arXiv subjects

Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 415 records · Page 23Linked to original sources

arXiv2013

Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation

We provide a detailed study of the estimation of probability distributions---discrete and continuous---in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental tradeoffs between privacy and convergence rate, as well as providing tools to allow movement along the privacy-statistical efficiency continuum. One of the consequences of our results is that Warner's classical work on randomized response is an optimal way to perform survey sampling while maintaining privacy of the respondents.

math.ST↗

arXiv2013

Cluster Forests

With inspiration from Random Forests (RF) in the context of classification, a new clustering ensemble method---Cluster Forests (CF) is proposed. Geometrically, CF randomly probes a high-dimensional data cloud to obtain "good local clusterings" and then aggregates via spectral clustering to obtain cluster assignments for the whole dataset. The search for good local clusterings is guided by a cluster quality measure kappa. CF progressively improves each local clustering in a fashion that resembles the tree growth in RF. Empirical studies on several real-world datasets under two different performance metrics show that CF compares favorably to its competitors. Theoretical analysis reveals that the kappa measure makes it possible to grow the local clustering in a desirable way---it is "noise-resistant". A closed-form expression is obtained for the mis-clustering rate of spectral clustering under a perturbation model, which yields new insights into some aspects of spectral clustering.

stat.ME↗

arXiv2013

MAD-Bayes: MAP-based Asymptotic Derivations from Bayes

The classical mixture of Gaussians model is related to K-means via small-variance asymptotics: as the covariances of the Gaussians tend to zero, the negative log-likelihood of the mixture of Gaussians model approaches the K-means objective, and the EM algorithm approaches the K-means algorithm. Kulis & Jordan (2012) used this observation to obtain a novel K-means-like algorithm from a Gibbs sampler for the Dirichlet process (DP) mixture. We instead consider applying small-variance asymptotics directly to the posterior in Bayesian nonparametric models. This framework is independent of any specific Bayesian inference algorithm, and it has the major advantage that it generalizes immediately to a range of models beyond the DP mixture. To illustrate, we apply our framework to the feature learning setting, where the beta process and Indian buffet process provide an appropriate Bayesian nonparametric prior. We obtain a novel objective function that goes beyond clustering to learn (and penalize new) groupings for which we relax the mutual exclusivity and exhaustivity assumptions of clustering. We demonstrate several other algorithms, all of which are scalable and simple to implement. Empirical results demonstrate the benefits of the new framework.

stat.ML↗

arXiv2013

Computing Upper and Lower Bounds on Likelihoods in Intractable Networks

We present deterministic techniques for computing upper and lower bounds on marginal probabilities in sigmoid and noisy-OR networks. These techniques become useful when the size of the network (or clique size) precludes exact computations. We illustrate the tightness of the bounds by numerical experiments.

cs.AI↗

arXiv2013

Mixture Representations for Inference and Learning in Boltzmann Machines

Boltzmann machines are undirected graphical models with two-state stochastic variables, in which the logarithms of the clique potentials are quadratic functions of the node states. They have been widely studied in the neural computing literature, although their practical applicability has been limited by the difficulty of finding an effective learning algorithm. One well-established approach, known as mean field theory, represents the stochastic distribution using a factorized approximation. However, the corresponding learning algorithm often fails to find a good solution. We conjecture that this is due to the implicit uni-modality of the mean field approximation which is therefore unable to capture multi-modality in the true distribution. In this paper we use variational methods to approximate the stochastic distribution using multi-modal mixtures of factorized distributions. We present results for both inference and learning to demonstrate the effectiveness of this approach.

cs.LG↗

arXiv2013

Feature allocations, probability functions, and paintboxes

The problem of inferring a clustering of a data set has been the subject of much research in Bayesian analysis, and there currently exists a solid mathematical foundation for Bayesian approaches to clustering. In particular, the class of probability distributions over partitions of a data set has been characterized in a number of ways, including via exchangeable partition probability functions (EPPFs) and the Kingman paintbox. Here, we develop a generalization of the clustering problem, called feature allocation, where we allow each data point to belong to an arbitrary, non-negative integer number of groups, now called features or topics. We define and study an "exchangeable feature probability function" (EFPF)---analogous to the EPPF in the clustering setting---for certain types of feature models. Moreover, we introduce a "feature paintbox" characterization---analogous to the Kingman paintbox for clustering---of the class of exchangeable feature models. We provide a further characterization of the subclass of feature allocations that have EFPF representations.

