A Neural Scaling Law from Lottery Ticket Ensembling
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911769409093632 |
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| author | Liu, Ziming Tegmark, Max |
| author_facet | Liu, Ziming Tegmark, Max |
| contents | Neural scaling laws (NSL) refer to the phenomenon where model performance improves with scale. Sharma & Kaplan analyzed NSL using approximation theory and predict that MSE losses decay as $N^{-α}$, $α=4/d$, where $N$ is the number of model parameters, and $d$ is the intrinsic input dimension. Although their theory works well for some cases (e.g., ReLU networks), we surprisingly find that a simple 1D problem $y=x^2$ manifests a different scaling law ($α=1$) from their predictions ($α=4$). We opened the neural networks and found that the new scaling law originates from lottery ticket ensembling: a wider network on average has more "lottery tickets", which are ensembled to reduce the variance of outputs. We support the ensembling mechanism by mechanistically interpreting single neural networks, as well as studying them statistically. We attribute the $N^{-1}$ scaling law to the "central limit theorem" of lottery tickets. Finally, we discuss its potential implications for large language models and statistical physics-type theories of learning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02258 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Neural Scaling Law from Lottery Ticket Ensembling Liu, Ziming Tegmark, Max Machine Learning Artificial Intelligence Data Analysis, Statistics and Probability Neural scaling laws (NSL) refer to the phenomenon where model performance improves with scale. Sharma & Kaplan analyzed NSL using approximation theory and predict that MSE losses decay as $N^{-α}$, $α=4/d$, where $N$ is the number of model parameters, and $d$ is the intrinsic input dimension. Although their theory works well for some cases (e.g., ReLU networks), we surprisingly find that a simple 1D problem $y=x^2$ manifests a different scaling law ($α=1$) from their predictions ($α=4$). We opened the neural networks and found that the new scaling law originates from lottery ticket ensembling: a wider network on average has more "lottery tickets", which are ensembled to reduce the variance of outputs. We support the ensembling mechanism by mechanistically interpreting single neural networks, as well as studying them statistically. We attribute the $N^{-1}$ scaling law to the "central limit theorem" of lottery tickets. Finally, we discuss its potential implications for large language models and statistical physics-type theories of learning. |
| title | A Neural Scaling Law from Lottery Ticket Ensembling |
| topic | Machine Learning Artificial Intelligence Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2310.02258 |