[{"date_created":"2026-09-08T13:40:08Z","day":"08","corr_author":"1","type":"dissertation","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"12537"},{"status":"public","relation":"part_of_dissertation","id":"18972"},{"relation":"part_of_dissertation","id":"18973","status":"public"},{"id":"21324","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"22894"},{"status":"public","relation":"part_of_dissertation","id":"19627"},{"relation":"part_of_dissertation","id":"12859","status":"public"}]},"acknowledgement":"This project was partially supported by the 2019 Lopez-Loreta prize,\r\nthe European Union (ERC, INF2\r\n, project number 101161364), the Austrian Science Fund\r\n(FWF) 10.55776/COE12, and a Google PhD fellowship in machine intelligence. Furthermore,\r\nthe candidate acknowledges the support from the Scientific Service Units of the Institute of\r\nScience and Technology Austria through resources provided by Scientific Computing.","citation":{"chicago":"Bombari, Simone. “Trustworthy Machine Learning in High Dimensions.” Institute of Science and Technology Austria, 2026. <a href=\"https://doi.org/10.15479/AT-ISTA-22857\">https://doi.org/10.15479/AT-ISTA-22857</a>.","mla":"Bombari, Simone. <i>Trustworthy Machine Learning in High Dimensions</i>. Institute of Science and Technology Austria, 2026, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-22857\">10.15479/AT-ISTA-22857</a>.","ama":"Bombari S. Trustworthy machine learning in high dimensions. 2026. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-22857\">10.15479/AT-ISTA-22857</a>","ieee":"S. Bombari, “Trustworthy machine learning in high dimensions,” Institute of Science and Technology Austria, 2026.","ista":"Bombari S. 2026. Trustworthy machine learning in high dimensions. Institute of Science and Technology Austria.","short":"S. Bombari, Trustworthy Machine Learning in High Dimensions, Institute of Science and Technology Austria, 2026.","apa":"Bombari, S. (2026). <i>Trustworthy machine learning in high dimensions</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-22857\">https://doi.org/10.15479/AT-ISTA-22857</a>"},"project":[{"_id":"92099302-16d5-11f0-9cad-f9a785f54fbd","name":"Trustworthy Deep Learning Theory: Private Over-Parameterized Models and Robust LLMs"},{"name":"Inference in High Dimensions: Light-speed Algorithms and Information Limits","grant_number":"101161364","_id":"911e6d1f-16d5-11f0-9cad-c5c68c6a1cdf"},{"grant_number":"COE12","_id":"74caaef7-b034-11f1-8f2d-e0e993bb422e","name":"Bilateral Artificial Intelligence (Mondelli)"}],"publication_status":"published","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","status":"public","OA_place":"publisher","acknowledged_ssus":[{"_id":"ScienComp"}],"publication_identifier":{"isbn":["978-3-99078-091-6"],"issn":["2663-337X"]},"date_published":"2026-09-08T00:00:00Z","degree_awarded":"PhD","oa_version":"Published Version","title":"Trustworthy machine learning in high dimensions","supplementarymaterial":"no","fulldoi":"https://doi.org/10.15479/AT-ISTA-22857","keyword":["machine learning","high-dimensional statistics","deep learning theory","privacy","memorization","robustness"],"alternative_title":["ISTA Thesis"],"year":"2026","doi_confirm":"1","publisher":"Institute of Science and Technology Austria","month":"09","date_updated":"2026-09-21T13:07:01Z","file_date_updated":"2026-09-10T10:03:48Z","file":[{"relation":"source_file","file_name":"Thesis copy.zip","file_id":"22859","creator":"sbombari","content_type":"application/zip","date_created":"2026-09-08T13:28:38Z","file_size":99154325,"checksum":"8eab99d6dc6e4476826bdfc7c00fdebb","date_updated":"2026-09-08T13:28:38Z","access_level":"closed"},{"access_level":"open_access","date_updated":"2026-09-10T10:03:48Z","file_size":12856211,"checksum":"007033bafe4622ff4c2e758301354df1","date_created":"2026-09-10T10:03:48Z","creator":"sbombari","file_id":"22898","content_type":"application/pdf","file_name":"2026_Bombari_Simone_Thesis.pdf","relation":"main_file","success":1}],"ddc":["519"],"has_accepted_license":"1","oa":1,"das_tickbox":"0","language":[{"iso":"eng"}],"_id":"22857","supervisor":[{"full_name":"Mondelli, Marco","orcid":"0000-0002-3242-7020","first_name":"Marco","id":"27EB676C-8706-11E9-9510-7717E6697425","last_name":"Mondelli"}],"author":[{"full_name":"Bombari, Simone","last_name":"Bombari","first_name":"Simone","id":"ca726dda-de17-11ea-bc14-f9da834f63aa"}],"page":"446","article_processing_charge":"No","researchdata_availability":"no","department":[{"_id":"GradSch"},{"_id":"MaMo"}],"abstract":[{"lang":"eng","text":"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."