximum-entropy principle and is characterized by a temperature parameter. It bridges the hinge function and the logit function in logistic regression. The limit of the coherence function at zero temperature corresponds to the hinge function, and the limit of the minimizer of its expected error is the minimizer of the expected error of the hinge loss. We refer to the use of the coherence function in large-margin classification as C-learning, and we present efficient coordinate descent algorithms for the training of regularized ${\cal C}$-learning models.

stat.ML↗

arXiv2012

Modeling Events with Cascades of Poisson Processes

We present a probabilistic model of events in continuous time in which each event triggers a Poisson process of successor events. The ensemble of observed events is thereby modeled as a superposition of Poisson processes. Efficient inference is feasible under this model with an EM algorithm. Moreover, the EM algorithm can be implemented as a distributed algorithm, permitting the model to be applied to very large datasets. We apply these techniques to the modeling of Twitter messages and the revision history of Wikipedia.

cs.LG↗

arXiv2011

Joint Modeling of Multiple Related Time Series via the Beta Process

We propose a Bayesian nonparametric approach to the problem of jointly modeling multiple related time series. Our approach is based on the discovery of a set of latent, shared dynamical behaviors. Using a beta process prior, the size of the set and the sharing pattern are both inferred from data. We develop efficient Markov chain Monte Carlo methods based on the Indian buffet process representation of the predictive distribution of the beta process, without relying on a truncated model. In particular, our approach uses the sum-product algorithm to efficiently compute Metropolis-Hastings acceptance probabilities, and explores new dynamical behaviors via birth and death proposals. We examine the benefits of our proposed feature-based model on several synthetic datasets, and also demonstrate promising results on unsupervised segmentation of visual motion capture data.

stat.ME↗

arXiv2011

Learning Dependency-Based Compositional Semantics

Suppose we want to build a system that answers a natural language question by representing its semantics as a logical form and computing the answer given a structured database of facts. The core part of such a system is the semantic parser that maps questions to logical forms. Semantic parsers are typically trained from examples of questions annotated with their target logical forms, but this type of annotation is expensive. Our goal is to learn a semantic parser from question-answer pairs instead, where the logical form is modeled as a latent variable. Motivated by this challenging learning problem, we develop a new semantic formalism, dependency-based compositional semantics (DCS), which has favorable linguistic, statistical, and computational properties. We define a log-linear distribution over DCS logical forms and estimate the parameters using a simple procedure that alternates between beam search and numerical optimization. On two standard semantic parsing benchmarks, our system outperforms all existing state-of-the-art systems, despite using no annotated logical forms.

cs.AI↗

arXiv2011

Beta processes, stick-breaking, and power laws

The beta-Bernoulli process provides a Bayesian nonparametric prior for models involving collections of binary-valued features. A draw from the beta process yields an infinite collection of probabilities in the unit interval, and a draw from the Bernoulli process turns these into binary-valued features. Recent work has provided stick-breaking representations for the beta process analogous to the well-known stick-breaking representation for the Dirichlet process. We derive one such stick-breaking representation directly from the characterization of the beta process as a completely random measure. This approach motivates a three-parameter generalization of the beta process, and we study the power laws that can be obtained from this generalized beta process. We present a posterior inference algorithm for the beta-Bernoulli process that exploits the stick-breaking representation, and we present experimental results for a discrete factor-analysis model.

stat.ME↗

arXiv2011

A sticky HDP-HMM with application to speaker diarization

We consider the problem of speaker diarization, the problem of segmenting an audio recording of a meeting into temporal segments corresponding to individual speakers. The problem is rendered particularly difficult by the fact that we are not allowed to assume knowledge of the number of people participating in the meeting. To address this problem, we take a Bayesian nonparametric approach to speaker diarization that builds on the hierarchical Dirichlet process hidden Markov model (HDP-HMM) of Teh et al. [J. Amer. Statist. Assoc. 101 (2006) 1566--1581]. Although the basic HDP-HMM tends to over-segment the audio data---creating redundant states and rapidly switching among them---we describe an augmented HDP-HMM that provides effective control over the switching rate. We also show that this augmentation makes it possible to treat emission distributions nonparametrically. To scale the resulting architecture to realistic diarization problems, we develop a sampling algorithm that employs a truncated approximation of the Dirichlet process to jointly resample the full state sequence, greatly improving mixing rates. Working with a benchmark NIST data set, we show that our Bayesian nonparametric architecture yields state-of-the-art speaker diarization results.

