@phdthesis{22857,
  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.
This 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. \\
In 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.
First, 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.
Second, 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.
Finally, 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.},
  author       = {Bombari, Simone},
  isbn         = {978-3-99078-091-6},
  issn         = {2663-337X},
  keywords     = {machine learning, high-dimensional statistics, deep learning theory, privacy, memorization, robustness},
  pages        = {446},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Trustworthy machine learning in high dimensions}},
  doi          = {10.15479/AT-ISTA-22857},
  year         = {2026},
}

@inproceedings{21326,
  abstract     = {Neural Collapse is a phenomenon where the last-layer representations of a well-trained neural network converge to a highly structured geometry. In this paper, we focus on its first (and most basic) property, known as NC1: the within-class variability vanishes. While prior theoretical studies establish the occurrence of NC1 via the data-agnostic unconstrained features model, our work adopts a data-specific perspective, analyzing NC1 in a three-layer neural network, with the first two layers operating in the mean-field regime and followed by a linear layer. In particular, we establish a fundamental connection between NC1 and the loss landscape: we prove that points with small empirical loss and gradient norm (thus, close to being stationary) approximately satisfy NC1, and the closeness to NC1 is controlled by the residual loss and gradient norm. We then show that (i) gradient flow on the mean squared error converges to NC1 solutions with small empirical loss, and (ii) for well-separated data distributions, both NC1 and vanishing test loss are achieved simultaneously. This aligns with the empirical observation that NC1 emerges during training while models attain near-zero test error. Overall, our results demonstrate that NC1 arises from gradient training due to the properties of the loss landscape, and they show the co-occurrence of NC1 and small test error for certain data distributions.},
  author       = {Wu, Diyuan and Mondelli, Marco},
  booktitle    = {Proceedings of the 42nd International Conference on Machine Learning},
  issn         = {2640-3498},
  location     = {Vancouver, Canada},
  pages        = {67499--67536},
  publisher    = {ML Research Press},
  title        = {{Neural collapse beyond the unconstrained features model: Landscape, dynamics, and generalization in the mean-field regime}},
  volume       = {267},
  year         = {2025},
}

@article{19627,
  abstract     = {Differentially private gradient descent (DP-GD) is a popular algorithm to train deep learning models with provable guarantees on the privacy of the training data. In the last decade, the problem of understanding its performance cost with respect to standard GD has received remarkable attention from the research community, which formally derived upper bounds on the excess population risk  RP  in different learning settings. However, existing bounds typically degrade with over-parameterization, i.e., as the number of parameters  p  gets larger than the number of training samples  n  -- a regime which is ubiquitous in current deep-learning practice. As a result, the lack of theoretical insights leaves practitioners without clear guidance, leading some to reduce the effective number of trainable parameters to improve performance, while others use larger models to achieve better results through scale. In this work, we show that in the popular random features model with quadratic loss, for any sufficiently large  p , privacy can be obtained for free, i.e.,  |RP|=o(1) , not only when the privacy parameter  ε  has constant order, but also in the strongly private setting  ε=o(1) . This challenges the common wisdom that over-parameterization inherently hinders performance in private learning.},
  author       = {Bombari, Simone and Mondelli, Marco},
  issn         = {1091-6490},
  journal      = {Proceedings of the National Academy of Sciences},
  number       = {15},
  publisher    = {National Academy of Sciences},
  title        = {{Privacy for free in the overparameterized regime}},
  doi          = {10.1073/pnas.2423072122},
  volume       = {122},
  year         = {2025},
}