arXiv subjects

Maurizio Filippone

Publications and source records attributed to Maurizio Filippone.

68 records · Page 4Linked to original sources

arXiv2017

Entropic Trace Estimates for Log Determinants

The scalable calculation of matrix determinants has been a bottleneck to the widespread application of many machine learning methods such as determinantal point processes, Gaussian processes, generalised Markov random fields, graph models and many others. In this work, we estimate log determinants under the framework of maximum entropy, given information in the form of moment constraints from stochastic trace estimation. The estimates demonstrate a significant improvement on state-of-the-art alternative methods, as shown on a wide variety of UFL sparse matrices. By taking the example of a general Markov random field, we also demonstrate how this approach can significantly accelerate inference in large-scale learning methods involving the log determinant.

math.NA↗

arXiv2017

Bayesian Inference of Log Determinants

The log-determinant of a kernel matrix appears in a variety of machine learning problems, ranging from determinantal point processes and generalized Markov random fields, through to the training of Gaussian processes. Exact calculation of this term is often intractable when the size of the kernel matrix exceeds a few thousand. In the spirit of probabilistic numerics, we reinterpret the problem of computing the log-determinant as a Bayesian inference problem. In particular, we combine prior knowledge in the form of bounds from matrix theory and evidence derived from stochastic trace estimation to obtain probabilistic estimates for the log-determinant and its associated uncertainty within a given computational budget. Beyond its novelty and theoretic appeal, the performance of our proposal is competitive with state-of-the-art approaches to approximating the log-determinant, while also quantifying the uncertainty due to budget-constrained evidence.

stat.ML↗

arXiv2017

AutoGP: Exploring the Capabilities and Limitations of Gaussian Process Models

We investigate the capabilities and limitations of Gaussian process models by jointly exploring three complementary directions: (i) scalable and statistically efficient inference; (ii) flexible kernels; and (iii) objective functions for hyperparameter learning alternative to the marginal likelihood. Our approach outperforms all previously reported GP methods on the standard MNIST dataset; performs comparatively to previous kernel-based methods using the RECTANGLES-IMAGE dataset; and breaks the 1% error-rate barrier in GP models using the MNIST8M dataset, showing along the way the scalability of our method at unprecedented scale for GP models (8 million observations) in classification problems. Overall, our approach represents a significant breakthrough in kernel methods and GP models, bridging the gap between deep learning approaches and kernel machines.

stat.ML↗

arXiv2017

Random Feature Expansions for Deep Gaussian Processes

The composition of multiple Gaussian Processes as a Deep Gaussian Process (DGP) enables a deep probabilistic nonparametric approach to flexibly tackle complex machine learning problems with sound quantification of uncertainty. Existing inference approaches for DGP models have limited scalability and are notoriously cumbersome to construct. In this work, we introduce a novel formulation of DGPs based on random feature expansions that we train using stochastic variational inference. This yields a practical learning framework which significantly advances the state-of-the-art in inference for DGPs, and enables accurate quantification of uncertainty. We extensively showcase the scalability and performance of our proposal on several datasets with up to 8 million observations, and various DGP architectures with up to 30 hidden layers.

stat.ML↗

arXiv2017

Disease Progression Modeling and Prediction through Random Effect Gaussian Processes and Time Transformation

The development of statistical approaches for the joint modelling of the temporal changes of imaging, biochemical, and clinical biomarkers is of paramount importance for improving the understanding of neurodegenerative disorders, and for providing a reference for the prediction and quantification of the pathology in unseen individuals. Nonetheless, the use of disease progression models for probabilistic predictions still requires investigation, for example for accounting for missing observations in clinical data, and for accurate uncertainty quantification. We tackle this problem by proposing a novel Gaussian process-based method for the joint modeling of imaging and clinical biomarker progressions from time series of individual observations. The model is formulated to account for individual random effects and time reparameterization, allowing non-parametric estimates of the biomarker evolution, as well as high flexibility in specifying correlation structure, and time transformation models. Thanks to the Bayesian formulation, the model naturally accounts for missing data, and allows for uncertainty quantification in the estimate of evolutions, as well as for probabilistic prediction of disease staging in unseen patients. The experimental results show that the proposed model provides a biologically plausible description of the evolution of Alzheimer's pathology across the whole disease time-span as well as remarkable predictive performance when tested on a large clinical cohort with missing observations.