}],"doi":"10.15479/AT-ISTA-22857"},{"quality_controlled":"1","conference":{"end_date":"2025-07-19","location":"Vancouver, Canada","start_date":"2025-07-13","name":"ICML: International Conference on Machine Learning"},"oa_version":"Published Version","title":"Neural collapse beyond the unconstrained features model: Landscape, dynamics, and generalization in the mean-field regime","date_published":"2025-07-30T00:00:00Z","OA_place":"publisher","OA_type":"gold","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","publication":"Proceedings of the 42nd International Conference on Machine Learning","publication_identifier":{"eissn":["2640-3498"]},"acknowledgement":"This research was funded in whole or in part by the Austrian Science Fund (FWF) 10.55776/COE12. For the purpose of open access, the authors have applied a CC BY public\r\ncopyright license to any Author Accepted Manuscript version arising from this submission. The authors would like to thank Peter Sukenık for general helpful discussions and for pointing out that all the stationary points are approximately proportional in the case without entropic regularization. ","project":[{"name":"Bilateral Artificial Intelligence (Mondelli)","grant_number":"COE12","_id":"74caaef7-b034-11f1-8f2d-e0e993bb422e"}],"citation":{"ama":"Wu D, Mondelli M. Neural collapse beyond the unconstrained features model: Landscape, dynamics, and generalization in the mean-field regime. In: <i>Proceedings of the 42nd International Conference on Machine Learning</i>. Vol 267. ML Research Press; 2025:67499-67536.","mla":"Wu, Diyuan, and Marco Mondelli. “Neural Collapse beyond the Unconstrained Features Model: Landscape, Dynamics, and Generalization in the Mean-Field Regime.” <i>Proceedings of the 42nd International Conference on Machine Learning</i>, vol. 267, ML Research Press, 2025, pp. 67499–536.","chicago":"Wu, Diyuan, and Marco Mondelli. “Neural Collapse beyond the Unconstrained Features Model: Landscape, Dynamics, and Generalization in the Mean-Field Regime.” In <i>Proceedings of the 42nd International Conference on Machine Learning</i>, 267:67499–536. ML Research Press, 2025.","apa":"Wu, D., &#38; Mondelli, M. (2025). Neural collapse beyond the unconstrained features model: Landscape, dynamics, and generalization in the mean-field regime. In <i>Proceedings of the 42nd International Conference on Machine Learning</i> (Vol. 267, pp. 67499–67536). Vancouver, Canada: ML Research Press.","short":"D. Wu, M. Mondelli, in:, Proceedings of the 42nd International Conference on Machine Learning, ML Research Press, 2025, pp. 67499–67536.","ista":"Wu D, Mondelli M. 2025. Neural collapse beyond the unconstrained features model: Landscape, dynamics, and generalization in the mean-field regime. Proceedings of the 42nd International Conference on Machine Learning. ICML: International Conference on Machine Learning, PMLR, vol. 267, 67499–67536.","ieee":"D. Wu and M. Mondelli, “Neural collapse beyond the unconstrained features model: Landscape, dynamics, and generalization in the mean-field regime,” in <i>Proceedings of the 42nd International Conference on Machine Learning</i>, Vancouver, Canada, 2025, vol. 267, pp. 67499–67536."},"corr_author":"1","arxiv":1,"date_created":"2026-02-18T12:02:45Z","day":"30","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"type":"conference","page":"67499-67536","author":[{"full_name":"Wu, Diyuan","last_name":"Wu","first_name":"Diyuan","id":"1a5914c2-896a-11ed-bdf8-fb80621a0635"},{"last_name":"Mondelli","first_name":"Marco","id":"27EB676C-8706-11E9-9510-7717E6697425","orcid":"0000-0002-3242-7020","full_name":"Mondelli, Marco"}],"abstract":[{"text":"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.","lang":"eng"}],"department":[{"_id":"MaMo"}],"article_processing_charge":"No","external_id":{"arxiv":["2501.19104"]},"language":[{"iso":"eng"}],"oa":1,"has_accepted_license":"1","intvolume":"       267","ddc":["000"],"_id":"21326","date_updated":"2026-09-16T07:25:42Z","file":[{"file_id":"21337","creator":"dernst","content_type":"application/pdf","file_name":"2025_ICML_Wu.pdf","relation":"main_file","success":1,"date_created":"2026-02-19T08:28:22Z","file_size":3994385,"checksum":"c5ce8b1c83e33dc3a11122f4910deb67","access_level":"open_access","date_updated":"2026-02-19T08:28:22Z"}],"file_date_updated":"2026-02-19T08:28:22Z","volume":267,"year":"2025","month":"07","publisher":"ML Research Press","alternative_title":["PMLR"]},{"citation":{"ama":"Bombari S, Mondelli M. Privacy for free in the overparameterized regime. <i>Proceedings of the National Academy of Sciences</i>. 