---
res:
  bibo_abstract:
  - "Artificial intelligence and machine learning have undergone an unprecedented
    evolution in the past decade, motivating a research effort toward a theory able
    to capture the qualitative behavior of large-scale neural systems. A central puzzle
    has been the clear benefit of scaling architecture size and overfitting the training
    set in supervised learning tasks. This evidence, in apparent contradiction with
    classical statistical learning theory, pushed researchers to develop a new theory
    capturing the interplay between the algorithmic and architectural bias of training
    and the specific target function, differently from previous methods rooted in
    uniform stability.\r\nThis approach has enabled a grounded understanding of novel
    learning regimes, typically through formal limits where the number of training
    samples $n$, data dimensions $d$, and model parameters $p$ grow to infinity at
    different rates. \\\\\r\nIn this thesis, we follow this approach, focusing on
    the trustworthiness of high-dimensional models: properties that are difficult
    to control during training or deployment and often emerge under unpredictable
    or adversarial conditions. In such settings, it is crucial to formally ensure
    a priori the reliability of machine learning systems.\r\nFirst, we study data
    memorization, both as label fitting and as the storage of private information
    about training samples in trained parameters. We prove that $p = \\Omega(n)$ parameters
    are sufficient for a deep neural network to memorize a generic set of labels,
    and for a model to memorize spurious features across training data. We then give
    evidence that $p = \\Omega(dn)$ parameters are instead necessary for an adversary
    to reconstruct the full training set from the trained parameters.\r\nSecond, we
    study robustness, both to adversarial perturbations and to distribution shift.
    We first prove that $p = \\Omega(dn)$ parameters can be sufficient for a class
    of neural networks to overfit the training data while guaranteeing robustness
    to adversarial perturbations. Then, we focus on spurious correlations learning
    in high-dimensional regression, studying the effect of the ridge regularization
    parameter in the proportional regime $n = \\Theta(d)$, and connecting it via an
    equivalence argument to the role of over-parameterization $p = \\Omega(n)$ in
    neural networks. We also investigate the architectural bias of attention-based
    networks, showing that they are sensitive to the replacement of individual words
    in an embedded sentence, allowing them to generalize on sentences where the contextual
    meaning depends on one or few words.\r\nFinally, we study differentially private
    optimization in high-dimensional regimes. We prove that standard private gradient
    methods do not suffer in the over-parameterized regime $p = \\Omega(n)$, challenging
    the current wisdom based on stability-derived generalization bounds. We then consider
    linear regression in the proportional regime $n = \\Theta(d)$, showing that standard
    private gradient descent can achieve optimal rates under appropriate hyper-parameter
    scaling, such as sufficiently small gradient clipping constants, whose role is
    still debated in practice.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Simone
      foaf_name: Bombari, Simone
      foaf_surname: Bombari
      foaf_workInfoHomepage: http://www.librecat.org/personId=ca726dda-de17-11ea-bc14-f9da834f63aa
  bibo_doi: 10.15479/AT-ISTA-22857
  dct_date: 2026^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2663-337X
  - http://id.crossref.org/issn/978-3-99078-091-6
  dct_language: eng
  dct_publisher: Institute of Science and Technology Austria@
  dct_subject:
  - machine learning
  - high-dimensional statistics
  - deep learning theory
  - privacy
  - memorization
  - robustness
  dct_title: Trustworthy machine learning in high dimensions@
...