stat.AP↗

arXiv2016

Mini-Batch Spectral Clustering

The cost of computing the spectrum of Laplacian matrices hinders the application of spectral clustering to large data sets. While approximations recover computational tractability, they can potentially affect clustering performance. This paper proposes a practical approach to learn spectral clustering based on adaptive stochastic gradient optimization. Crucially, the proposed approach recovers the exact spectrum of Laplacian matrices in the limit of the iterations, and the cost of each iteration is linear in the number of samples. Extensive experimental validation on data sets with up to half a million samples demonstrate its scalability and its ability to outperform state-of-the-art approximate methods to learn spectral clustering for a given computational budget.

stat.ML↗

arXiv2016

Preconditioning Kernel Matrices

The computational and storage complexity of kernel machines presents the primary barrier to their scaling to large, modern, datasets. A common way to tackle the scalability issue is to use the conjugate gradient algorithm, which relieves the constraints on both storage (the kernel matrix need not be stored) and computation (both stochastic gradients and parallelization can be used). Even so, conjugate gradient is not without its own issues: the conditioning of kernel matrices is often such that conjugate gradients will have poor convergence in practice. Preconditioning is a common approach to alleviating this issue. Here we propose preconditioned conjugate gradients for kernel machines, and develop a broad range of preconditioners particularly useful for kernel matrices. We describe a scalable approach to both solving kernel machines and learning their hyperparameters. We show this approach is exact in the limit of iterations and outperforms state-of-the-art approximations for a given computational budget.

stat.ML↗

arXiv2016

Adaptive Multiple Importance Sampling for Gaussian Processes

In applications of Gaussian processes where quantification of uncertainty is a strict requirement, it is necessary to accurately characterize the posterior distribution over Gaussian process covariance parameters. Normally, this is done by means of standard Markov chain Monte Carlo (MCMC) algorithms. Motivated by the issues related to the complexity of calculating the marginal likelihood that can make MCMC algorithms inefficient, this paper develops an alternative inference framework based on Adaptive Multiple Importance Sampling (AMIS). This paper studies the application of AMIS in the case of a Gaussian likelihood, and proposes the Pseudo-Marginal AMIS for non-Gaussian likelihoods, where the marginal likelihood is unbiasedly estimated. The results suggest that the proposed framework outperforms MCMC-based inference of covariance parameters in a wide range of scenarios and remains competitive for moderately large dimensional parameter spaces.

stat.CO↗

arXiv2015

Enabling scalable stochastic gradient-based inference for Gaussian processes by employing the Unbiased LInear System SolvEr (ULISSE)

In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes an adaptation of the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear systems solver based on parallelizable covariance matrix-vector products is developed to accelerate the unbiased estimation of gradients. The results demonstrate the possibility to enable scalable and exact (in a Monte Carlo sense) quantification of uncertainty in Gaussian processes without imposing any special structure on the covariance or reducing the number of input vectors.

stat.ME↗

arXiv2015

MCMC for Variationally Sparse Gaussian Processes

Gaussian process (GP) models form a core part of probabilistic machine learning. Considerable research effort has been made into attacking three issues with GP models: how to compute efficiently when the number of data is large; how to approximate the posterior when the likelihood is not Gaussian and how to estimate covariance function parameter posteriors. This paper simultaneously addresses these, using a variational approximation to the posterior which is sparse in support of the function but otherwise free-form. The result is a Hybrid Monte-Carlo sampling scheme which allows for a non-Gaussian approximation over the function values and covariance parameters simultaneously, with efficient computations based on inducing-point sparse GPs. Code to replicate each experiment in this paper will be available shortly.

stat.ML↗

arXiv2015

Bayesian nonparametric disclosure risk estimation via mixed effects log-linear models