stat.ME↗

arXiv2011

Bayesian inference for queueing networks and modeling of internet services

Modern Internet services, such as those at Google, Yahoo!, and Amazon, handle billions of requests per day on clusters of thousands of computers. Because these services operate under strict performance requirements, a statistical understanding of their performance is of great practical interest. Such services are modeled by networks of queues, where each queue models one of the computers in the system. A key challenge is that the data are incomplete, because recording detailed information about every request to a heavily used system can require unacceptable overhead. In this paper we develop a Bayesian perspective on queueing models in which the arrival and departure times that are not observed are treated as latent variables. Underlying this viewpoint is the observation that a queueing model defines a deterministic transformation between the data and a set of independent variables called the service times. With this viewpoint in hand, we sample from the posterior distribution over missing data and model parameters using Markov chain Monte Carlo. We evaluate our framework on data from a benchmark Web application. We also present a simple technique for selection among nested queueing models. We are unaware of any previous work that considers inference in networks of queues in the presence of missing data.

stat.ML↗

arXiv2011

Support union recovery in high-dimensional multivariate regression

In multivariate regression, a $K$-dimensional response vector is regressed upon a common set of $p$ covariates, with a matrix $B^*\in\mathbb{R}^{p\times K}$ of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the $\ell_1/\ell_2$ norm is used for support union recovery, or recovery of the set of $s$ rows for which $B^*$ is nonzero. Under high-dimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that is specified by the sample complexity parameter $θ(n,p,s):=n/[2ψ(B^*)\log(p-s)]$. Here $n$ is the sample size, and $ψ(B^*)$ is a sparsity-overlap function measuring a combination of the sparsities and overlaps of the $K$-regression coefficient vectors that constitute the model. We prove that the multivariate group Lasso succeeds for problem sequences $(n,p,s)$ such that $θ(n,p,s)$ exceeds a critical level $θ_u$, and fails for sequences such that $θ(n,p,s)$ lies below a critical level $θ_{\ell}$. For the special case of the standard Gaussian ensemble, we show that $θ_{\ell}=θ_u$ so that the characterization is sharp. The sparsity-overlap function $ψ(B^*)$ reveals that, if the design is uncorrelated on the active rows, $\ell_1/\ell_2$ regularization for multivariate regression never harms performance relative to an ordinary Lasso approach and can yield substantial improvements in sample complexity (up to a factor of $K$) when the coefficient vectors are suitably orthogonal. For more general designs, it is possible for the ordinary Lasso to outperform the multivariate group Lasso. We complement our analysis with simulations that demonstrate the sharpness of our theoretical results, even for relatively small problems.

stat.ML↗

arXiv2011

Multiway Spectral Clustering: A Margin-Based Perspective

Spectral clustering is a broad class of clustering procedures in which an intractable combinatorial optimization formulation of clustering is "relaxed" into a tractable eigenvector problem, and in which the relaxed solution is subsequently "rounded" into an approximate discrete solution to the original problem. In this paper we present a novel margin-based perspective on multiway spectral clustering. We show that the margin-based perspective illuminates both the relaxation and rounding aspects of spectral clustering, providing a unified analysis of existing algorithms and guiding the design of new algorithms. We also present connections between spectral clustering and several other topics in statistics, specifically minimum-variance clustering, Procrustes analysis and Gaussian intrinsic autoregression.

stat.ME↗