arXiv subjects

Rocco Caprio

Publications and source records attributed to Rocco Caprio.

5 recordsLinked to original sources

arXiv2026

Wasserstein Contraction of Coordinate Ascent Variational Inference

We study the non-asymptotic contraction in Wasserstein distance of the sequential, parallel, and random-scan coordinate ascent variational inference algorithms. This is shown to hold under a functional smoothness condition of the optimality maps and a transportation-information inequality at their fixed points. Our results are sharp and general, and as opposed to those based on global strong log-concavity assumptions, they allow for local convergence on smooth, non-smooth, and discrete manifolds, including within the context of data augmentation. We consider many applications in statistical physics and Bayesian statistics. These include pairwise Markov Random field models such as Ising and Curie-Weiss, unbalanced Bayesian Gaussian Mixture Models, high-dimensional Bayesian Probit Regression, and high-dimensional Logistic Regression with P\'olya--Gamma random variables (i.e. Jaakkola-Jordan's algorithm). In many of these models, these represent the first available convergence results of their kind.

stat.ML↗

arXiv2025

Analysis of Multiple-try Metropolis via Poincar\'e inequalities

We study the Multiple-try Metropolis algorithm using the framework of Poincar\'e inequalities. We describe the Multiple-try Metropolis as an auxiliary variable implementation of a resampling approximation to an ideal Metropolis--Hastings algorithm. Under suitable moment conditions on the importance weights, we derive explicit Poincar\'e comparison results between the Multiple-try algorithm and the ideal algorithm. We characterize the spectral gap of the latter, and finally in the Gaussian case prove explicit non-asymptotic convergence bounds for Multiple-try Metropolis by comparison.

stat.CO↗

arXiv2024

Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality

We present a new framework for analysing the Expectation Maximization (EM) algorithm. Drawing on recent advances in the theory of gradient flows over Euclidean-Wasserstein spaces, we extend techniques from alternating minimization in Euclidean spaces to the EM algorithm, via its representation as coordinate-wise minimization of the free energy. In so doing, we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of the log-Sobolev inequality. We further show that this framework naturally extends to several variants of EM, offering a unified approach for studying such algorithms.

stat.ML↗

arXiv2024

Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities

We prove non-asymptotic error bounds for particle gradient descent (PGD, Kuntz et al., 2023), a recently introduced algorithm for maximum likelihood estimation of large latent variable models obtained by discretizing a gradient flow of the free energy. We begin by showing that the flow converges exponentially fast to the free energy's minimizers for models satisfying a condition that generalizes both the log-Sobolev and the Polyak--{\L}ojasiewicz inequalities (LSI and P{\L}I, respectively). We achieve this by extending a result well-known in the optimal transport literature (that the LSI implies the Talagrand inequality) and its counterpart in the optimization literature (that the P{\L}I implies the so-called quadratic growth condition), and applying the extension to our new setting. We also generalize the Bakry--\'Emery Theorem and show that the LSI/P{\L}I extension holds for models with strongly concave log-likelihoods. For such models, we further control PGD's discretization error and obtain the non-asymptotic error bounds. While we are motivated by the study of PGD, we believe that the inequalities and results we extend may be of independent interest.

cs.LG↗

arXiv2023

A calculus for Markov chain Monte Carlo: studying approximations in algorithms

Markov chain Monte Carlo (MCMC) algorithms are based on the construction of a Markov chain with transition probabilities leaving invariant a probability distribution of interest. In this work, we look at these transition probabilities as functions of their invariant distributions, and we develop a notion of derivative in the invariant distribution of a MCMC kernel. We build around this concept a set of tools that we refer to as Markov chain Monte Carlo Calculus. This allows us to compare Markov chains with different invariant distributions within a suitable class via what we refer to as mean value inequalities. We explain how MCMC Calculus provides a natural framework to study algorithms using an approximation of an invariant distribution, and we illustrate this by using the tools developed to prove convergence of interacting and sequential MCMC algorithms. Finally, we discuss how similar ideas can be used in other frameworks.

math.PR↗