math.PR↗

arXiv2013

Loopy Belief Propagation for Approximate Inference: An Empirical Study

Recently, researchers have demonstrated that loopy belief propagation - the use of Pearls polytree algorithm IN a Bayesian network WITH loops OF error- correcting codes.The most dramatic instance OF this IS the near Shannon - limit performance OF Turbo Codes codes whose decoding algorithm IS equivalent TO loopy belief propagation IN a chain - structured Bayesian network. IN this paper we ask : IS there something special about the error - correcting code context, OR does loopy propagation WORK AS an approximate inference schemeIN a more general setting? We compare the marginals computed using loopy propagation TO the exact ones IN four Bayesian network architectures, including two real - world networks : ALARM AND QMR.We find that the loopy beliefs often converge AND WHEN they do, they give a good approximation TO the correct marginals.However,ON the QMR network, the loopy beliefs oscillated AND had no obvious relationship TO the correct posteriors. We present SOME initial investigations INTO the cause OF these oscillations, AND show that SOME simple methods OF preventing them lead TO the wrong results.

cs.AI↗

arXiv2013

Evolutionary Inference via the Poisson Indel Process

We address the problem of the joint statistical inference of phylogenetic trees and multiple sequence alignments from unaligned molecular sequences. This problem is generally formulated in terms of string-valued evolutionary processes along the branches of a phylogenetic tree. The classical evolutionary process, the TKF91 model, is a continuous-time Markov chain model comprised of insertion, deletion and substitution events. Unfortunately this model gives rise to an intractable computational problem---the computation of the marginal likelihood under the TKF91 model is exponential in the number of taxa. In this work, we present a new stochastic process, the Poisson Indel Process (PIP), in which the complexity of this computation is reduced to linear. The new model is closely related to the TKF91 model, differing only in its treatment of insertions, but the new model has a global characterization as a Poisson process on the phylogeny. Standard results for Poisson processes allow key computations to be decoupled, which yields the favorable computational profile of inference under the PIP model. We present illustrative experiments in which Bayesian inference under the PIP model is compared to separate inference of phylogenies and alignments.

q-bio.PE↗

arXiv2013

A Nested HDP for Hierarchical Topic Models

We develop a nested hierarchical Dirichlet process (nHDP) for hierarchical topic modeling. The nHDP is a generalization of the nested Chinese restaurant process (nCRP) that allows each word to follow its own path to a topic node according to a document-specific distribution on a shared tree. This alleviates the rigid, single-path formulation of the nCRP, allowing a document to more easily express thematic borrowings as a random effect. We demonstrate our algorithm on 1.8 million documents from The New York Times.

stat.ML↗

arXiv2013

PEGASUS: A Policy Search Method for Large MDPs and POMDPs

We propose a new approach to the problem of searching a space of policies for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP), given a model. Our approach is based on the following observation: Any (PO)MDP can be transformed into an "equivalent" POMDP in which all state transitions (given the current state and action) are deterministic. This reduces the general problem of policy search to one in which we need only consider POMDPs with deterministic transitions. We give a natural way of estimating the value of all policies in these transformed POMDPs. Policy search is then simply performed by searching for a policy with high estimated value. We also establish conditions under which our value estimates will be good, recovering theoretical results similar to those of Kearns, Mansour and Ng (1999), but with "sample complexity" bounds that have only a polynomial rather than exponential dependence on the horizon time. Our method applies to arbitrary POMDPs, including ones with infinite state and action spaces. We also present empirical results for our approach on a small discrete problem, and on a complex continuous state/continuous action problem involving learning to ride a bicycle.

cs.AI↗

arXiv2013

Variational MCMC

We propose a new class of learning algorithms that combines variational approximation and Markov chain Monte Carlo (MCMC) simulation. Naive algorithms that use the variational approximation as proposal distribution can perform poorly because this approximation tends to underestimate the true variance and other features of the data. We solve this problem by introducing more sophisticated MCMC algorithms. One of these algorithms is a mixture of two MCMC kernels: a random walk Metropolis kernel and a blockMetropolis-Hastings (MH) kernel with a variational approximation as proposaldistribution. The MH kernel allows one to locate regions of high probability efficiently. The Metropolis kernel allows us to explore the vicinity of these regions. This algorithm outperforms variationalapproximations because it yields slightly better estimates of the mean and considerably better estimates of higher moments, such as covariances. It also outperforms standard MCMC algorithms because it locates theregions of high probability quickly, thus speeding up convergence. We demonstrate this algorithm on the problem of Bayesian parameter estimation for logistic (sigmoid) belief networks.

cs.LG↗

arXiv2013

Efficient Stepwise Selection in Decomposable Models

In this paper, we present an efficient way of performing stepwise selection in the class of decomposable models. The main contribution of the paper is a simple characterization of the edges that canbe added to a decomposable model while keeping the resulting model decomposable and an efficient algorithm for enumerating all such edges for a given model in essentially O(1) time per edge. We also discuss how backward selection can be performed efficiently using our data structures.We also analyze the complexity of the complete stepwise selection procedure, including the complexity of choosing which of the eligible dges to add to (or delete from) the current model, with the aim ofminimizing the Kullback-Leibler distance of the resulting model from the saturated model for the data.

