Angelo Gnazzo

Angelo Gnazzo

PhD @ TU Wien - AITHYRA

Publications

Covariance Estimation for High-Dimensional Distributions

Published in ETH Zürich (Master’s Thesis), 2023

This thesis focuses on the estimation of covariance matrices for a variety of high-dimensional distributions, a problem pivotal in several areas such as multivariate statistics, machine learning, or Markowitz’s modern portfolio theory. The dissertation aims to study, through a random matrix theory approach, the theoretical guarantees for the convergence of the sample covariance matrix to the true covariance matrix. The results presented in this thesis provide valuable insights into the non-asymptotic theory of random matrices and several key ideas and concepts from high-dimensional probability.

Recommended citation: Gnazzo, Angelo. (2023). "Covariance Estimation for High-Dimensional Distributions." Master’s Thesis, ETH Zürich.
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Generalization Error Bounds via Information-Theoretic Methods and Convex Analysis

Published in ETH Zürich (Semester Paper), 2022

Generalization error bounds are critical to understanding the performance of machine learning models. The last few years have witnessed important new advancements in the study of these bounds. The purpose of this work is to offer an overview of some of these developments and of the related theoretical tools. Russo and Zou, Xu and Raginsky were the first to obtain bounds that provided an information-theoretic understanding of generalization in learning problems. In this survey we study the generalization error of supervised learning algorithms, first in terms of the mutual information between their input and the output and then, we study the recent generalization of the results of Neu and Lugosi beyond the choice of the mutual information to measure the dependence between the input and the output. They showed that it is possible to replace the mutual information by any strongly convex function of the joint input-output distribution. This new approach provides us generalization bounds that improve the previously known ones or are entirely new. Moreover, we show some bounds constructed in terms of the mutual information between each individual training sample and the output of the learning algorithm, which can be tighter and easier to evaluate in several applications.

Recommended citation: Gnazzo, Angelo. (2022). "Generalization Error Bounds via Information-Theoretic Methods and Convex Analysis." Se ester Paper, ETH Zürich.
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