arXiv subjects

Giacomo Zanella

Publications and source records attributed to Giacomo Zanella.

At least 37 records · Page 2Linked to original sources

arXiv2024

Convergence rate of random scan Coordinate Ascent Variational Inference under log-concavity

The Coordinate Ascent Variational Inference scheme is a popular algorithm used to compute the mean-field approximation of a probability distribution of interest. We analyze its random scan version, under log-concavity assumptions on the target density. Our approach builds on the recent work of M. Arnese and D. Lacker, \emph{Convergence of coordinate ascent variational inference for log-concave measures via optimal transport} [arXiv:2404.08792] which studies the deterministic scan version of the algorithm, phrasing it as a block-coordinate descent algorithm in the space of probability distributions endowed with the geometry of optimal transport. We obtain tight rates for the random scan version, which imply that the total number of factor updates required to converge scales linearly with the condition number and the number of blocks of the target distribution. By contrast, available bounds for the deterministic scan case scale quadratically in the same quantities, which is analogue to what happens for optimization of convex functions in Euclidean spaces.

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arXiv2024

Robust Approximate Sampling via Stochastic Gradient Barker Dynamics

Stochastic Gradient (SG) Markov Chain Monte Carlo algorithms (MCMC) are popular algorithms for Bayesian sampling in the presence of large datasets. However, they come with little theoretical guarantees and assessing their empirical performances is non-trivial. In such context, it is crucial to develop algorithms that are robust to the choice of hyperparameters and to gradients heterogeneity since, in practice, both the choice of step-size and behaviour of target gradients induce hard-to-control biases in the invariant distribution. In this work we introduce the stochastic gradient Barker dynamics (SGBD) algorithm, extending the recently developed Barker MCMC scheme, a robust alternative to Langevin-based sampling algorithms, to the stochastic gradient framework. We characterize the impact of stochastic gradients on the Barker transition mechanism and develop a bias-corrected version that, under suitable assumptions, eliminates the error due to the gradient noise in the proposal. We illustrate the performance on a number of high-dimensional examples, showing that SGBD is more robust to hyperparameter tuning and to irregular behavior of the target gradients compared to the popular stochastic gradient Langevin dynamics algorithm.

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arXiv2023

Dimension-free mixing times of Gibbs samplers for Bayesian hierarchical models

Gibbs samplers are popular algorithms to approximate posterior distributions arising from Bayesian hierarchical models. Despite their popularity and good empirical performances, however, there are still relatively few quantitative results on their convergence properties, e.g. much less than for gradient-based sampling methods. In this work we analyse the behaviour of total variation mixing times of Gibbs samplers targeting hierarchical models using tools from Bayesian asymptotics. We obtain dimension-free convergence results under random data-generating assumptions, for a broad class of two-level models with generic likelihood function. Specific examples with Gaussian, binomial and categorical likelihoods are discussed.

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arXiv2023

Scalable Bayesian computation for crossed and nested hierarchical models

We develop sampling algorithms to fit Bayesian hierarchical models, the computational complexity of which scales linearly with the number of observations and the number of parameters in the model. We focus on crossed random effect and nested multilevel models, which are used ubiquitously in applied sciences. The posterior dependence in both classes is sparse: in crossed random effects models it resembles a random graph, whereas in nested multilevel models it is tree-structured. For each class we identify a framework for scalable computation, building on previous work. Methods for crossed models are based on extensions of appropriately designed collapsed Gibbs samplers, where we introduce the idea of local centering; while methods for nested models are based on sparse linear algebra and data augmentation. We provide a theoretical analysis of the proposed algorithms in some simplified settings, including a comparison with previously proposed methodologies and an average-case analysis based on random graph theory. Numerical experiments, including two challenging real data analyses on predicting electoral results and real estate prices, compare with off-the-shelf Hamiltonian Monte Carlo, displaying drastic improvement in performance.

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arXiv2023

Robust leave-one-out cross-validation for high-dimensional Bayesian models

Leave-one-out cross-validation (LOO-CV) is a popular method for estimating out-of-sample predictive accuracy. However, computing LOO-CV criteria can be computationally expensive due to the need to fit the model multiple times. In the Bayesian context, importance sampling provides a possible solution but classical approaches can easily produce estimators whose asymptotic variance is infinite, making them potentially unreliable. Here we propose and analyze a novel mixture estimator to compute Bayesian LOO-CV criteria. Our method retains the simplicity and computational convenience of classical approaches, while guaranteeing finite asymptotic variance of the resulting estimators. Both theoretical and numerical results are provided to illustrate the improved robustness and efficiency. The computational benefits are particularly significant in high-dimensional problems, allowing to perform Bayesian LOO-CV for a broader range of models, and datasets with highly influential observations. The proposed methodology is easily implementable in standard probabilistic programming software and has a computational cost roughly equivalent to fitting the original model once.

