The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problem Article Swipe
Matthew J. Colbrook
,
Vegard Antun
,
Anders C. Hansen
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.1073/pnas.2107151119
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.1073/pnas.2107151119
Significance Instability is the Achilles’ heel of modern artificial intelligence (AI) and a paradox, with training algorithms finding unstable neural networks (NNs) despite the existence of stable ones. This foundational issue relates to Smale’s 18th mathematical problem for the 21st century on the limits of AI. By expanding methodologies initiated by Gödel and Turing, we demonstrate limitations on the existence of (even randomized) algorithms for computing NNs. Despite numerous existence results of NNs with great approximation properties, only in specific cases do there also exist algorithms that can compute them. We initiate a classification theory on which NNs can be trained and introduce NNs that—under suitable conditions—are robust to perturbations and exponentially accurate in the number of hidden layers.
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Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1073/pnas.2107151119
- OA Status
- green
- Cited By
- 118
- References
- 86
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4220974656
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W4220974656Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1073/pnas.2107151119Digital Object Identifier
- Title
-
The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problemWork title
- Type
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articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
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2022Year of publication
- Publication date
-
2022-03-16Full publication date if available
- Authors
-
Matthew J. Colbrook, Vegard Antun, Anders C. HansenList of authors in order
- Landing page
-
https://doi.org/10.1073/pnas.2107151119Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2101.08286Direct OA link when available
- Concepts
-
Artificial neural network, Computer science, Stability (learning theory), Randomized algorithm, Inverse, Algorithm, Deep neural networks, Artificial intelligence, Mathematics, Machine learning, GeometryTop concepts (fields/topics) attached by OpenAlex
- Cited by
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118Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 26, 2024: 33, 2023: 32, 2022: 23, 2021: 4Per-year citation counts (last 5 years)
- References (count)
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86Number of works referenced by this work
- Related works (count)
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10Other works algorithmically related by OpenAlex
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