Neural Networks Generalize on Low Complexity Data
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910036026982400 |
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| author | Chatterjee, Sourav Sudijono, Timothy |
| author_facet | Chatterjee, Sourav Sudijono, Timothy |
| contents | We show that feedforward neural networks with ReLU activation generalize on low complexity data, suitably defined. Given i.i.d.~data generated from a simple programming language, the minimum description length (MDL) feedforward neural network which interpolates the data generalizes with high probability. We define this simple programming language, along with a notion of description length of such networks. We provide several examples on basic computational tasks, such as checking primality of a natural number. For primality testing, our theorem shows the following and more. Suppose that we draw an i.i.d.~sample of $n$ numbers uniformly at random from $1$ to $N$. For each number $x_i$, let $y_i = 1$ if $x_i$ is a prime and $0$ if it is not. Then, the interpolating MDL network accurately answers, with probability $1- O((\ln N)/n)$, whether a newly drawn number between $1$ and $N$ is a prime or not. Note that the network is not designed to detect primes; minimum description learning discovers a network which does so. Extensions to noisy data are also discussed, suggesting that MDL neural network interpolators can demonstrate tempered overfitting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Neural Networks Generalize on Low Complexity Data Chatterjee, Sourav Sudijono, Timothy Machine Learning Artificial Intelligence Statistics Theory We show that feedforward neural networks with ReLU activation generalize on low complexity data, suitably defined. Given i.i.d.~data generated from a simple programming language, the minimum description length (MDL) feedforward neural network which interpolates the data generalizes with high probability. We define this simple programming language, along with a notion of description length of such networks. We provide several examples on basic computational tasks, such as checking primality of a natural number. For primality testing, our theorem shows the following and more. Suppose that we draw an i.i.d.~sample of $n$ numbers uniformly at random from $1$ to $N$. For each number $x_i$, let $y_i = 1$ if $x_i$ is a prime and $0$ if it is not. Then, the interpolating MDL network accurately answers, with probability $1- O((\ln N)/n)$, whether a newly drawn number between $1$ and $N$ is a prime or not. Note that the network is not designed to detect primes; minimum description learning discovers a network which does so. Extensions to noisy data are also discussed, suggesting that MDL neural network interpolators can demonstrate tempered overfitting. |
| title | Neural Networks Generalize on Low Complexity Data |
| topic | Machine Learning Artificial Intelligence Statistics Theory |
| url | https://arxiv.org/abs/2409.12446 |