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arXiv2023

Improving multiple-try Metropolis with local balancing

Multiple-try Metropolis (MTM) is a popular Markov chain Monte Carlo method with the appealing feature of being amenable to parallel computing. At each iteration, it samples several candidates for the next state of the Markov chain and randomly selects one of them based on a weight function. The canonical weight function is proportional to the target density. We show both theoretically and empirically that this weight function induces pathological behaviours in high dimensions, especially during the convergence phase. We propose to instead use weight functions akin to the locally-balanced proposal distributions of Zanella (2020), thus yielding MTM algorithms that do not exhibit those pathological behaviours. To theoretically analyse these algorithms, we study the high-dimensional performance of ideal schemes that can be thought of as MTM algorithms which sample an infinite number of candidates at each iteration, as well as the discrepancy between such schemes and the MTM algorithms which sample a finite number of candidates. Our analysis unveils a strong distinction between the convergence and stationary phases: in the former, local balancing is crucial and effective to achieve fast convergence, while in the latter, the canonical and novel weight functions yield similar performance. Numerical experiments include an application in precision medicine involving a computationally-expensive forward model, which makes the use of parallel computing within MTM iterations beneficial.

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arXiv2023

Bayesian conjugacy in probit, tobit, multinomial probit and extensions: A review and new results

A broad class of models that routinely appear in several fields can be expressed as partially or fully discretized Gaussian linear regressions. Besides including basic Gaussian response settings, this class also encompasses probit, multinomial probit and tobit regression, among others, thereby yielding to one of the most widely-implemented families of models in applications. The relevance of such representations has stimulated decades of research in the Bayesian field, mostly motivated by the fact that, unlike for Gaussian linear regression, the posterior distribution induced by such models does not seem to belong to a known class, under the commonly-assumed Gaussian priors for the coefficients. This has motivated several solutions for posterior inference relying on sampling-based strategies or on deterministic approximations that, however, still experience computational and accuracy issues, especially in high dimensions. The scope of this article is to review, unify and extend recent advances in Bayesian inference and computation for this class of models. To address such a goal, we prove that the likelihoods induced by these formulations share a common analytical structure that implies conjugacy with a broad class of distributions, namely the unified skew-normals (SUN), that generalize Gaussians to skewed contexts. This result unifies and extends recent conjugacy properties for specific models within the class analyzed, and opens avenues for improved posterior inference, under a broader class of formulations and priors, via novel closed-form expressions, i.i.d. samplers from the exact SUN posteriors, and more accurate and scalable approximations from VB and EP. Such advantages are illustrated in simulations and are expected to facilitate the routine-use of these core Bayesian models, while providing a novel framework to study theoretical properties and develop future extensions.

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arXiv2022

Clustering consistency with Dirichlet process mixtures

Dirichlet process mixtures are flexible non-parametric models, particularly suited to density estimation and probabilistic clustering. In this work we study the posterior distribution induced by Dirichlet process mixtures as the sample size increases, and more specifically focus on consistency for the unknown number of clusters when the observed data are generated from a finite mixture. Crucially, we consider the situation where a prior is placed on the concentration parameter of the underlying Dirichlet process. Previous findings in the literature suggest that Dirichlet process mixtures are typically not consistent for the number of clusters if the concentration parameter is held fixed and data come from a finite mixture. Here we show that consistency for the number of clusters can be achieved if the concentration parameter is adapted in a fully Bayesian way, as commonly done in practice. Our results are derived for data coming from a class of finite mixtures, with mild assumptions on the prior for the concentration parameter and for a variety of choices of likelihood kernels for the mixture.