Statistical agencies and other institutions collect data under the promise to protect the confidentiality of respondents. When releasing microdata samples, the risk that records can be identified must be assessed. To this aim, a widely adopted approach is to isolate categorical variables key to the identification and analyze multi-way contingency tables of such variables. Common disclosure risk measures focus on sample unique cells in these tables and adopt parametric log-linear models as the standard statistical tools for the problem. Such models often have to deal with large and extremely sparse tables that pose a number of challenges to risk estimation. This paper proposes to overcome these problems by studying nonparametric alternatives based on Dirichlet process random effects. The main finding is that the inclusion of such random effects allows us to reduce considerably the number of fixed effects required to achieve reliable risk estimates. This is studied on applications to real data, suggesting, in particular, that our mixed models with main effects only produce roughly equivalent estimates compared to the all two-way interactions models, and are effective in defusing potential shortcomings of traditional log-linear models. This paper adopts a fully Bayesian approach that accounts for all sources of uncertainty, including that about the population frequencies, and supplies unconditional (posterior) variances and credible intervals.

stat.AP↗

arXiv2014

Bayesian Inference for Gaussian Process Classifiers with Annealing and Pseudo-Marginal MCMC

Kernel methods have revolutionized the fields of pattern recognition and machine learning. Their success, however, critically depends on the choice of kernel parameters. Using Gaussian process (GP) classification as a working example, this paper focuses on Bayesian inference of covariance (kernel) parameters using Markov chain Monte Carlo (MCMC) methods. The motivation is that, compared to standard optimization of kernel parameters, they have been systematically demonstrated to be superior in quantifying uncertainty in predictions. Recently, the Pseudo-Marginal MCMC approach has been proposed as a practical inference tool for GP models. In particular, it amounts in replacing the analytically intractable marginal likelihood by an unbiased estimate obtainable by approximate methods and importance sampling. After discussing the potential drawbacks in employing importance sampling, this paper proposes the application of annealed importance sampling. The results empirically demonstrate that compared to importance sampling, annealed importance sampling can reduce the variance of the estimate of the marginal likelihood exponentially in the number of data at a computational cost that scales only polynomially. The results on real data demonstrate that employing annealed importance sampling in the Pseudo-Marginal MCMC approach represents a step forward in the development of fully automated exact inference engines for GP models.

stat.ME↗

arXiv2014

On User Availability Prediction and Network Applications

User connectivity patterns in network applications are known to be heterogeneous, and to follow periodic (daily and weekly) patterns. In many cases, the regularity and the correlation of those patterns is problematic: for network applications, many connected users create peaks of demand; in contrast, in peer-to-peer scenarios, having few users online results in a scarcity of available resources. On the other hand, since connectivity patterns exhibit a periodic behavior, they are to some extent predictable. This work shows how this can be exploited to anticipate future user connectivity and to have applications proactively responding to it. We evaluate the probability that any given user will be online at any given time, and assess the prediction on six-month availability traces from three different Internet applications. Building upon this, we show how our probabilistic approach makes it easy to evaluate and optimize the performance in a number of diverse network application models, and to use them to optimize systems. In particular, we show how this approach can be used in distributed hash tables, friend-to-friend storage, and cache pre-loading for social networks, resulting in substantial gains in data availability and system efficiency at negligible costs.

cs.DC↗

arXiv2014

Pseudo-Marginal Bayesian Inference for Gaussian Processes

The main challenges that arise when adopting Gaussian Process priors in probabilistic modeling are how to carry out exact Bayesian inference and how to account for uncertainty on model parameters when making model-based predictions on out-of-sample data. Using probit regression as an illustrative working example, this paper presents a general and effective methodology based on the pseudo-marginal approach to Markov chain Monte Carlo that efficiently addresses both of these issues. The results presented in this paper show improvements over existing sampling methods to simulate from the posterior distribution over the parameters defining the covariance function of the Gaussian Process prior. This is particularly important as it offers a powerful tool to carry out full Bayesian inference of Gaussian Process based hierarchic statistical models in general. The results also demonstrate that Monte Carlo based integration of all model parameters is actually feasible in this class of models providing a superior quantification of uncertainty in predictions. Extensive comparisons with respect to state-of-the-art probabilistic classifiers confirm this assertion.

stat.ML↗