2025;122(15). doi:<a href=\"https://doi.org/10.1073/pnas.2423072122\">10.1073/pnas.2423072122</a>","mla":"Bombari, Simone, and Marco Mondelli. “Privacy for Free in the Overparameterized Regime.” <i>Proceedings of the National Academy of Sciences</i>, vol. 122, no. 15, e2423072122, National Academy of Sciences, 2025, doi:<a href=\"https://doi.org/10.1073/pnas.2423072122\">10.1073/pnas.2423072122</a>.","chicago":"Bombari, Simone, and Marco Mondelli. “Privacy for Free in the Overparameterized Regime.” <i>Proceedings of the National Academy of Sciences</i>. National Academy of Sciences, 2025. <a href=\"https://doi.org/10.1073/pnas.2423072122\">https://doi.org/10.1073/pnas.2423072122</a>.","apa":"Bombari, S., &#38; Mondelli, M. (2025). Privacy for free in the overparameterized regime. <i>Proceedings of the National Academy of Sciences</i>. National Academy of Sciences. <a href=\"https://doi.org/10.1073/pnas.2423072122\">https://doi.org/10.1073/pnas.2423072122</a>","short":"S. Bombari, M. Mondelli, Proceedings of the National Academy of Sciences 122 (2025).","ista":"Bombari S, Mondelli M. 2025. Privacy for free in the overparameterized regime. Proceedings of the National Academy of Sciences. 122(15), e2423072122.","ieee":"S. Bombari and M. Mondelli, “Privacy for free in the overparameterized regime,” <i>Proceedings of the National Academy of Sciences</i>, vol. 122, no. 15. National Academy of Sciences, 2025."},"project":[{"name":"Prix Lopez-Loretta 2019 - Marco Mondelli","_id":"059876FA-7A3F-11EA-A408-12923DDC885E"},{"_id":"92099302-16d5-11f0-9cad-f9a785f54fbd","name":"Trustworthy Deep Learning Theory: Private Over-Parameterized Models and Robust LLMs"},{"name":"Bilateral Artificial Intelligence (Mondelli)","_id":"74caaef7-b034-11f1-8f2d-e0e993bb422e","grant_number":"COE12"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"22857"}]},"acknowledgement":"This research was funded in whole, or in part, by the Austrian Science Fund (FWF) Grant number COE 12. For the purpose of open access, the author has applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission. The authors were also supported by the 2019 Lopez-Loreta prize, and Simone Bombari was supported by a Google PhD fellowship. We thank Diyuan Wu, Edwige Cyffers, Francesco Pedrotti, Inbar Seroussi, Nikita P. Kalinin, Pietro Pelliconi, Roodabeh Safavi, Yizhe Zhu, and Zhichao Wang for helpful discussions.","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"day":"15","scopus_import":"1","date_created":"2025-04-27T22:02:13Z","article_type":"original","arxiv":1,"isi":1,"corr_author":"1","date_published":"2025-04-15T00:00:00Z","article_number":"e2423072122","title":"Privacy for free in the overparameterized regime","oa_version":"Published Version","quality_controlled":"1","publication_identifier":{"issn":["0027-8424"],"eissn":["1091-6490"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","publication":"Proceedings of the National Academy of Sciences","status":"public","OA_type":"hybrid","OA_place":"publisher","file_date_updated":"2025-05-05T07:27:54Z","file":[{"creator":"dernst","file_id":"19648","content_type":"application/pdf","relation":"main_file","file_name":"2025_PNAS_Bombari.pdf","success":1,"date_created":"2025-05-05T07:27:54Z","file_size":2328320,"checksum":"1ac6f78e368d35a0cafb4d2d9bd63443","date_updated":"2025-05-05T07:27:54Z","access_level":"open_access"}],"date_updated":"2026-09-21T13:07:01Z","fulldoi":"https://doi.org/10.1073/pnas.2423072122","pmid":1,"month":"04","publisher":"National Academy of Sciences","year":"2025","volume":122,"article_processing_charge":"Yes (in subscription journal)","abstract":[{"lang":"eng","text":"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."}],"department":[{"_id":"MaMo"}],"doi":"10.1073/pnas.2423072122","author":[{"full_name":"Bombari, Simone","last_name":"Bombari","first_name":"Simone","id":"ca726dda-de17-11ea-bc14-f9da834f63aa"},{"last_name":"Mondelli","orcid":"0000-0002-3242-7020","id":"27EB676C-8706-11E9-9510-7717E6697425","first_name":"Marco","full_name":"Mondelli, Marco"}],"_id":"19627","issue":"15","ddc":["000"],"has_accepted_license":"1","intvolume":"       122","oa":1,"language":[{"iso":"eng"}],"APC_amount":"2754,32 EUR","external_id":{"arxiv":["2410.14787"],"pmid":["40215275"],"isi":["001471214000001"]}}]