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arXiv2022

Scalable and Accurate Variational Bayes for High-Dimensional Binary Regression Models

Modern methods for Bayesian regression beyond the Gaussian response setting are often computationally impractical or inaccurate in high dimensions. In fact, as discussed in recent literature, bypassing such a trade-off is still an open problem even in routine binary regression models, and there is limited theory on the quality of variational approximations in high-dimensional settings. To address this gap, we study the approximation accuracy of routinely-used mean-field variational Bayes solutions in high-dimensional probit regression with Gaussian priors, obtaining novel and practically relevant results on the pathological behavior of such strategies in uncertainty quantification, point estimation and prediction. Motivated by these results, we further develop a new partially-factorized variational approximation for the posterior of the probit coefficients which leverages a representation with global and local variables but, unlike for classical mean-field assumptions, it avoids a fully factorized approximation, and instead assumes a factorization only for the local variables. We prove that the resulting approximation belongs to a tractable class of unified skew-normal densities that crucially incorporates skewness and, unlike for state-of-the-art mean-field solutions, converges to the exact posterior density as p goes to infinity. To solve the variational optimization problem, we derive a tractable CAVI algorithm that easily scales to p in the tens of thousands, and provably requires a number of iterations converging to 1 as p goes to infinity. Such findings are also illustrated in extensive empirical studies where our novel solution is shown to improve the approximation accuracy of mean-field variational Bayes for any n and p, with the magnitude of these gains being remarkable in those high-dimensional p>n settings where state-of-the-art methods are computationally impractical.

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arXiv2022

Optimal design of the Barker proposal and other locally-balanced Metropolis-Hastings algorithms

We study the class of first-order locally-balanced Metropolis--Hastings algorithms introduced in Livingstone & Zanella (2021). To choose a specific algorithm within the class the user must select a balancing function $g:\mathbb{R} \to \mathbb{R}$ satisfying $g(t) = tg(1/t)$, and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a universal limiting optimal acceptance rate of 57% and scaling of $n^{-1/3}$ as the dimension $n$ tends to infinity among all members of the class under mild smoothness assumptions on $g$ and when the target distribution for the algorithm is of the product form. In particular we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimise this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, optimal choice of balancing function under a Gaussian noise distribution, and optimal choice of first-order locally-balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings and in particular show that a bi-modal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.

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arXiv2021

A fresh take on 'Barker dynamics' for MCMC

We study a recently introduced gradient-based Markov chain Monte Carlo method based on 'Barker dynamics'. We provide a full derivation of the method from first principles, placing it within a wider class of continuous-time Markov jump processes. We then evaluate the Barker approach numerically on a challenging ill-conditioned logistic regression example with imbalanced data, showing in particular that the algorithm is remarkably robust to irregularity (in this case a high degree of skew) in the target distribution.

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arXiv2020

The Barker proposal: combining robustness and efficiency in gradient-based MCMC

There is a tension between robustness and efficiency when designing Markov chain Monte Carlo (MCMC) sampling algorithms. Here we focus on robustness with respect to tuning parameters, showing that more sophisticated algorithms tend to be more sensitive to the choice of step-size parameter and less robust to heterogeneity of the distribution of interest. We characterise this phenomenon by studying the behaviour of spectral gaps as an increasingly poor step-size is chosen for the algorithm. Motivated by these considerations, we propose a novel and simple gradient-based MCMC algorithm, inspired by the classical Barker accept-reject rule, with improved robustness properties. Extensive theoretical results, dealing with robustness to tuning, geometric ergodicity and scaling with dimension, suggest that the novel scheme combines the robustness of simple schemes with the efficiency of gradient-based ones. We show numerically that this type of robustness is particularly beneficial in the context of adaptive MCMC, giving examples where our proposed scheme significantly outperforms state-of-the-art alternatives.

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arXiv2020

Random Partition Models for Microclustering Tasks

Traditional Bayesian random partition models assume that the size of each cluster grows linearly with the number of data points. While this is appealing for some applications, this assumption is not appropriate for other tasks such as entity resolution, modeling of sparse networks, and DNA sequencing tasks. Such applications require models that yield clusters whose sizes grow sublinearly with the total number of data points -- the microclustering property. Motivated by these issues, we propose a general class of random partition models that satisfy the microclustering property with well-characterized theoretical properties. Our proposed models overcome major limitations in the existing literature on microclustering models, namely a lack of interpretability, identifiability, and full characterization of model asymptotic properties. Crucially, we drop the classical assumption of having an exchangeable sequence of data points, and instead assume an exchangeable sequence of clusters. In addition, our framework provides flexibility in terms of the prior distribution of cluster sizes, computational tractability, and applicability to a large number of microclustering tasks. We establish theoretical properties of the resulting class of priors, where we characterize the asymptotic behavior of the number of clusters and of the proportion of clusters of a given size. Our framework allows a simple and efficient Markov chain Monte Carlo algorithm to perform statistical inference. We illustrate our proposed methodology on the microclustering task of entity resolution, where we provide a simulation study and real experiments on survey panel data.