cs.AI↗

arXiv2012

Tree-dependent Component Analysis

We present a generalization of independent component analysis (ICA), where instead of looking for a linear transform that makes the data components independent, we look for a transform that makes the data components well fit by a tree-structured graphical model. Treating the problem as a semiparametric statistical problem, we show that the optimal transform is found by minimizing a contrast function based on mutual information, a function that directly extends the contrast function used for classical ICA. We provide two approximations of this contrast function, one using kernel density estimation, and another using kernel generalized variance. This tree-dependent component analysis framework leads naturally to an efficient general multivariate density estimation technique where only bivariate density estimation needs to be performed.

cs.LG↗

arXiv2012

Loopy Belief Propogation and Gibbs Measures

We address the question of convergence in the loopy belief propagation (LBP) algorithm. Specifically, we relate convergence of LBP to the existence of a weak limit for a sequence of Gibbs measures defined on the LBP s associated computation tree.Using tools FROM the theory OF Gibbs measures we develop easily testable sufficient conditions FOR convergence.The failure OF convergence OF LBP implies the existence OF multiple phases FOR the associated Gibbs specification.These results give new insight INTO the mechanics OF the algorithm.

cs.AI↗

arXiv2012

Computational and Statistical Tradeoffs via Convex Relaxation

In modern data analysis, one is frequently faced with statistical inference problems involving massive datasets. Processing such large datasets is usually viewed as a substantial computational challenge. However, if data are a statistician's main resource then access to more data should be viewed as an asset rather than as a burden. In this paper we describe a computational framework based on convex relaxation to reduce the computational complexity of an inference procedure when one has access to increasingly larger datasets. Convex relaxation techniques have been widely used in theoretical computer science as they give tractable approximation algorithms to many computationally intractable tasks. We demonstrate the efficacy of this methodology in statistical estimation in providing concrete time-data tradeoffs in a class of denoising problems. Thus, convex relaxation offers a principled approach to exploit the statistical gains from larger datasets to reduce the runtime of inference algorithms.

math.ST↗

arXiv2012

Ancestor Sampling for Particle Gibbs

We present a novel method in the family of particle MCMC methods that we refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing PG with backward simulation (PG-BS) procedure, we use backward sampling to (considerably) improve the mixing of the PG kernel. Instead of using separate forward and backward sweeps as in PG-BS, however, we achieve the same effect in a single forward sweep. We apply the PG-AS framework to the challenging class of non-Markovian state-space models. We develop a truncation strategy of these models that is applicable in principle to any backward-simulation-based method, but which is particularly well suited to the PG-AS framework. In particular, as we show in a simulation study, PG-AS can yield an order-of-magnitude improved accuracy relative to PG-BS due to its robustness to the truncation error. Several application examples are discussed, including Rao-Blackwellized particle smoothing and inference in degenerate state-space models.

stat.CO↗

arXiv2012

A Generalized Mean Field Algorithm for Variational Inference in Exponential Families

The mean field methods, which entail approximating intractable probability distributions variationally with distributions from a tractable family, enjoy high efficiency, guaranteed convergence, and provide lower bounds on the true likelihood. But due to requirement for model-specific derivation of the optimization equations and unclear inference quality in various models, it is not widely used as a generic approximate inference algorithm. In this paper, we discuss a generalized mean field theory on variational approximation to a broad class of intractable distributions using a rich set of tractable distributions via constrained optimization over distribution spaces. We present a class of generalized mean field (GMF) algorithms for approximate inference in complex exponential family models, which entails limiting the optimization over the class of cluster-factorizable distributions. GMF is a generic method requiring no model-specific derivations. It factors a complex model into a set of disjoint variable clusters, and uses a set of canonical fix-point equations to iteratively update the cluster distributions, and converge to locally optimal cluster marginals that preserve the original dependency structure within each cluster, hence, fully decomposed the overall inference problem. We empirically analyzed the effect of different tractable family (clusters of different granularity) on inference quality, and compared GMF with BP on several canonical models. Possible extension to higher-order MF approximation is also discussed.

cs.LG↗

arXiv2012

Ergodic Mirror Descent

We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, but instead receive samples that are coupled over time. We show that as long as the source of randomness is suitably ergodic---it converges quickly enough to a stationary distribution---the method enjoys strong convergence guarantees, both in expectation and with high probability. This result has implications for stochastic optimization in high-dimensional spaces, peer-to-peer distributed optimization schemes, decision problems with dependent data, and stochastic optimization problems over combinatorial spaces.

math.OC↗