Neural Networks Generalize on Low Complexity Data

• URL: http://arxiv.org/abs/2409.12446v4
• Date: Tue, 01 Jul 2025 01:09:51 GMT
• Title: Neural Networks Generalize on Low Complexity Data
• Authors: Sourav Chatterjee, Timothy Sudijono,
• Abstract summary: We show that feedforward neural networks with ReLU activation generalize on low complexity data.<n>We define this simple programming language, along with a notion of description length of such networks.
• Score: 5.678271181959529
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 error 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.

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