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arXiv2019

Scalable Importance Tempering and Bayesian Variable Selection

We propose a Monte Carlo algorithm to sample from high dimensional probability distributions that combines Markov chain Monte Carlo and importance sampling. We provide a careful theoretical analysis, including guarantees on robustness to high dimensionality, explicit comparison with standard Markov chain Monte Carlo methods and illustrations of the potential improvements in efficiency. Simple and concrete intuition is provided for when the novel scheme is expected to outperform standard schemes. When applied to Bayesian variable-selection problems, the novel algorithm is orders of magnitude more efficient than available alternative sampling schemes and enables fast and reliable fully Bayesian inferences with tens of thousand regressors.

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arXiv2019

Multilevel linear models, Gibbs samplers and multigrid decompositions

We study the convergence properties of the Gibbs Sampler in the context of posterior distributions arising from Bayesian analysis of conditionally Gaussian hierarchical models. We develop a multigrid approach to derive analytic expressions for the convergence rates of the algorithm for various widely used model structures, including nested and crossed random effects. Our results apply to multilevel models with an arbitrary number of layers in the hierarchy, while most previous work was limited to the two-level nested case. The theoretical results provide explicit and easy-to-implement guidelines to optimize practical implementations of the Gibbs Sampler, such as indications on which parametrization to choose (e.g. centred and non-centred), which constraint to impose to guarantee statistical identifiability, and which parameters to monitor in the diagnostic process. Simulations suggest that the results are informative also in the context of non-Gaussian distributions and more general MCMC schemes, such as gradient-based ones.implementation of Gibbs samplers on conditionally Gaussian hierarchical models.

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arXiv2018

Scalable inference for crossed random effects models

We analyze the complexity of Gibbs samplers for inference in crossed random effect models used in modern analysis of variance. We demonstrate that for certain designs the plain vanilla Gibbs sampler is not scalable, in the sense that its complexity is worse than proportional to the number of parameters and data. We thus propose a simple modification leading to a collapsed Gibbs sampler that is provably scalable. Although our theory requires some balancedness assumptions on the data designs, we demonstrate in simulated and real datasets that the rates it predicts match remarkably the correct rates in cases where the assumptions are violated. We also show that the collapsed Gibbs sampler, extended to sample further unknown hyperparameters, outperforms significantly alternative state of the art algorithms.

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arXiv2017

Informed proposals for local MCMC in discrete spaces

There is a lack of methodological results to design efficient Markov chain Monte Carlo (MCMC) algorithms for statistical models with discrete-valued high-dimensional parameters. Motivated by this consideration, we propose a simple framework for the design of informed MCMC proposals (i.e. Metropolis-Hastings proposal distributions that appropriately incorporate local information about the target) which is naturally applicable to both discrete and continuous spaces. We explicitly characterize the class of optimal proposal distributions under this framework, which we refer to as locally-balanced proposals, and prove their Peskun-optimality in high-dimensional regimes. The resulting algorithms are straightforward to implement in discrete spaces and provide orders of magnitude improvements in efficiency compared to alternative MCMC schemes, including discrete versions of Hamiltonian Monte Carlo. Simulations are performed with both simulated and real datasets, including a detailed application to Bayesian record linkage. A direct connection with gradient-based MCMC suggests that locally-balanced proposals may be seen as a natural way to extend the latter to discrete spaces.

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arXiv2017

A note on MCMC for nested multilevel regression models via belief propagation

In the quest for scalable Bayesian computational algorithms we need to exploit the full potential of existing methodologies. In this note we point out that message passing algorithms, which are very well developed for inference in graphical models, appear to be largely unexplored for scalable inference in Bayesian multilevel regression models. We show that nested multilevel regression models with Gaussian errors lend themselves very naturally to the combined use of belief propagation and MCMC. Specifically, the posterior distribution of the regression parameters conditionally on covariance hyperparameters is a high-dimensional Gaussian that can be sampled exactly (as well as marginalized) using belief propagation at a cost that scales linearly in the number of parameters and data. We derive an algorithm that works efficiently even for conditionally singular Gaussian distributions, e.g., when there are linear constraints between the parameters at different levels. We show that allowing for such non-invertible Gaussians is critical for belief propagation to be applicable to a large class of nested multilevel models. From a different perspective, the methodology proposed can be seen as a generalization of forward-backward algorithms for sampling to multilevel regressions with tree-structure graphical models, as opposed to single-branch trees used in classical Kalman filter contexts.

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