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Andrea Agazzi

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Machine Learning math.PR Computation cond-mat.mes-hall eess.SY Information Retrieval math-ph math.MP math.SP Methodology Molecular Networks Systems and Control

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preprint2026arXiv

Scalable Bayesian Inference for Generalized Linear Mixed Models via Stochastic Gradient MCMC

The generalized linear mixed model (GLMM) is widely used for analyzing correlated data, particularly in large-scale biomedical and social science applications. Scalable Bayesian inference for GLMMs is challenging because the marginal likelihood is intractable and conventional Markov chain Monte Carlo (MCMC) methods become computationally prohibitive as the number of subjects grows. We develop a stochastic gradient MCMC (SGMCMC) algorithm tailored to GLMMs that enables accurate posterior inference in the large-sample regime. Our approach uses Fisher's identity to construct an unbiased Monte Carlo estimator of the gradient of the marginal log-likelihood, making SGMCMC feasible when direct gradient computation is impossible. We analyze the additional variability introduced by both minibatching and gradient approximation, and derive a post-hoc covariance correction that yields properly calibrated posterior uncertainty. Through simulations, we show that the proposed method provides accurate posterior means and variances, outperforming existing approaches, including control variate methods, in large-$n$ settings. We further demonstrate the method's practical utility in an analysis of electronic health records data, where accounting for variance inflation materially changes scientific conclusions.

preprint2026arXiv

Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

preprint2021arXiv

Large deviations for Markov jump processes with uniformly diminishing rates

We prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further discuss the optimality of our assumptions on the decay of the jump rates. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of mass action kinetics.

preprint2021arXiv

Urgency-aware Optimal Routing in Repeated Games through Artificial Currencies

When people choose routes minimizing their individual delay, the aggregate congestion can be much higher compared to that experienced by a centrally-imposed routing. Yet centralized routing is incompatible with the presence of self-interested agents. How can we reconcile the two? In this paper we address this question within a repeated game framework and propose a fair incentive mechanism based on artificial currencies that routes selfish agents in a system-optimal fashion, while accounting for their temporal preferences. We instantiate the framework in a parallel-network whereby agents commute repeatedly (e.g., daily) from a common start node to the end node. Thereafter, we focus on the specific two-arcs case whereby, based on an artificial currency, the agents are charged when traveling on the first, fast arc, whilst they are rewarded when traveling on the second, slower arc. We assume the agents to be rational and model their choices through a game where each agent aims at minimizing a combination of today's discomfort, weighted by their urgency, and the average discomfort encountered for the rest of the period (e.g., a week). We show that, if prices of artificial currencies are judiciously chosen, the routing pattern converges to a system-optimal solution, while accommodating the agents' urgency. We complement our study through numerical simulations. Our results show that it is possible to achieve a system-optimal solution whilst reducing the agents' perceived discomfort by 14-20% when compared to a centralized optimal but urgency-unaware policy.

preprint2015arXiv

Diffusion Fingerprints

We introduce, test and discuss a method for classifying and clustering data modeled as directed graphs. The idea is to start diffusion processes from any subset of a data collection, generating corresponding distributions for reaching points in the network. These distributions take the form of high-dimensional numerical vectors and capture essential topological properties of the original dataset. We show how these diffusion vectors can be successfully applied for getting state-of-the-art accuracies in the problem of extracting pathways from metabolic networks. We also provide a guideline to illustrate how to use our method for classification problems, and discuss important details of its implementation. In particular, we present a simple dimensionality reduction technique that lowers the computational cost of classifying diffusion vectors, while leaving the predictive power of the classification process substantially unaltered. Although the method has very few parameters, the results we obtain show its flexibility and power. This should make it helpful in many other contexts.

preprint2014arXiv

The Colored Hofstadter Butterfly for the Honeycomb Lattice

We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number $σ_H$ is given as the winding number of an eigenvector of a $2 \times 2$ transfer matrix, as a function of the quasi-momentum $k \in (0,2 π)$. This method is computationally efficient (of order $O(n^4)$ in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for $σ_H $ for flux $p/q$ in the $r$-th gap conforms with the Diophantine equation $r=σ_H\cdot p+ s\cdot q$, which determines $σ_H \mod q$. A window such as $σ_H \in(-q/2,q/2)$, or possibly shifted, provides a natural further condition for $σ_H$, which however turns out not to be met. Based on extensive numerical calculations, we conjecture that the solution conforms with the relaxed condition $σ_H\in(-q,q)$.