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When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the data size. Perhaps the best known example of such scaling laws are for transformer-based large language models (**LLMs**), where networks with billions of parameters are trained on trillions of tokens of text. Yet, despite sustained widespread interest, a rigorous understanding of why transformer scaling laws exist is still missing. To answer this question, we establish novel statistical estimation and mathematical approximation theories for transformers when the input data are concentrated on a low-dimensional manifold. Our theory predicts a power law between the generalization error and both the training data size and the network size for transformers, where the power depends on the intrinsic dimension d of the training data. Notably, the constructed model architecture is shallow, requiring only logarithmic depth in d. By leveraging low-dimensional data structures under a manifold hypothesis, we are able to explain transformer scaling laws in a way which respects the data geometry. Moreover, we test our theory with empirical observation by training LLMs on natural language datasets. We find the observed empirical scaling laws closely agree with our theoretical predictions. Taken together, these results rigorously show the intrinsic dimension of data to be a crucial quantity affecting transformer scaling laws in both theory and practice.
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Neural Scaling Laws From Large-N Field Theory: Solvable Model Beyond the Ridgeless Limit
Abstract Many machine learning models based on neural networks exhibit scaling laws: their performance scales as power laws with respect to the sizes of the model and training data set. We use large-N field theory methods to solve a model recently proposed by Maloney, Roberts and Sully which provides a simplified setting to study neural scaling laws. Our solution extends the result in this latter paper to general nonzero values of the ridge parameter, which are essential to regularize the behavior of the model. In addition to obtaining new and more precise scaling laws, we also uncover a duality transformation at the diagrams level which explains the symmetry between model and training data set sizes. The same duality underlies recent efforts to design neural networks to simulate quantum field theories.
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- Award ID(s):
- 2412880
- PAR ID:
- 10580736
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Machine Learning: Science and Technology
- ISSN:
- 2632-2153
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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